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Bibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Dur´ an, R.G., and Muschietti A.M., 2006, Solutions of the Diver- gence Operator on John Domains, Adv. Math., 206, 373-401 [III.3] Adams, R.A. 1975, Sobolev Spaces, Academic Press, New York [ II.3, Notes for II, III.4] Adams, R.A., and Fournier J.J.F, 2003, Sobolev Spaces (second edition), Aca- demic Press, New York [Notes for II] Agmon, S., Douglis, A., and Nirenberg, L., 1959, Estimates Near the Boun- dary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions I, Comm. Pure Appl. Math., 12, 623 -727 [II.11, III.1, III.9, Introduction to IV, IV.5, IV.6, IV.7] Agmon, S., Douglis, A., and Nirenberg, L., 1964, Estimates Near the Bound- ary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions II, Comm. Pure Appl. Math., 17, 35 -92 [Introduction to IV, IV.7] Alekseev G. V., and Tereshko D. A., 1998, Stationary Optimal Control Prob- lems for Equations of Viscous Heat Conducting Fluid, Sibirsk. Zh. Industr. Mat., 1, 24-44 (in Russian) [IX.4] Ames, K.A., and Payne, L.E., 1989, Decay Estimate in Steady Pipe Flow, SIAM J. Math. Anal., 20, 789-815 [Notes for VI, Notes for XIII] Amick, C.J., 1976, Decomposition Theorems for Solenoidal Vector Fields, J. Lon- don Math. Soc., 15, 288-296 [III.3, Notes for III, Notes for IV] Amick, C.J., 1977, Steady Solutions of the Navier-Stokes Equations in Unbounded Channels and Pipes, Ann. Scuola Norm. Pisa, (4) 4, 473-513 [I.3, Introduction to VI, Notes for VI, XIII.3, Notes for XIII] Amick, C.J., 1978, Properties of Steady Solutions of the Navier-Stokes Equations for Certain Unbounded Channels and Pipes, Nonlin. Anal., Theory, Meth. Appl., 2, 689-720 [I.3, Introduction to VI, Notes for VI, Notes for XIII] Amick, C.J., 1979, Steady Solutions of the Navier-Stokes Equations Representing Plane Flow in Channels of Various Types, Approximation Methods for Navier- Stokes Problems, Rautmann, R., Ed., Lecture Notes in Mathematics, Vol. 771, Springer-Verlag, 1-11 [Introduction to VI, Notes for XIII] 975 G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Springer Monographs in Mathematics, DOI 10.1007/978-0-387-09620-9, © Springer Science+Business Media, LLC 2011
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Page 1: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography

Numbers in square brackets denote the chapter and section where the reference isquoted.

Acosta, G., Duran, R.G., and Muschietti A.M., 2006, Solutions of the Diver-gence Operator on John Domains, Adv. Math., 206, 373-401 [III.3]

Adams, R.A. 1975, Sobolev Spaces, Academic Press, New York [ II.3, Notes for II,III.4]

Adams, R.A., and Fournier J.J.F, 2003, Sobolev Spaces (second edition), Aca-demic Press, New York [Notes for II]

Agmon, S., Douglis, A., and Nirenberg, L., 1959, Estimates Near the Boun-dary for Solutions of Elliptic Partial Differential Equations Satisfying GeneralBoundary Conditions I, Comm. Pure Appl. Math., 12, 623 -727 [II.11, III.1,III.9, Introduction to IV, IV.5, IV.6, IV.7]

Agmon, S., Douglis, A., and Nirenberg, L., 1964, Estimates Near the Bound-ary for Solutions of Elliptic Partial Differential Equations Satisfying GeneralBoundary Conditions II, Comm.Pure Appl. Math., 17, 35 -92 [Introduction toIV, IV.7]

Alekseev G. V., and Tereshko D. A., 1998, Stationary Optimal Control Prob-lems for Equations of Viscous Heat Conducting Fluid, Sibirsk. Zh. Industr.Mat., 1, 24-44 (in Russian) [IX.4]

Ames, K.A., and Payne, L.E., 1989, Decay Estimate in Steady Pipe Flow, SIAMJ. Math. Anal., 20, 789-815 [Notes for VI, Notes for XIII]

Amick, C.J., 1976, Decomposition Theorems for Solenoidal Vector Fields, J. Lon-don Math. Soc., 15, 288-296 [III.3, Notes for III, Notes for IV]

Amick, C.J., 1977, Steady Solutions of the Navier-Stokes Equations in UnboundedChannels and Pipes, Ann. Scuola Norm. Pisa, (4) 4, 473-513 [I.3, Introductionto VI, Notes for VI, XIII.3, Notes for XIII]

Amick, C.J., 1978, Properties of Steady Solutions of the Navier-Stokes Equationsfor Certain Unbounded Channels and Pipes, Nonlin. Anal., Theory, Meth.Appl., 2, 689-720 [I.3, Introduction to VI, Notes for VI, Notes for XIII]

Amick, C.J., 1979, Steady Solutions of the Navier-Stokes Equations RepresentingPlane Flow in Channels of Various Types, Approximation Methods for Navier-Stokes Problems, Rautmann, R., Ed., Lecture Notes in Mathematics, Vol. 771,Springer-Verlag, 1-11 [Introduction to VI, Notes for XIII]

975G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Springer Monographs in Mathematics, DOI 10.1007/978-0-387-09620-9, © Springer Science+Business Media, LLC 2011

Page 2: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

976 Bibliography

Amick, C.J., 1984, Existence of Solutions to the Nonhomogeneous Steady Navier-Stokes Equations, Indiana Univ. Math. J., 33, 817-830 [IX.4, Notes for IX]

Amick, C. J., 1986, On Steady Navier-Stokes Flow Past a Body in the Plane, Proc.Symposia Pure Math., 45, Amer. Math. Soc., 37-50 [I.2, XII.3]

Amick, C. J., 1988, On Leray’s Problem of Steady Navier-Stokes Flow Past aBody in the Plane, Acta Math., 161, 71-130 [I.2, Introduction to XII, XII.3,XII.4, Notes for XII]

Amick, C. J., 1991, On the Asymptotic Form of Navier-Stokes Flow Past a Bodyin the Plane, J. Differential Equations, 91, 149-167 [I.2, Notes for XII]

Amick, C.J., and Fraenkel, L.E., 1980, Steady Solutions of the Navier-StokesEquations Representing Plane Flow in Channels of Various Types, Acta Math.,144, 83-152 [I.3, Introduction to VI, VI.3, VI.4, Notes for XIII]

Amrouche, C., and Girault, V., 1990, Proprietes Fonctionnelles d’Operateurs.Application au Probleme de Stokes en Dimension Quelconque, Publ. Laborat.Anal. Num., Universite P. et M. Curie, [R] 90025 [Notes for IV]

Amrouche, C., and Girault, V., 1991, On the Existence and Regularity of theSolution of Stokes Problem in Arbitrary Dimension, Proc. Japan Acad., 67,171-175 [Notes for IV]

Arai, T., 1995, A Note on Stokes’ Solutions for a Flow Past a Sphere, Fluid Dy-nam. Res., 16, 109-113 [Notes for X]

Arnold, D.N., Scott, L.R., and Vogelius, M., 1988, Regular Inversion of theDivergence Operator with Dirichlet Boundary Conditions on a Polygon, Ann.Scuola Norm. Pisa, 15, 169-192 [Notes for III]

Avudainayagam, A., Jothiram, B., and Ramakrishna, J., 1986, A NecessaryCondition for the Existence of a Class of Plane Stokes Flows, Q. Jl. Mech. Appl.Math., 39, 425-434 [Notes for V]

Babenko, K.I., 1970, The Asymptotic Behavior of a Vortex Far Away from a Bodyin a Plane Flow of Viscous Fluid, Prikl. Math. Meh., 34, (5) 911-925; EnglishTransl.: J. Appl. Math. Mech., 34, 1970, 869-881 [XII.8]

Babenko, K.I., 1973, On Stationary Solutions of the Problem of Flow Past a Bodyof a Viscous Incompressible Fluid, Mat. Sb., 91 (133), 3-27; English Transl.:Math. SSSR Sbornik, 20 1973, 1-25 [Notes for VII, Introduction to X, Notesfor X]

Babenko, K.I., 1976, Perturbation Theory of Stationary Flows of a Viscous In-compressible Fluid for Small Reynolds Numbers, Dokl. Akad. Nauk. SSSR,227, 592-595; English Transl.: Sov. Phys. Dokl., 1976, 21, 143-145 [X.10]

Babenko, K.I., 1980, On Properties of Steady Viscous Incompressible Fluid Flows,Approximation Methods for Navier-Stokes Problems, Rautmann R., Ed., Lec-ture Notes in Mathematics, Vol. 771, Springer-Verlag, 12-42 [V.6]

Babenko, K.I., and Vasil’ev, M.M., 1973, Asymptotic Behaviour of the Solu-tions of the Problem of the Flow of a Viscous Fluid Around a Finite BodyPrikl. Mat. Meh., 37, 690-705; English Transl.: J. Appl. Math. Mech., 1973,37, 651-665 [X.8]

Babuska, I., and Aziz, A.K., 1972, Survey Lectures on the Mathematical Foun-dations of the Finite Element Method, The Mathematical Foundations of theFinite Element Method with Applications to Partial Differential Equations,Aziz A. K., Ed., Academic Press, New York [Notes for III]

Batchelor G.K., 1999, An Introduction to Fluid Mechanics, Second PaperbackEdition, Cambridge University Press [I.2]

Page 3: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 977

Bateman, H., Dryden, H.L., and Murnagham, F.D., 1932, Hydrodynamics,Bull. National Res. Council No. 84 [I.1]

Beirao da Veiga, H., 1997, A New Approach to the L2-Regularity Theorems forLinear Stationary Nonhomogeneous Stokes Systems, Portugal. Math., 54, 271-286 [Notes for IV]

Beirao Da Veiga, H., 2004, Regularity for Stokes and Generalized Stokes Sys-tems Under Nonhomogeneous Slip-Type Boundary Conditions, Adv. Differen-tial Equations, 9, 1079-1114 [Notes for I]

Beirao da Veiga, H., 2005, Regularity of Solutions to a Non-Homogeneous Bo-undary Value Problem for General Stokes Systems in R

n+, Math. Ann., 331,

203-217 [Notes for I]Beirao da Veiga, H., 2006, Concerning Time-Periodic Solutions of the Navier-

Stokes Equations in Cylindrical Domains under Navier Boundary Conditions,J. Partial Differential Equations, 19, 369-376 [Notes for I]

Beirao da Veiga, H., 2010, A Challenging Open Problem: the Inviscid Limit un-der Slip-Type Boundary Conditions, Discrete Contin. Dyn. Syst., 3, 231-236[Notes for I]

Bers, L., John, F. and Schechter, M., 1964, Partial Differential Equations,Interscience Publ., New York [II.11, IV.7]

Berker, R., 1952, Sur les Forces Exercees par un Fluide Visqueux sur un Obstacle,Rend. Circ. Mat. Palermo, (2) 1, 260-280 [X.8]

Besov, O.V., 1967, Continuation of Functions from Lpl and Wp

l, Trudy Mat. Inst.Steklov, 89, 5-17 (in Russian) [II.6, Notes for II]

Besov, O.V., 1969, On the Behaviour of Differentiable Functions at Infinity andthe Density of Functions with Compact Support, Trudy Mat. Inst. Steklov,105, 1-15; English Transl.: Proc. Steklov Inst. Math., 105, 1969, 1-15 [Notesfor II]

Bickley, W.G., 1938, The Secondary Flow due to a Spherical Rotating in a Vis-cous Fluid, Phil. Mag., 25, (7) 746-752 [Introduction to V]

Bird, R.B., Armstrong, R.C., and Hassager, O., 1987,Dynamics of Polymer-ic Liquids, Volume I,, John Wiley & Sons, 2nd ed. [Notes for I]

Birman, M.S., and Solomjak, M.Z., 1974, Quantitative Analysis in SobolevImbedding Theorems and Applications to Spectral Theory, Tenth SummerSchool on Mathematical Physics, Kiev Univ. Press; English Transl.: Amer.Math. Soc. Transl., 114, 1980 [Notes for II]

Bjorland, C., Brandolese, L., Iftimie, D., and Schonbek M.E., 2011, Lp–Solutions of the Steady-State Navier-Stokes Equations with Rough ExternalForces, Comm. Partial Differential Equations, 36, 216-246 [X.6]

Bjorland, C., and Schonbek, M.E., 2009, Existence and Stability of Steady-State Solutions with Finite Energy for the NavierStokes Equation in the WholeSpace, Nonlinearity, 22, 1615-1637 [X.6]

Boggio, T., 1910, Sul Moto Stazionario di una Sfera in un Liquido Viscoso, Rend.Circ. Mat. Palermo, 30, 65-81 [Notes for IV, Notes for V]

Bogovskii, M.E., 1979, Solution of the First Boundary Value Problem for theEquation of Continuity of an Incompressible Medium, Dokl. Akad. Nauk SSSR,248, (5), 1037-1040; English Transl.: Soviet Math Dokl., 20 (1979), 1094-1098[III.3, Notes for III]

Bogovskii, M.E., 1980, Solution of Some Vector Analysis Problems Connectedwith Operators Div and Grad, Trudy Seminar S.L.Sobolev, #1, 80, Akademia

Page 4: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

978 Bibliography

Nauk SSSR, Sibirskoe Otdelnie Matematiki, Nowosibirsk, 5-40 (in Russian)[III.3, III.4, Notes for III].

Bogovskii, M.E., 1986,Decomposition of Lp(Ω; Rn) infto the Direct Sum of Sub-spaces of Solenoidal and Potential Vector Fields, Dokl. Akad. Nauk SSSR,286, 662-666; English Transl.: Soviet Math. Dokl., 33, 161-165 [III.1]

Bogovskii, M.E., and Maslennikova, V.N., 1978, On the Density of Finite So-lenoidal Vector Fields, Uspehi Mat. Nauk, 33, 152 (in Russian) [III.4, Notesfor III]

Bolik, J., and von Wahl, W., 1997, Estimating ∇u in Terms of divu, curlu,either (ν,u) or ν×u and the Topology, Math. Methods Appl. Sci., 20, 737-744[Notes for III]

Borchers, W., 1992, Zur Stabilitat und Faktorisierungsmethode fur die Navier-Stokes Gleichungen inkompressibler viskoser Flussigkeiten, Habilitationsschrift,Universitat Paderborn [I.2, V.2, Notes for VIII, Introduction to XI, Notes forXI]

Borchers, W., Galdi, G.P., and Pileckas, K., 1993, On the Uniqueness of Le-ray-Hopf Solutions for the Flow through an Aperture, Arch. Rational Mech.Anal., 122, 19-33 [Notes for XIII]

Borchers, W., and Miyakawa, T., 1988, L2 Decay for the Navier-Stokes Flowin Halfspaces, Math. Ann., 282, 139-155 [Notes for IV]

Borchers, W., and Miyakawa, T., 1990, Algebraic L2 Decay for Navier-StokesFlows in Exterior Domains, Acta Math., 165, 189-227 [Notes for IV, Notes forV]

Borchers, W., and Miyakawa, T., 1992, On Some Coercive Estimates for theStokes Problem in Unbounded Domains, Navier-Stokes Equations: Theory andNumerical Methods, Heywood J.G., Masuda, K., Rautmann R., & Solonnikov,V.A., Eds., Lecture Notes in Mathematics, Vol. 1530, Springer-Verlag, 71-84[Notes for V]

Borchers, W., and Miyakawa,T., 1995, On Stability of Exterior Stationary Na-vier-Stokes Flows, Acta Math., 174, 311-382 [Notes for X]

Borchers, W., and Pileckas, K., 1992, Existence, Uniqueness and Asympto-tics of Steady Jets, Arch. Rational Mech. Anal., 120, 1-49 [I.3, Notes for VI,Notes for XIII]

Borchers, W., and Pileckas, K., 1994, Note on the Flux Problem for Station-ary Incompressible Navier-Stokes Equations in Domains with a Multiply Con-nected Boundary, Acta. Appl. Math., 37, 21-30 [Introduction to IX, Notes forIX]

Borchers, W., and Sohr, H., 1987, On the Semigroup of the Stokes Operatorfor Exterior Domains in Lq-Spaces, Math. Z., 196, 415-425 [Notes for V]

Borchers, W., and Sohr, H., 1990, On the Equations rotv=g and divu=f withZero Boundary Conditions, Hokkaido Math. J., 19, 67-87 [Notes for III]

Bourgain, J., and Brezis, H., 2003, On the Equation div Y = f and Applica-tion to Control of Phases, J. Amer. Math. Soc. , 16, 393-426 [III.3, Notes forIII]

Brezis, H., 1983, Analyse Fonctionnelle, Theorie et Applications, Masson et Cie

[II.2, III.5]Burenkov, V.I., 1976, On the Extension of Functions with Preservation of Semi-

norm, Dokl. Akad. Nauk SSSR, 228, 971-976; English Transl.: Soviet MathDokl., 17, 806-810 [II.6, Notes for II]

Page 5: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 979

Bykhovski, E.B., and Smirnov, V.I., 1960, On Orthogonal Expansions of theSpace of Vector Functions which are Square Summable over a Given Domain,Trudy Mat. Inst. Steklov, 59, 6-36 (in Russian) [III.1]

Caccioppoli, R., 1937, Sui Teoremi di Esistenza di Riemann, Ann. Scuola Norm.Pisa, (3) 6, 177-187 [IV.2, VIII.2]

Calderon, A.P., and Zygmund, A., 1956,´ On Singular Integrals, Amer. J. Math.,78, 289-309 [II.11]

Calderon, A.P., and Zygmund, A., 1957, Singular Integral Operators and Dif-ferential Equations, Amer. J. Math., 79, 901-921 [II.11]

Cattabriga, L., 1961, Su un Problema al Contorno Relativo al Sistema di Equa-zioni di Stokes, Rend. Sem. Mat. Padova, 31, 308-340 [Notes for III, Introduc-tion to IV, IV.3, IV.5. IV.8, Notes for IV]

Chandrasekhar, S., 1981, Hydrodynamic and Hydromagnetic Stability,Dover Pu-bl. Inc., New York [I.1]

Chang, H., 1992, The Steady Navier-Stokes Problem for Low Reynolds NumberViscous Jets into a Half Space, Navier-Stokes Equations: Theory and NumericalMethods, Heywood J.G., Masuda, K., Rautmann R., & Solonnikov, V.A., Eds.,Lecture Notes in Mathematics, Vol. 1530, Springer-Verlag, 85-96 [I.3, Notes forXIII]

Chang, H., 1993, The Steady Navier-Stokes Problem for Low Reynolds NumberViscous Jets, Stability and Appl. Anal. Cont. Media, 2, 203-228 [I.3, Notes forXIII]

Chang, I-D. and Finn, R., 1961, On the Solutions of a Class of Equations Occur-ring in Continuum Mechanics, with Application to the Stokes Paradox, Arch.Rational Mech. Anal., 7, 388-401 [Notes for V, Notes for VII]

Chasles, M., 1830, Notes sur les Proprietes Generales de deux Corps SemblablesEntr’eux et Places d’une Maniere Quelconque dans l’Espace, et sur le Deplace-ment Fini ou Infiniment Petit d’un Corps Solide Libre, Bull. Sci. Math. Astron.Phys. et Chim. A, 14, 321-326 [Introduction to VIII]

Choe, H., and Kim, H., 2000, Isolated Singularity for the Stationary Navier-Stok-es System, J. Math. Fluid Mech., 2, 151-184 [Notes for IX]

Choquet-Bruhat, Y., and Christodoulou, D. 1981, Elliptic Systems in Hs,δ-Spaces on Manifolds which are Euclidic at Infinity, Acta Math., 146, 129-150[Notes for V]

Cimmino, G., 1938a, Sulle Equazioni Lineari alle Derivate Parziali del SecondoOrdine di Tipo Ellittico Sopra una Superficie Chiusa, Ann. Scuola Norm. Pisa(3) 7, 73-96 [IV.2, VIII.2]

Cimmino, G., 1938b, Nuovo Tipo di Condizioni al Contorno e Nuovo Metododi Trattazione per il Problema Generalizzato di Dirichlet, Rend. Circ. Mat.Palermo, 61, 177-221 [IV.2, VIII.2]

Clark, D., 1971, The Vorticity at Infinity for Solutions of the Stationary Navier-Stokes Equations in Exterior Domains, Indiana Univ. Math. J., 20, 633-654[VII.6, X.8, XII.8]

Conca, C., 1989, Stokes Equations with Nonsmooth Data, Rev. Mat. Apl., 10,115–122 [Notes for IV]

Coscia, V., and Galdi, G.P., 1997, Existence of Steady-State Solutions of theEquations of a Fluid of Second Grade with Non-homogeneous Boundary Con-ditions, Zapiski Nauk. Sem. POMI., 243, 117-130 [III.3]

Page 6: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

980 Bibliography

Coscia, V., and Patria, M.C., 1992, Existence, Uniqueness and Asymptotic Be-haviour of Stationary Navier-Stokes Equations in a Half Space, Stability andAppl. Anal. Cont. Media, 2, 101-127 [I.3, II.7, XIII.9, Notes for XIII]

Courant, R., and Hilbert, D., 1937, Methoden der MathematischenPhysik, Ba-nd II, Springer-Verlag, Berlin [II.5, III.3]

Crispo, F. and Maremonti, P., 2004, An Interpolation Inequality in ExteriorDomains, Rend. Sem. Mat. Univ. Padova, 112, 11-39 [Notes for II]

Crudeli, U., 1925a, Metodo di Risoluzione di un Problema Fondamentale nellaTeoria del Moto Lento Stazionario di Liquidi Viscosi, Atti Accad. Naz. Lincei,(IV), 25, 247-251 [Notes for IV]

Crudeli, U., 1925b, Sopra un Problema Fondamentale nella Teoria del MotoLento Stazionario dei Liquidi Viscosi, Rivista di Mat. e Fis., Circolo Mat.-Fis.Univ. Messina [Notes for IV]

Dacorogna, B., 2002, Existence and Regularity of Solutions of dω = f withDirichlet Boundary Conditions, Nonlinear Problems in Mathematical Physicsand Related Topics, Int. Math. Ser. (N. Y.), Kluwer/Plenum, New York, I,67-82 [Notes for III]

Dacorogna, B., Fusco, N., and Tartar, L., 2004, On the Solvability of Theequation div u = f in L1 and in C0, Atti Accad. Naz. Lincei Cl. Sci. Fis.Mat. Natur. Rend. Lincei (9) Mat. Appl., 14, 239-245 [III.3]

Darrigol, O., 2002,, Between Hydrodynamics and Elasticity Theory: The FirstFive Births of the Navier-Stokes Equation, Arch. Hist. Exact Sci., 56, 95-150[Introduction to I]

Dauge, M., 1989, Stationary Stokes and Navier-Stokes Equations on Two or Three-Dimensional Domains with Corners, SIAM J. Math. Anal., 20, 74-97 [Notesfor IV]

Deny, J., and Lions, J.L., 1954, Les Espaces du Type de Beppo Levi, Ann. Inst.Fourier, 5, 305-370 [II.7, Notes for II]

de Saint-Venant, B., 1843, Comptes Rendus (Paris), 17, 1240-1243 [Introduc-tion to I]

Deuring, P., 1990a, An Integral Operator Related to the Stokes System in Exte-rior Domains, Math. Meth. in the Appl. Sci., 13, 323-333; Addendum, Ibid.,14, 445 [Notes for V]

Deuring, P., 1990b, The Resolvent Problem for the Stokes System in ExteriorDomains: An Elementary Approach, Math. Meth. in the Appl. Sci., 13, 335-349 [Notes for V]

Deuring, P., 1990c, The Stokes System in Exterior Domains: Lp-Estimates forSmall Values of a Resolvent Parameter, J. Appl. Math. Phys. (ZAMP), 41,829-842 [Notes for V]

Deuring, P., 1991, The Stokes System in Exterior Domains: Existence, Uniqu-eness, and Regularity of Solutions in Lp-Spaces, Commun. in Partial Diff.Equations, 16, 1513-1528 [Notes for V]

Deuring, P., 1994, The Stokes System in an Infinite Cone, Mathematical Re-search, Vol. 78, Akademie Verlag [Notes for IV]

Deuring, P., 2005, Exterior Stationary Navier-Stokes Flows in 3D with NonzeroVelocity at Infinity: Asymptotic Behavior of the Second Derivatives of theVelocity, Comm. Partial Diff. Equations, 30, 987-1020 [X.8]

Deuring, P., and Galdi, G.P., 2000, On the Asymptotic Behavior of PhysicallyReasonable Solutions to the Stationary Navier-Stokes System in Threedimen-

Page 7: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 981

sional Exterior Domains with Zero Velocity at Infinity, J. Math. Fluid Mech.,2, 353-364 [I.2, X.9]

Deuring, P., Kracmar, S., and Necasova, S., 2009, A representation Formulafor Linearized Stationary Incompressible Viscous Flows Around Rotating andTranslating Bodies, Preprint, [VIII.3]

Deuring, P., and von Wahl, W., 1989, Das Lineare Stokes System in R3. II.

Das Außenraumproblem, Bayreuth. Math. Schr., 28, 1-109 [Notes for V]Deuring, P., von Wahl, W., and Weidemaier, P., 1988,Das Lineare Stokes Sy-

stem in R3. I. Vorlesung uber das Inneraumproblem, Bayreuth. Math. Schr.,

27, 1-252 [Notes for IV]De Vito, L., 1958, Sulle Funzioni ad Integrale di Dirichlet Finito, Ann. Scuola

Norm. Sup. Pisa, (3) 12, 55-127 [II.4]Duran, R.G., and Lopez Garcıa, F., 2010,´ ´ ´ Solutions of the Divergence and Anal-

ysis of the Stokes Equations in Planar Holder-α Domains, Math. Models andMethods in Appl. Sci., 20, 95-120 [Notes for III, Notes for IV]

Duran, R.G., and Lopez Garcıa, F., 2010,´ ´ ´ Solutions of the Divergence and KornInequalities on Domains with an External Cusp, Ann. Acad. Sci. FennicaeMathematica, 35, 421-438 [III.3. Notes for III]

Dyer, R.H., and Edmunds, D.E., 1970, Removable Singularities of Solutions ofthe NavierStokes Equations, J. London Math. Soc., 2, 535-538 [Notes for IX]

Ebmeyer, C., and Frehse, J., 2001, Steady Navier-Stokes Equations with MixedBoundary Value Conditions in Three-Dimensional Lipschitzian Domains, Math.Ann., 319, 349-381 [Notes for I]

Elcrat, A.R., and MacLean, H.A., 1980, Weighted Wirtinger and Poincare In-equalities on Unbounded Domains, Indiana Univ. Math. J., (3) 29, 321-332[Notes for II]

Erig, W., 1982, Die Gleichungen von Stokes un die Bogovskii-Formel, Diplomar-beit, Universitat Paderborn [Notes for II, Notes for III]

Evett, A.A., 1966, Permutation Symbol Approach to Elementary Vector Analy-sis, Amer. J. Phys., 34, 503-507 [VIII.2]

Fabes, E.B., Kenig, C.E., and Verchota, G.C., 1988, The Dirichlet Problemfor the Stokes System on Lipschitz Domains, Duke Math. J., 57, 769-793 [Notesfor IX]

Fabes, E., Mendez, O., and Mitrea, M., 1998, Boundary Layers on Sobolev-Besov Spaces and Poisson’s Equation for the Laplacian in Lipschitz Domains,J. Funct. Anal., 159, 323-368 [III.1, Notes for III]

Farwig, R., 1990, Das Stationare 3D-Außenraumproblem der Navier-Stokes Gle-ichungen bei Nichtwerschwindender Anstromgeschwindigkeit in Anisotrop Ge-wichteten Sobolevraumen, Universitat Bonn, SFB 256, #110 [Notes for V,Notes for X]

Farwig, R., 1992a, A Variational Approach in Weighted Sobolev Spaces to theOperator −∆ + ∂/∂x1 in Exterior Domains of R

3, Math. Z., 210, 449–464[Notes for VII, Notes for X]

Farwig, R., 1992b, The Stationary Exterior 3D-Problem of Oseen and Navier-Stokes Equations in Anisotropically Weighted Sobolev Spaces, Math. Z., 211,409–447 [Notes for VII, VIII.3, Notes for X]

Farwig, R., 1993, The Weak Neumann Problem and the Helmholtz Decomposi-tion in General Aperture Domains, Progress in Partial Differential Equations:The Metz Surveys 2, Chipot, M., Ed., Pitman Research Notes in MathematicsSeries, Longman Scientific & Technical, Vol. 296, 86-96 [Notes for III]

Page 8: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

982 Bibliography

Farwig, R., 2003, Weighted Lq-Helmholtz Decompositions in Infinite Cylindersand in Infinite Layers, Adv. Differential Equations, 8, 357-384 [Notes for III]

Farwig, R., 2005, Estimates of Lower Order Derivatives of Viscous Fluid Flowpast a Rotating Obstacle, Banach Center Publ., Polish Acad. Sci., Warsaw,70, 73-84 [Notes for VIII]

Farwig, R., 2006, An Lq-Analysis of Viscous Fluid Flow Past a Rotating Obstacle,Tohoku Math. J., 58, 129-147 [VIII.8, Notes for VIII]

Farwig, R., Galdi, G.P., and Kyed, M., 2010, Asymptotic Structure of a LeraySolution to the Navier-Stokes Flow Around a Rotating Body, Pacific J. Math.,in press [Introduction to XI, XI.7]

Farwig, R., Galdi, G.P., and Sohr, H, 2006, Very Weak Solutions and LargeUniqueness Classes of Stationary Navier-Stokes Equations in Bounded Do-mains of R

2, J. Differential Equations, 227, 564-580 [Notes for IX]Farwig, R., and Hishida, T., 2007, Stationary Navier-Stokes Flow Around a Ro-

tating Obstacle, Funkcial. Ekvac., 50, 371-403 [Notes for VIII]Farwig, R., and Hishida, T., 2009, Asymptotic Profile of Steady Stokes Flow

Around a Rotating Obstacle, Preprint, TU Darmstadt [VIII.6, XI.7, Intro-duction to XI]

Farwig, R., and Hishida, T., 2009, Leading Term at Infinity of Steady Navier-Stokes Flow around a Rotating Obstacle, Preprint 2591, TU Darmstadt, 20pp. [XI.7]

Farwig, R., Hishida, T., and Muller, D., 2004,¨ Lq-Theory of a Singular “Wind-ing” Integral Operator Arising from Fluid Dynamics, Pacific J. Math., 215,297-312 [Notes for VIII]

Farwig, R., Kozono, H. and Yanagisawa, T., 2010, Leray’s Inequality in Gen-eral Multi-Connected Domains in R

n, Preprint, TU Darmstadt [IX.4]Farwig, R., Krbec, M., and Necasova, S., 2008, A Weighted Lq-Approach to

Stokes Flow Around a Rotating Body, Ann. Univ. Ferrara Sez. VII (N.S.), 54,61-84 [Notes for VIII]

Farwig, R., Novotny, A., and Pokorny, M., 2000, The Fundamental Solutionof a Modified Oseen Problem, Z. Anal. Anwendungen, 19, 713-728 [Notes forVII]

Farwig, R., Simader, C.G., and Sohr, H., 1993, An Lq-Theory for Weak Solu-tions of the Stokes System in Exterior Domains, Math. Meth. in the Appl. Sci.,16, 707-723 [Notes for V]

Farwig, R., and Sohr, H., 1994a, The Stationary and Non-stationary Stokes Sys-tem in Exterior Domains with Non-zero Divergence and Non-zero BoundaryValues, Math. Meth. Appl. Sci., 17, 269-291 [Notes for III]

Farwig, R., and Sohr, H., 1994b, On the Stokes and Navier–Stokes System forDomains with Noncompact Boundary in Lq–Spaces, Math. Nachr., 170, 53–77[Notes for VI]

Farwig, R., and Sohr, H., 1995, Weighted Energy Inequalities for the Navier-Stokes Equations in Exterior Domains, Appl. Anal., 58, 157-173 [Notes for X,Notes for XII]

Farwig, R., and Sohr, H., 1996, Helmholtz Decomposition and Stokes ResolventSystem for Aperture Domains in Lq–Spaces, Analysis, 16, 1-26 [Notes for III]

Farwig, R., & Sohr, H., 1998, Weighted Estimates for the Oseen Equations andthe Navier-Stokes Equations in Exterior Domains, Theory of the Navier-Stokesequations,Heywood J.G., Masuda, K., Rautmann R., & Solonnikov, V.A., Eds.,

Page 9: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 983

Ser. Adv. Math. Appl. Sci., Vol. 47, World Sci. Publ., River Edge, NJ 11-30[I.2. Introduction to X, Notes for X, Notes for XII]

Farwig, R., and Sohr, H., 2009, Existence, Uniqueness and Regularity of Sta-tionary Solutions to Inhomogeneous Navier-Stokes Equations in R

n, Czechoslo-vak Math. J., 59, 61-79 [Notes for IX]

Faxen, H., 1928/1929, Fredholm’sche Integraleichungen zu der Hydrodynamik Za-her Flussigkeiten, Ark. Mat. Astr. Fys., (14) 21, 1-20 [Notes for VII]

Filon, L.G.N., 1928, On the Second Approximation to the Oseen Solution for theMotion of a Viscous Fluid, Phil. Trans Roy. Soc. London, Ser. A., 227, 93-135[Introduction to VII]

Finn, R. 1959a, On Steady-State Solutions of the Navier-Stokes Partial Differen-tial Equations, Arch. Rational Mech. Anal., 3, 381-396 [I.2, Notes for X]

Finn, R. 1959b, Estimates at Infinity for Stationary Solutions of the Navier-StokesEquations, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine, 3 (51) 387-418[X.8, Notes for X]

Finn, R., 1960, An Energy Theorem for Viscous Fluid Motions, Arch. RationalMech. Anal., 6, 371-381 [X.6, Notes for X]

Finn, R., 1961a, On the Steady-State Solutions of the Navier-Stokes Equations.III, Acta Math., 105, 197-244 [IX.6, Notes for IX, X.4, X.6, Notes for X]

Finn, R., 1961b, Estimates at Infinity for Steady State Solutions of the Navier-Stokes Equations, Proc. Symp. Appl. Math., 4, Amer. Math. Soc., Providence.R. I., 143-148 [Notes for X]

Finn, R., 1963, On the Stokes Paradox and Related Questions, Nonlinear Prob-lems, Langer, R.E., Ed., University of Wisconsin Press, 121-135 [Notes for X]

Finn, R., 1965a, On the Exterior Stationary Problem for the Navier-Stokes Equa-tions, and Associated Perturbation Problems, Arch. Rational Mech. Anal., 19,363-406 [II.6, Notes for II, V.6, Notes for V, VII.2, Notes for VII, X.8, X.9,Notes for X, Introduction to XI]

Finn, R., 1965b, Stationary Solutions of the Navier-Stokes Equations, Proc. Symp.Appl. Math. 17, Amer. Math. Soc., Providence. R. I., 121-153 [Introduction toIX, Notes for IX, Notes for X]

Finn, R., 1970, The Exterior Problem for the Navier-Stokes Equations, Actes,Congres Intern. Math., 3, 85-94 [Notes for X, Notes for XII]

Finn, R., 1973, Mathematical Questions Relating to Viscous Fluid Flow in an Ex-terior Domain, Rocky Mountain J. Math., 3 (1), 107-140 [Notes for X]

Finn, R., and Gilbarg, D., 1957, Three-Dimensional Subsonic Flows, and Asym-ptotic Estimates for Elliptic Partial Differential Equations, Acta Math., 98,265-296 [II.5]

Finn, R., and Noll, W., 1957, On the Uniqueness and Non-Existence of StokesFlows, Arch. Rational Mech. Anal., 1, 97-106 [Notes for V]

Finn, R., and Smith, D.R., 1967a,, On the Linearized Hydrodynamical Equa-tions in Two Dimensions, Arch. Rational Mech. Anal., 25, 1-25 [VII.5, VII.8,Notes for XII]

Finn, R., and Smith, D.R., 1967b, On the Stationary Solution of the Navier-Stokes Equations in Two Dimensions, Arch. Rational Mech. Anal., 25, 26-39[I.2, Introduction to XII, XII.9, Notes for XII]

Finn, R., and Solonnikov, V.A., 1997, Gradient Estimates for Solutions of theNavier-Stokes Equations, Topol. Methods Nonlinear Anal., 9, 29-39 [IX.4]

Page 10: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

984 Bibliography

Fischer, T.M., Hsiao, G.C., and Wendland, W.L., 1985, Singular Perturba-tions for the Exterior Three-Dimensional Slow Viscous Flow Problem, J. Math.Anal. and Appl., 110, 583-603 [Notes for VII, X.10]

Foias, C., and Temam, R., 1978, Remarques sur les Equations de Navier-StokesStationnaires et les Phenomenes Successifs de Bifurcation, Ann. Scuola Norm.Pisa, (4) 5, 29-63 [Notes for IX]

Fraenkel, L.E., 1973, On a Theory of Laminar Flow in Channels of a CertainClass, Proc. Cambridge Phil. Soc., 73, 361-390 [Notes for VI]

Fraenkel, L.E., 1979, On Regularity of the Boundary in the Theory of SobolevSpaces, Proc. London Math. Soc., 39, 385-427 [II.5, III.3]

Frehse, J., and Ruzicka, M., 1994a, On the Regularity of the Stationary Solu-tions to the Navier-Stokes Equations, Ann. Scu. Norm. Pisa, 21, 63-95 [Intro-duction to IX]

Frehse, J., and Ruzicka, M., 1994b, Regularity for the Stationary Navier-StokesEquations in Bounded Domains, Arch. Rational Mech. Anal., 128, 361-381[Introduction to IX]

Frehse, J., and Ruzicka, M., 1995, Existence of Regular Solutions to the Sta-tionary Navier-Stokes equations, Math. Annalen, 302, 699-717 [IntroductionIX]

Frehse, J., and Ruzicka, M.ˇ ˇ , 1996, Weighted Estimates for the Stationary Navier-Stokes Equations, Mathematical Theory in Fluid Mechanics, Galdi, G.P.,Malek, J., and Necas, J., Eds., Pitman Research Notes in Mathematics Se-ries, Longman Scientific & Technical, Vol. 354, 1-29 [Introduction to IX]

Friedman, A., 1958, On the Regularity of the Solutions of Nonlinear Elliptic andParabolic Systems of Partial Differential Equations, J. Math. Mech, 7, 43-60[Notes for IX]

Friedman, A., 1969, Partial Differential Equations, Holt, Rinehart & WinstonInc., New York [II.3, Notes for II]

Friedrichs, K.O., 1933, Spektraltheorie Halbbeschrankter Operatoren und An-wendungen auf die Spektralzerlegung von Differentialoperatoren, Math. Ann.,109, 465-482 [II.5]

Friedrichs, K.O., 1955, Differential Forms on Riemannian Manifolds, Comm.Pure Appl. Math., 8, 551-590 [III.1]

Frohlich, A., 2000, The Helmholtz Decomposition of Weighted Lq-Spaces forMuckenhoupt Weights, Ann. Univ. Ferrara Sez. VII (N.S.), 46, 11-19 [Notesfor III]

Fujita, H., 1961, On the Existence and Regularity of the Steady-State Solutionsof the Navier-Stokes Equation, J. Fac. Sci. Univ. Tokyo, (1A) 9, 59-102 [V.3,Notes for V, Notes for VII, VII.6, Introduction to IX, IX.3, Notes for IX,Introduction to X, X.4]

Fujita, H., 1998, On Stationary Solutions to Navier-Stokes Equation in Symmet-ric Plane Domains Under General Outflow Condition, Navier-StokesEquations:Theory and Numerical Methods, Salvi, R., Ed., Pitman Res. Notes Math. Ser.,Vol. 388, Longman, Harlow, 16-30 [Notes for IX]

Fujita, H., and Morimoto, H., 1997, A Remark on the Existence of the Navier-Stokes Flow with Non-Vanishing Outflow Condition, GAKUTO Internat. Ser.Math. Sci. Appl., 10, 53-61 [Notes for IX]

Fujiwara, D., and Morimoto, H., 1977 An Lr-Theorem of the Helmholtz De-composition of Vector Fields, J. Fac. Sci. Univ. Tokyo (1A) 24,685-700 [III.1,Notes for III]

Page 11: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 985

Fujiwara, H., and Yamazaki, M., 2007, The Helmholtz Decomposition in Sobo-lev and Besov Spaces, Adv. Stud. Pure Math., 47-1, 99-116 [Notes for III]

Gagliardo, E., 1957, Caratterizzazione delle Tracce sulla Frontiera Relative adAlcune Classi di Funzioni in n Variabili, Rend. Sem. Mat. Padova, 27, 284-305[II.4]

Gagliardo, E., 1958, Proprieta di Alcune Classi di Funzioni in Piu Variabili,Ricerche Mat., 7, 102-137 [II.3, III.3]

Gagliardo, E., 1959, Ulteriori Proprieta di Alcune Classi di Funzioni in Piu Vari-abili, Ricerche Mat., 8, 24-51 [II.3]

Galdi, G.P., 1991, On the Existence of Steady Motions of a Viscous Flow withNon-homogeneous Boundary Conditions, Le Matematiche, 66, 503-524 [Intro-duction to IX, IX.4]

Galdi, G.P., 1992, On the Oseen Boundary-Value Problem in Exterior Domains,Navier-Stokes Equations: Theory and Numerical Methods, Heywood J.G., Ma-suda, K., Rautmann R., & Solonnikov, V.A., Eds., Lecture Notes in Math-ematics, Vol. 1530, Springer-Verlag, 111-131 [Introduction to VII, Notes forVII]

Galdi, G.P., 1992a, On the Energy Equation and on the Uniqueness for D-Solutions to Steady Navier-Stokes Equations in Exterior Domains, Mathemati-cal Problems Related to the Navier-Stokes Equation, Galdi G.P., Ed., Advancesin Mathematics for Applied Science, 11, World Scientific, 36-80 [Notes for III,Introduction to X, X.3, X.6, Notes for X]

Galdi, G.P., 1992b, On the Asymptotic Structure ofD-Solutions to Steady Navier-Stokes Equations in Exterior Domains, Mathematical Problems Related to theNavier-Stokes Equation, Galdi G.P., Ed., Advances in Mathematics for AppliedSciences, 11, World Scientific, 81-105 [I.2, Introduction to X, X.7, X.9, Notesfor X]

Galdi, G.P., 1992c, On the Asymptotic Properties of Leray’s Solution to the Ex-terior Stationary Three-Dimensional Navier-Stokes Equations with Zero Ve-locity at Infinity, Degenerate Diffusions, IMA Volumes in Mathematics andIts Applications Vol. 47, Ni, W.-M., Peletier, L.A., & Vasquez, J.L., Eds.,Springer-Verlag, 95-103 [I.2, Introduction to X, X.3, X.9, Notes for X]

Galdi, G.P., 1993, Existence and Uniqueness at Low Reynolds Number of Sta-tionary Plane Flow of a Viscous Fluid in Exterior Domains, Recent Develop-ments in Theoretical Fluid Mechanics, Galdi, G.P., and Necas, J., Eds., PitmanResearch Notes in Mathematics Series, Longman Scientific & Technical, Vol.291, 1-33 [I.2, Introduction to XII, Notes for XII]

Galdi, G.P., 1994a, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Volume I. Linearized Steady Problems, Springer Tracts inNatural Philosophy, Vol. 38. Springer-Verlag, New York [Notes for II]

Galdi, G.P., 1994b, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Volume II. Nonlinear Steady Problems, Springer Tracts inNatural Philosophy, Vol. 39. Springer-Verlag, New York [IX.4, Notes for IX]

Galdi, G.P., 1998, Mathematical Questions Relating to the Plane Steady Motionof a Navier-Stokes Fluid Past a Body, Lecture Notes Numer. Appl. Anal., 16,117-160 [Notes for XII]

Galdi, G.P, 1999a, On the Steady Self-Propelled Motion of a Body in a ViscousIncompressible Fluid, Arch. Rational Mech. Anal., 148, 53-88 [V.7, Notes forV, Introduction to X, X.6, XII.9]

Page 12: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

986 Bibliography

Galdi, G.P., 1999b, On the Existence of Symmetric Steady-State Solutions to thePlane Exterior Navier-Stokes Problem for Arbitrary Large Reynolds Number,Advances in fluid dynamics, Quad. Mat., Aracne, Rome, 4, 1-25 [I.2, Introduc-tion to XII, XII.6, Notes for XII]

Galdi, G.P., 2002, On the Motion of a Rigid Body in a Viscous Liquid: A Mathe-matical Analysis with Applications, Handbook of Mathematical Fluid Mechan-ics, Vol. 1, Friedlander, S., & Serre, D., Eds., Elsevier Science, 653-791 [I.2,Notes for V, Introduction to VII, Introduction to VIII, Notes for VIII, Intro-duction to X, X.6, Notes for XI]

Galdi, G.P., 2003, Steady Flow of a Navier-Stokes Exterior Fluid Around a Ro-tating Obstacle, Journal of Elasticity, 71, 1-31 [Introduction to VIII, VIII.6,Introduction to XI, XI.7]

Galdi, G.P., 2004, Stationary Navier-Stokes Problem in a Two-Dimensional Ex-terior Domain, Handbook of Differential Equations, Stationary Partial Differ-ential Equations, Vol. 1, Chipot, M., & Quittner, P., Eds., Elsevier Science,71-155 [I.2, XII.3, XII.5, XII.6, XII.9, Notes for XII]

Galdi, G.P., 2007, Further Properties of Steady-State Solutions to the Navier-Stokes Problem Past a Three-Dimensional Obstacle, J. Math. Phys., 48, 43 pp[II.5, II.6, III.3]

Galdi, G.P., 2008, Mathematical Problems in Classical and Non-Newtonian FluidMechanics, Hemodynamical Flows: Modeling, Analysis and Simulation, Ober-wolfach Seminars, Vol. 37, Galdi, G.P., Robertson, A.M., Rannacher, R., &Turek S., Eds., Birkhaeuser-Verlag, Basel, 121-273 [Notes for I]

Galdi, G.P., 2009, A Steady-State Exterior Navier-Stokes Problem that is notWell-Posed, Proc. Amer. Math. Soc., 137, 679-684 [I.2, Notes for X, Notes forXII]

Galdi, G.P., and Grisanti, C.R., 2010, Existence and Regularity of Steady Flo-ws for Shear-Thinning Liquids in Exterior Two-Dimensional Domains, Arch.Rational Mech. Anal., Online First, September 10, 27 pp. [Notes for I]

Galdi, G.P., and Kyed, M., 2010, Asymptotic Behavior of a Leray Solution ar-ound a Rotating Obstacle, Nonlinear Parabolic Problems: Herbert AmannFestschrift, Birkhauser-Verlag [I.2, Introduction to XI, XI.7, Notes for XI]

Galdi, G.P., and Kyed, M., 2011a, Steady-State NavierStokes Flows Past a Ro-tating Body: Leray Solutions are Physically Reasonable, Arch. Rational Mech.Anal., 200, 21-58 [I.2, Notes for VIII, Introduction to XI, Notes for XI]

Galdi, G.P., and Kyed, M., 2011b, On Lq-Estimates for the Stationary OseenEquations in a Rotating Frame, Preprint, TU Darmstadt, 12 pp. [VIII.7]

Galdi, G.P., and Layton, W.J., 2000, Approximation of the Larger Eddies inFluid Motions. II. A Model for Space-Filtered Flow, Math. Models MethodsAppl. Sci., 10, 343-350 [I.1]

Galdi, G.P., and Maremonti, P., 1986, Monotonic Decreasing and AsymptoticBehavior of the Kinetic Energy for Weak Solutions of the Navier-Stokes Equa-tions in Exterior Domains, Arch. Ratl Mech. Anal., 94, 253-266 [II.6, Notes forII]

Galdi, G.P., and Padula, M., 1991 Existence of Steady Incompressible FlowsPast an Obstacle, Mathematical Analysis of Phenomena in Fluid and PlasmaDynamics, RIMS Kokyuroku, Kyoto, vol. 745, 87-101 [Notes for X]

Galdi, G.P., Padula, M., and Passerini, A., 1995, Existence and AsymptoticDecay of Plane-Steady Flow in an Aperture Domain, Advances in Geomet-ric Analysis and Continuum Mechanics, Concus, P., and Lancaster, K., Eds.,

Page 13: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 987

International Press, Cambridge MA, 81-99 [I.3, Introduction to XIII, XIII.5,XIII.7]

Galdi, G.P., Padula, M., and Solonnikov, V.A., 1996, Existence, Uniqueness,and Asymptotic Behaviour of Solutions of Steady-State Navier-Stokes Equa-tions in a Plane Aperture Domain, Indiana Univ. Math. J., 45, 961-997 [I.3,VI.5, Introduction to XIII, XIII.6, XIII.9, Notes for XIII]

Galdi, G.P., Payne, L.E., Proctor, M.R., and Straughan, B., 1987, On theStability of Thawing Subsea Permafrost, Proc. Roy. Soc. London, Ser. A, 414,83-104 [Notes for II]

Galdi, G.P., and Rionero, S., 1985, Weighted Energy Methods in Fluid Dynam-ics and Elasticity, Springer Lecture Notes in Mathematics, Vol. 1134 [Notes forIX]

Galdi, G.P., and Robertson, A.M., 2005, On Flow of a Navier-Stokes Fluid inCurved Pipes. I. Steady flow, Appl. Math. Lett., 18, 1116-1124 [Notes for XIII]

Galdi, G.P., and Silvestre, A.L., 2007a, The Steady Motion of a Navier-StokesLiquid Around a Rigid Body, Arch. Rational Mech. Anal., 184, 371-400 [II.6,Introduction to VIII, Notes for VIII, Introduction to XI]

Galdi, G.P., and Silvestre, A. L., 2007b, Further Results on Steady-State Flowof a Navier-Stokes Liquid Around a Rigid Body. Existence of the Wake, RIMS,Kokyuroku Bessatsu Kyoto Univ., 127-143 [Introduction to VIII, VIII.6, Notesfor VIII, Introduction to XI]

Galdi, G.P., and Simader, C.G., 1990, Existence, Uniqueness and Lq-Estimat-es for the Stokes Problem in an Exterior Domain, Arch. Rational Mech. Anal.,112, 291-318 [II.7, Introduction to IV, Notes for IV, V.5, Notes for V]

Galdi, G.P., and Simader C.G., 1994, New Estimates for the Stationary StokesProblem in Exterior Domains with Applications to Navier-Stokes Problem,Differential and Integral Equations, 7, 847-861 [V.7, Notes for X]

Galdi, G.P., Simader, C.G., and Sohr, H., 1994, On the Stokes Problem in Lip-schitz Domains, Ann. Mat. Pura Appl., 167, 147-163 [IV.5, Notes for IV]

Galdi, G.P., Simader, C.G., and Sohr, H., 2005, A Class of Solutions to Sta-tionary Stokes and Navier-Stokes Equations with Boundary Data in W−1/q,q ,Math. Ann., 331, 41-74 [Notes for IV, Notes for IX]

Galdi, G.P., and Sohr, H., 1992, Existence, Uniqueness and Asymptotics of So-lutions to the Stationary Navier-Stokes Equations in Certain Domains withNoncompact Boundaries, Preprint #5, Istituto di Ingegneria dell’Universita diFerrara [I.3, Notes for VI, Notes for XIII]

Galdi, G.P., and Sohr, H., 1995, On the Asymptotic Structure of Plane SteadyFlow of a Viscous Fluid in Exterior Domains, Arch. Rational Mech. Anal., 131,101-119 [I.2, Notes for X, Introduction to XII, XII.7, Notes for XII]

Galdi, G.P., and Straughan, S., 1985, Exchange of Stabilities, Symmetry andNonlinear Stability, Arch. Rational Mech. Anal., 89, 211-228 [IX.2]

Galdi, G.P., and Varnhorn, W., 1996, The Maximum Modulus Theorem forthe Stokes System, manuscript form [Notes for IV]

Galerkin, B.G., 1915, Rods and Plates. Series on Some Problems of Elastic Equi-librium of Rods and Plates, Vestn. Inzh. Tech, 19, 897- 908 (in Russian) [VII.2,IX.3]

Geissert, M., Heck, H., and Hieber, M., 2006, On the Equation div u = g andBogovskiı’s Operator in Sobolev Spaces of Negative Order, Partial differen-tial equations and functional analysis; Oper. Theory Adv. Appl., 168, 113-121[Notes for III]

Page 14: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

988 Bibliography

Gerhardt, C., 1979, Stationary Solutions to the Navier-Stokes Equations in Di-mension Four, Math. Z., 165, 193-197 [Notes for IX]

Geymonat, G., 1965, Sui Problemi ai Limiti per i Sistemi Lineari Ellittici, Ann.Mat. Pura Appl., 69, 207-284 [Notes for IV]

Ghidaglia, J.M., 1984, Regularite des Solutions de Certains Problems aux Lim-ites Lineaires Liees aux Equations d’Euler, Comm. in Partial Diff. Eqs., 9,1265-1298 [Notes for IV]

Giaquinta, M., and Modica, G., 1982, Nonlinear Systems of the Type of Sta-tionary Navier-Stokes System, J. Reine Ang. Math., 330, 173-214 [Notes forIII, Notes for IX]

Giga, Y., 1981, Analyticity of the Semigroup Generated by the Stokes Operatorin Lr-Spaces, Math. Z., 178, 287-329 [Notes for IV]

Giga, Y., 1986, Solutions for Semilinear Parabolic Equations in Lp and Regularityof Weak Solutions of the Navier-Stokes System, J. Differential Equations, 62,186-212 [Notes for III]

Giga, Y., and Sohr, H., 1989, On the Stokes Operator in Exterior Domains, J.Fac. Sci. Univ. Tokyo, 1A Math., 36, 103-130 [Notes for III]

Gilbarg, D., and Trudinger, N.S., 1983, Elliptic Partial Differential Equationsof Second Order, Grundleheren der Mathematischen Wissenschaften, 224,Springer-Verlag, Berlin-Heidelberg-New York [V.4]

Gilbarg, D., and Weinberger, H.F., 1974, Asymptotic Properties of Leray’sSolution of the Stationary Two-Dimensional Navier-Stokes Equations, RussianMath.Surveys, 29, 109-123 [I.2, Introduction to XII, Notes for XII]

Gilbarg, D., and Weinberger, H.F., 1978, Asymptotic Properties of Steady Pl-ane Solutions of the Navier-Stokes Equations with Bounded Dirichlet Integral,Ann. Scuola Norm. Sup. Pisa, (4), 5, 381-404 [I.2, Notes for II, Notes for VI,Introduction to XII, XII.3, Notes for XII]

Girard, P.S., 1816, Memoire sur le Mouvement des Fluides dans les Tubes Capil-laires et L’Influence de la Temperature sur ce Mouvement, Institut Nationaldes Sciences et des Arts, Memoires de sciences Mathematiques et Physiques,249-380 [Notes for I]

Girault, V., and Raviart, P.A., 1986, Finite Element Methods for Navier-Stok-es Equations, Theory and Algorithms, Springer Series in Comp. Math. 5,Springer-Verlag, Berlin-Heidelberg-New York [Notes for III]

Girault, V., and Sequeira, A., 1991, A Well-Posed Problem for the ExteriorStokes Equations in Two and Three Dimensions, Arch. Rational Mech. Anal.,114, 313-333 [Notes for V]

Goldstein, S., 1929, The Steady Flow of Viscous Fluid Past a Fixed SphericalObstacle at Small Reynolds Numbers, Proc. Roy. Soc. London, Ser. A, 123,225-235 [Introduction to VII]

Goldstein, S., 1933a, On the Two-Dimensional Study of Flow of a Viscous FluidBehind a Solid Body, I. Proc. Roy. Soc. London Ser. A, 142, 545-562 [Notesfor XII]

Goldstein, S., 1933b, On the Two-Dimensional Study of Flow of a Viscous FluidBehind a Solid Body, II. Proc. Roy. Soc. London Ser. A, 142, 563-573 [Notesfor XII]

Gradshteyn, I.S., and Ryzhik, I.M., 1980, Table of Integrals, Series and Prod-ucts, Corrected and Enlarged Edition, Academic Press, Inc. [VIII.3]

Page 15: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 989

Griesinger, R., 1990a, On the Boundary Value Problem rotu=f , u Vanishing atthe Boundary and Related Decomposition of Lq and H1,q

0 : Existence, Ann.Univ. Ferrara Sez. VII, 35, 15-44 [Notes for III, IX.4]

Griesinger, R., 1990b, Decomposition of Lq and H1,q0 with Respect to the Oper-

ator rot, Math. Ann., 288, 245-262 [Notes for III]

Grisvard, P., 1985, Elliptic Problems in Non-Smooth Domains, Monographs andStudies in Math., Vol. 24, Pitman [Notes for IV]

Guirguis, G.H., 1986, On the Existence, Uniqueness and Regularity of the Exte-rior Stokes Problem in R

3, Commun. in Partial Diff. Equations, 11, 567-594[Notes for III]

Hamel, G., 1916, Spiralformige Bewegungen zaher Flussigkeiten, Jber. Dtsch.Math. Ver., 25, 34-60; English Trans.: NACA Tech. Memo. 1342, 1953 [I.2,XII.2]

Hardy, G.H., Littlewood, J.E., and Polya, G., 1934, Inequalities,CambridgeUniv. Press [II.3, II.4, II.5, II.6]

Helmholtz, H., 1870, Uber die Theorie der Elektrodynamik. Erste Abhandlung.Uber die Bewegungsgleichungen der Elektricitat fur Ruhende Leitende Korper,J. Reine Ang. Math., 72, 57-129 [III.1]

Heywood, J.G., 1974, On Some Paradoxes Concerning Two-dimensional StokesFlow Past an Obstacle, Indiana Univ. Math. J., 24, 443-450 [V.2]

Heywood, J.G., 1976, On Uniqueness Questions in the Theory of Viscous Flow,Acta Math., 136, 61-102 [I.3, Introduction to III, III.4, III.5, Notes for III,Introduction to VI, VI.3, VI.4, Notes for VI, Introduction to XIII, XIII.5,Notes for XIII]

Heywood, J.G., 1980, The Navier-Stokes Equations: On the Existence, Regular-ity and Decay of Solutions, Indiana Univ. Math. J., 29, 639-681 [Notes forV]

Heywood, J.G., 2010, On the Impossibility, in Some Cases, of the LerayHopf Con-dition for Energy Estimates, J. Math. Fluid Mech., Online First, June 9, 9 pp.[I.1,IX.4]

Hishida, T., 1999a, The Stokes Operator with Rotation Effect in Exterior Do-mains, Analysis (Munich), 19, 51-67 [Notes for VIII]

Hishida, T., 1999b, An Existence Theorem for the Navier-Stokes Flow in the Ex-terior of a Rotating Obstacle, Arch. Ration. Mech. Anal., 150, 307-348 [Notesfor VIII]

Hishida, T., 2006, Lq Estimates of Weak Solutions to the Stationary Stokes Equa-tions Around a Rotating Body, J. Math. Soc. Japan, 58, 743-767 [Introductionto VIII, Notes for VIII]

Hopf, E., 1941, Ein Allgemeiner Endlichkeitsatz der Hydrodynamik, Math. Ann.117, 764-775 [I.1, Introduction to IX, IX.4, Notes for IX]

Hopf, E., 1950/1951, Uber die Anfganswertaufgabe fur die HydrodynamischenGrundgleichungen, Math. Nachr., 4, 213-231 [Notes for III]

Hopf, E., 1952, A Remark on Linear Elliptic Differential Equations of the SecondOrder, Proc. Amer. Math. Soc., 3, 791-793 [XII.3]

Hopf, E., 1957, On Non-linear Partial Differential Equations, Lecture Series of theSymposium on Partial Differential Equations held at the University of Califor-nia at Berkley, June/July 1955, printed by the Department of Mathematics,the University of Kansas, 1-31 [I.1, Introduction to IX, IX.4, Notes for IX]

Page 16: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

990 Bibliography

Horgan, C.O., 1978, Plane Steady Flows and Energy Estimates for the Navier-Stokes Equations, Arch. Rational Mech. Anal., 68, 359-381 [Notes for VI, Notesfor XIII]

Horgan, C.O., and Payne, L.E., 1983, On Inequalities of Korn, Friederichs andBabuska-Aziz, Arch. Rational Mech. Anal., 82, 165-179 [Notes for III]

Horgan, C.O., and Wheeler, L.T., 1978, Spatial Decay Estimates for the Navi-er-Stokes Equations with Application to the Problem of Entry Flow, SIAM J.Appl. Math., 35, 97-116 [I.3, Notes for III, Notes for XIII]

Hormander, L., 1966, An Introduction to Complex Analysis in Several Variables,D. Van Nostrand Co. Inc. [XII.4]

Hsiao, G.C., and MacCamy, R.C., 1981, Singular Perturbations for the Two-Dimensional Viscous Flow Problem, Theory and Applications of Singular Per-turbations, Eckhaus, W., and de Jager, E.M. Eds., Lecture Notes in Mathe-matics, Vol. 942, Springer-Verlag, 229-244 [Notes for V]

Hurri, R., 1990, The Weighted Poincare Inequalities, Math. Scand., 67, 145-160[Notes for II]

Imai, I., 1951, On the Asymptotic Behavior of Viscous Fluid Flow at a Great Dis-tance from a Cylindrical Body, with Special Reference to Filon’s Paradox, Proc.Roy. Soc. London, Ser. A, 208, 487-516 [Introduction to VII]

Iosif’jan, G.A., 1978, An Analogue of Saint-Venant’s Principle and the Unique-ness of the Solutions of the First Boundary Value Problem for Stokes’ System inDomains with Noncompact Boundaries, Dokl. Akad. Nauk SSSR, 242, 36-39;English Transl.: Soviet Math Dokl., 19, 1048-1052 [Notes for VI]

Iosif‘jan, G.A., 1979, Saint-Venant’s Principle for the Flow of a Viscous Incom-pressible Liquid, Uspehi Mat. Nauk. 34, 191-192; English Trans.: Russian Math.Surveys 34, 1979, 166-167 [Notes for XIII]

Jones, F., 2001, Lebesgue Integration on Euclidean Space, Jones and BartlettPublishers Canada [II.2]

Joseph, D.D., 1976, Stability of Fluid Motions, Springer Tracts in Natural Philos-ophy, Vol. I [Notes for IX]

Kang, K., 2004, On Regularity of Stationary Stokes and Navier-Stokes EquationsNear Boundary, J. Math. Fluid Mech., 6, 78-101 [Notes for IV]

Kantorovich, L.V., and Akilov, G.P., 1964, FunctionalAnalysis in Normed Sp-aces, Pergamon Press [II.1, IX.3]

Kapitanskii, L.V., 1981, Coincidence of the Spaces J12 (Ω) and cJ1

2 (Ω) for PlaneDomains Ω, Having Exits at Infinity, Zap. Nauch. Sem. Len. Otdel. Mat. Inst.Steklov (LOMI), 110, 74-80; English Transl.: J. Soviet Math., 25, 1984, 850-854[III.4]

Kapitanskii, L.V., 1982, Stationary Solutions of the Navier-Stokes Equations inPeriodic Tubes, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI),115, 104-113; English Transl.: J.Soviet Math., 28, 1985, 689-695 [Notes forXIII]

Kapitanskii, L.V., and Pileckas, K., 1983, On Spaces of Solenoidal Vector Fi-elds and Boundary Value Problems for the Navier-Stokes Equations in Domainswith Noncompact Boundaries Trudy Mat. Inst. Steklov, 159 11-49; EnglishTrans.: Proc. Steklov Math Inst., 159, 1984, 3-34 [Notes for IX, Notes for XIII]

Kapitanskii, L.V., and Pileckas, K., 1984, Certain Problems of Vector Anal-ysis, Zap. Nauch. Sem. Len. Otdel. Mat. Inst. Steklov (LOMI), 138, 65-85;English Transl.: J. Soviet Math., 32, 1986, 469-483 [III.3, Notes for III]

Page 17: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 991

Kato, T., 1966, Perturbation Theory for Linear Operators, Die Grundlehren derMatematischen Wissenschaften, Springer-Verlag, Berlin-Heidelberg-New York[III.1]

Kellog, O.D., 1929, Foundations of Potential Theory, Federick Ungar Publ. Co.,New York [IV.8]

Kellogg, R.B., and Osborn, J.E., 1976, Regularity for the Stokes Problem ina Convex Polygon, J. Funct. Anal., 21, 397-431 [Notes for IV]

Kim, H., 2009, Existence and Regularity of Very Weak Solutions of the StationaryNavier-Stokes Equations, Arch. Rational Mech. Anal., 193, 117-152 [Notes forIX]

Konieczny, P., 2006, On a Steady Flow in a Three-Dimensional Infinite Pipe,Colloq. Math., 104, 33-56 [Notes for I]

Konieczny, P., 2009, On Nonhomogeneous Slip Boundary Conditions for 2D In-compressible Exterior Fluid Flows, Acta Appl. Math., 106, 61-77 [Notes forI]

Korn, A., 1908, Allgemeine Losung des Problems Kleiner, Stationaren Bewegun-gen in Reibenden Flussigkeiten, Rend. Circ. Mat. Palermo, 25, 253-271 [Notesfor IV]

Korolev, A., and Sverak, V., 2007, On the Large-Distance Asymptotics of Stea-dy State Solutions of the Navier-Stokes Equations in 3D Exterior Domains,published as arXiv:0711.0560 [I.2, X.9]

Korolev, A., and Sverak, V., 2011, On the Large-Distance Asymptotics of Stea-dy State Solutions of the Navier-Stokes Equations in 3D Exterior Domains,Ann. I. H. Poincare, 28, 303-313 [I.2, X.9]

Kotschin, N.J., Kibel, I.A., and Rose, N.W.,1954, TheoretischeHydromechan-ik, Vol. 2, Akademie-Verlag, Berlin [Introduction to V]

Kozlov, V.A., Maz’ja, V.G., and Schwab, C., 1994, On Singularities of Solu-tions to the Dirichlet Problem of Hydrodynamics Near the Vertex of a Cone,J. Reine Angew. Math., 456, 65-97 [Notes for IV]

Kozono, H., and Nakao, M., 1996, Periodic Solutions of the Navier-Stokes Equa-tions in Unbounded Domains, Tohoku Math. J., 48, 33-50 [Notes for X]

Kozono, H., and Sohr, H., 1991, New A Priori Estimates for the Stokes Equa-tions in Exterior Domains, Indiana Univ. Math. J., 40, 1-28 [II.9, Notes for II,Notes for IV, Notes for V]

Kozono, H., and Sohr, H., 1992a, Density Properties for Solenoidal Vector Fie-lds, with Applications to the Navier-Stokes Equations, J. Math. Soc. Japan,,300, 307-330 [Notes for III, Notes for X]

Kozono, H., and Sohr, H., 1992b, On a New Class of Generalized Solutions forthe Stokes Equations in Exterior Domains, Ann. Scuola Norm. Pisa (4), 19,155-181 [Notes for V]

Kozono, H., and Sohr, H., 1993, On Stationary Navier-Stokes Equations in Un-bounded Domains, Ricerche Mat., 42, 69-86 [X.3, Notes for X]

Kozono, H., Sohr, H., and Yamazaki, M., 1997, Representation Formula, NetForce and Energy Relation to the Stationary Navier-Stokes Equations in 3-Dimensional Exterior Domains, Kyushu J. Math., 51, 39-60 [X.9, Notes forX]

Kozono, H., and Yamazaki, M., 1998, Exterior Problem for the Stationary Navier-Stokes Equations in the Lorentz Space, Math. Ann. , 310, 279-305 [Notes forV, Notes for VIII, Notes for X]

Page 18: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

992 Bibliography

Kozono, H., and Yanagisawa, T., 2009a, Nonhomogeneous Boundary Value Pro-blems for Stationary Navier-Stokes Equations in a Multiply Connected BoundedDomain, Pacific J. Math., 243, 127-150 [Introduction to IX]

Kozono, H., and Yanagisawa, T., 2009b, Leray’s Problem on the StationaryNavier-Stokes Equations with Inhomogeneous Boundary Data, Math. Z. , 262,27-39 [Introduction to IX]

Kracmar, S., Novotny, A., and Pokorny, M., 1999, Estimates of Three-Dim-ensional Oseen Kernels in Weighted Lp Spaces., Applied Nonlinear Analysis,Vol 3., Sequeira, A., Beirao da Veiga, H., and Videman, J., Eds., Kluwer/PlenumNew York, 281-316 [Notes for VII]

Kracmar, S., Novotny, A., and Pokorny, M., 2001, Estimates of Oseen Ker-nels in Weighted Lp Spaces, J. Math. Soc. Japan, 53, 59-111 [Notes for VII,VIII.3]

Kracmar, S., Necasova, S., and Penel, P., 2005, Estimates of Weak Solutionsin Anisotropically Weighted Spaces to the Stationary Rotating Oseen Equa-tions, WSEAS Trans. Math., 2, 854-861 [Notes for VIII]

Kracmar, S., Necasova, S., and Penel, P., 2006, On the Weak Solution tothe Oseen-Type Problem Arising from Flow Around a Rotating Rigid Body inthe Whole Space, WSEAS Trans. Math., 5, 243-251 [Notes for VIII]

Kracmar, S., Necasova, S., and Penel, P., 2007, Anisotropic L2-Estimates ofWeak Solutions to the Stationary Oseen-Type Equations in R

3 for a RotatingBody, RIMS, Kokyuroku Bessatsu Kyoto Univ., 127-143 [Notes for VIII]

Kracmar, S., Necasova, S., and Penel, P., 2010, Anisotropic L2-Estimates ofWeak Solutions to the Stationary Oseen-Type Equations in 3D-Exterior Do-main for a Rotating Body, J. Math. Soc. Japan, 62, 239-268 [Notes for VIII]

Kratz, W., 1997a,, The Maximum Modulus Theorem for the Stokes System in aBall, Math. Z., 226, 389-403 [Notes for IV]

Kratz, W., 1997b,, On the Maximum Modulus Theorem for Stokes Functions inthe Plane, Analysis, 17, 219-225 [Notes for IV]

Krzywcky, A., 1961, An Application of Weyl’s Method to a Boundary ValueProblem of the Linearized Equations of Hydrodynamics, Prace Mat., 5, 15-26 (in Polish) [Notes for IV]

Kudrjavcev, L.D., 1966a, Imbedding Theorem for a Class of Functions Definedon the Entire Space or on a Half Space. I, Mat. SSSR Sbornik, 69, 616-636;English Transl.: Amer. Math. Soc. Transl., 74, 1968, 199-225 [II.10]

Kudrjavcev, L.D., 1966b, Imbedding Theorem for a Class of Functions Definedon the Entire Space or on a Half Space. II, Mat. SSSR Sbornik, 69, 638-652;English Transl.: Amer. Math. Soc. Transl., 74, 1968, 227-260 [II.10]

Ladyzhenskaya, O.A., 1958, Solution “in the Large” of the Boundary-Value Prob-lem for the Navier-Stokes System for the Case of Two Space Variables, Dokl.Akad. Nauk SSSR, 123, 427-431 (in Russian) [II.3]

Ladyzhenskaya, O.A., 1959a, Solution “in the Large” of the Nonstationary Bound-ary Value Problem for the Navier-Stokes System with Two Space Variables,Comm. Pure Appl. Math., 12, 427-433 [II.3, Notes for II]

Ladyzhenskaya, O.A., 1959b, Investigation of the Navier-Stokes Equation for aStationary Flow of an Incompressible Fluid, Uspehi Mat. Nauk., (3) 14, 75-97(in Russian) [I.3, Introduction to IV, IV.1, Notes for V, Introduction to VI,Introduction to IX, IX.1, Notes for IX, Introduction to X, Notes for X]

Page 19: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 993

Ladyzhenskaya, O.A., 1959c, Stationary Motion of a Viscous Incompressible Fluidin a Pipe, Dokl. Akad. Nauk SSSR, 124, 551-553 (in Russian) [I.3, Notes forXIII]

Ladyzhenskaya, O.A., 1967, New Equations for the Description of Motion of Vis-cous Incompressible Fluids and Solvability in the Large of Boundary ValueProblem for them, Trudy Mat. Inst. Steklov,102, 80-103; English Transl.: Proc.Steklov Inst. Math., 102, 1967, 95-118 [Notes for I]

Ladyzhenskaya, O.A., 1969, The Mathematical Theory of Viscous Incompress-ible Flow, Gordon and Breach [I.2, Notes for III, IV.4, Notes for IV, Notes forV, Notes for VIII, X.4, Notes for X, Notes for XII]

Ladyzhenskaya, O.A., and Solonnikov, V.A., 1976, Some Problems of VectorAnalysis and Generalized Formulations of Boundary-Value Problems for theNavier-Stokes Equations, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst.Steklov (LOMI), 59, 81-116; English Transl.: J. Soviet Math., 10, 1978, 257-286[III.4, III.5, Notes for III]

Ladyzhenskaya, O.A., and Solonnikov, V.A., 1977, On the Solvability of Bo-undary Value and Initial-Boundary Value Problems for the Navier-StokesEquations in Regions with Noncompact Boundaries, Vestnik Leningrad Univ.(Math.), 13 35-45; English Transl.: Vestnik Leningrad Univ. Math. 10 271-279[Notes for XI, Introduction to XIII, XIII.7, Notes for XIII]

Ladyzhenskaya, O.A., and Solonnikov, V.A., 1980,of Boundary Value Problems for Steady-State Stokes and Navier-Stokes Equa-tions in Domains Having an Unbounded Dirichlet Integral, Zap. Nauchn. Sem.Leningrad Otdel. Mat. Inst. Steklov (LOMI), 96, 117-160; English Transl.:J.Soviet Math., 21, 1983, 728-761 [I.3, Introduction to VI, VI.1, VI.2, VI.3,Notes for VI, XIII.3, XIII.7, Notes for XIII]

Lamb, H., 1911, On the Uniform Motion of a Sphere Through a Viscous Fluid,Phil. Mag., (6) 21, 112-121 [Introduction to VII]

Lamb, H., 1932, Hydrodynamics, Cambridge Univ. Press [Introduction to V, Notesfor V, Notes for VI]

Landau, L., 1944, A new Exact Solution of Navier-Stokes Equations, DokladyAcad. Sci. URSS, 43, 286-288 [I.2, X.9]

Lax, P.D., 1955, On Cauchy’s Problem for Hyperbolic Equations and the Differ-entiability of Solutions of Elliptic Equations, Comm. Pure Appl. Math., 8,615-633 [II.3, II.8]

Leray, J., 1933, Etude de Diverses Equations Integrales non Lineaires et de Quelq-ues Problemes que Pose l’Hydrodynamique, J. Math. Pures Appl., 12, 1-82 [I.1,I.2, II.6, Introduction to IX, IX.3, IX.4, Notes for IX, Introduction to X, X.4,Notes for X, Introduction to XI, Notes for XI, Notes for XII]

Leray, J., 1934, Sur le Mouvement d’un Liquide Visqueux Emplissant l’Espace,Acta Math., 63, 193-248 [II.6, X.3]

Leray, J., 1936, Les Problemes non Lineaires, Enseignement Math., 35, 139-151[I.1, Introduction to IX, IX.1]

Leray, J., and Schauder J., 1934, Topologie et Equations Fonctionnelles, Ann.Sc. Ecole Norm. Sup., 51, 45-78 [Introduction to IX]

Lichtenstein, L., 1928, Uber Einige Existenzeprobleme der Hydrodynamik, Math.Z., 28, 387-415 [Notes for IV]

Lions, J.L., 1962, Problemes aux Limites dans les Equations aux Derivees Par-tielles,Seminaire de Mathematiques Superieures, Vol. 1, Les Presses de l’Univer-site de Montreal [II.3]

Determination of Solutions

Page 20: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

994 Bibliography

Lions, J.L., 1969, Quelques Methodes de Resolution des Problemes aux LimitesNon Lineaires, Dunod, Paris [Notes for III, IX.3]

Lions, J.L., and Magenes, E., 1962, Problemi ai Limiti Non Omogenei (V), Ann.Scuola Norm. Pisa, (3) 16, 1-44 [III.1]

Lizorkin, P.I., 1963, (Lp, Lq) Multipliers of Fourier Integrals, Dokl. Akad NaukSSSR, 152, 808-811; English Transl.: Soviet Math Dokl., 4, 1963, 1420-1424[VII.4]

Lizorkin, P.I., 1967, On Multipliers of Fourier Integrals in the Spaces Lp,θ , TrudyMat. Inst. Steklov, 89, 269-290; English Transl.: Proc. Steklov Math Inst., 89,1968, 269-290 [VII.4]

Lockhart, R., and McOwen, R.C., 1983, On Elliptic Systems in Rn, Acta Math.,

150, 125-135 [Notes for III]Lorentz, H.A., 1896, Ein Allgemeiner Satz, die Bewegung Einer Reibenden Flussi-

gkeit Betreffend, Nebst Einegen Anwendungen Desselben, Zittingsverlag Akad.Wet. Amsterdam, 5, 168-175 [IV.2, Notes for IV]

MacRobert, T.M., 1966, Functions of a Complex Variable, MacMillan Students’Edition [VII.3, VIII.4]

Marcinkiewicz, J., 1939, Sur les Multiplicateurs des Series de Fourier, StudiaMath., 8, 78-91 [VII.4]

Maremonti, P., 1991, Existence and Stability of Time-Periodic Solutions to theNavier-Stokes Equations in the Whole Space, Nonlinearity, 4, 503-529 [Notesfor III, Notes for X]

Maremonti, P., 2000, On the Stokes Equations: The Maximum Modulus Theo-rem, Math. Models Methods Appl. Sci., 10, 1047-1072 [Notes for IV]

Maremonti, P., and Russo, R., 1994, On the Maximum Modulus Theorem forthe Stokes System, Ann. Scuola Norm. Pisa (4), 21, 629-643 [Notes for IV]

Maremonti, P., and Solonnikov, V.A., 1985, Estimates for the Laplace Oper-ator in Exterior Domains, Zap. Nauch. Sem. Len. Otdel. Mat. Inst. Steklov(LOMI), 146, 92-102 (in Russian) [Notes for V]

Maremonti, P., and Solonnikov, V.A., 1986, Su una Diseguaglianza per le So-luzioni del Problema di Stokes in Domini Esterni, Preprint, Dipartimento diMatematica, Universita di Napoli [Notes for V]

Maremonti, P., and Solonnikov, V.A., 1990, An Estimate for Solutions of theStokes System in Exterior Domains, Zap. Nauch. Sem. Len. Otdel. Mat. Inst.Steklov (LOMI), 180, 105-120 (in Russian) [Notes for V]

Marusıc-Paloka, E., 2000, Solvability of the Navier-Stokes System withL2 Boun-dary Data, Appl. Math. Optim., 41, 365-375 [Notes for IV, Notes for IX]

Maslennikova, V.N., 1973, The Rate of Decrease, for Large Time, of the Solutionof a Sobolev System, with Allowance Made for the Viscosity (Russian), Mat.Sb. (N.S.), 92, 589-610 [Introduction to VII]

Maslennikova, V.N., and Bogovskii, M.E., 1978, Denseness of Finite Soleno-idal Vector Fields, Sib. Mat. Zh., 19, 1092-1108 (in Russian) [III.4, Notes forIII]

Maslennikova, V.N., and Bogovskii, M.E., 1981a, On the Approximation ofSolenoidal and Potential Vector Fields, Uspehi Mat. Nauk., 36, 239-240 (inRussian) [III.4]

Maslennikova, V.N., and Bogovskii, M.E., 1981b, Sobolev Spaces of Soleno-idal Vector Fields, Sib. Mat. Zh., 22, 91-118; English Transl.: Sibirian Math.J. 22, 1982, 399-420 [III.4]

Page 21: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 995

Maslennikova, V.N., and Bogovskii, M.E., 1983, Approximation of Potentialand Solenoidal Vector Fields, Sib. Mat. Zh., 24, 149-171; English Transl.:Sibirian Math. J., 24, 1983, 768-787 [III.4]

Maslennikova, V.N., and Bogovskii, M.E., 1986ˇ a, Approximation of Solenoidaland Potential Vector Fields in Sobolev Spaces and Equations of Mathemati-cal Physics, Akad. Nauk. SSSR Sib. Otdel. Mat. Inst., Proc. Conference onDifferential Equations, 2, 129-138 (in Russian) [III.1]

Maslennikova, V.N., and Bogovskii, M.E., 1986b, Elliptic Boundary Value Pr-oblems in Unbounded Domains with Noncompact and Nonsmooth Boundaries,Rend. Sem. Mat. Fis. Milano, 56, 125-138 [III.1]

Maslennikova, V.N., and Bogovskii, M.E., 1993, On Non-Closure of Range ofValues of Elliptic Operator for a Plane Angle, Ann. Univ. Ferrara Sez. VII,39, 65-75 [III.1]

Maslennikova, V.N., and Timoshin, M.A., 1989, Lp Theory for the Stokes Sys-tem in Exterior Domains, Proc. Symposium on New Methods in Analysis andDifferential Equations, University of Voronez Press 63-77 (in Russian) [Notesfor V]

Maslennikova, V.N., and Timoshin, M.A., 1990, Lp Theory for the Stokes Eq-uations in Unbounded Domains with a Compact Boundary, Dokl. Akad. Nauk.SSSR, 313, 1341-1346 (in Russian) [Notes for IV, Notes for V]

Maslennikova, V.N., and Timoshin, M.A., 1994, Generalized Solutions withFirst Derivatives in Lp in the Flow Problem for the Stokes System, Sib. Mat.Zh., 35, 135-162; English Transl.: Sibirian Math. J., 35, 1994, 123-149 [Notesfor V]

Masuda, K., 1975, On the Stability of Incompressible Viscous Fluid Motions PastObjects, J. Math. Soc. Japan, 27, 294-327 [Notes for V]

Masuda, K., 1984, Weak Solutions of Navier-Stokes Equations, Tohoku Math.Journ., 36, 623-646 [Notes for III]

Maz’ja, V.G., 1985, Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-NewYork[Notes for II]

Maz’ja, V.G., Plamenevskii, B.A., and Stupyalis, L.I., 1974, The Three-Di-mensional Problem of Steady-State Motion of a Fluid with a Free-Surface, Diff.Uravn. Prim, Trudy Sem. Protsessy Optimal. Upravleniya I Sektsiya Vyp., 23;,English Transl.: Amer. Math. Soc. Transl., 123, 1984, 171-268 [Notes for I,IV.3]

McCracken, M., 1981, The Resolvent Problem for the Stokes Equation on Half-spaces in Lp, SIAM J. Math. Anal., 12, 201-228 [III.1]

McLachlan, N.W., 1961, Bessel Functions for Engineers, Oxford at ClarendonPress [IX.4]

McMullen, C. T., 1998, Lipschitz Maps and Nets in Euclidean space, Geom.Funct. Anal., 8, 304-314 [III.3].

Meyers, N., and Serrin, J.B., 1964, H=W, Proc. Nat. Acad. Sci. USA, 51, 1055-1056 [II.3]

Mikhlin, S.G., 1957, Fourier Integrals and Multiple Singular Integrals, Vestn.Leningrad. Univ. Ser. Mat. Meh. Astron., 12, 143-155 (in Russian) [VII.4]

Mikhlin, S.G., 1965, Multidimensional Singular Integrals and Integral Equations,Pergamon Press [V.3]

Miles, J.W., and Lee, Y.K., 1975, Helmholtz Resonance of Harbours, J. FluidMech., 67, 445-464 [Notes for VI]

Page 22: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

996 Bibliography

Milne-Thomson, L.M. 1938, Theoretical Hydrodynamics, Macmillan & Co., Lon-don [Notes for VI]

Miranda, C., 1940, Un’Osservazione su un Teorema di Brouwer, Boll. Un. Mat.Ital., 3, 5-7 [IX.3]

Miranda, C., 1978, Istituzioni di Analisi Funzionale Lineare, Unione MatematicaItaliana, Oderisi Gubbio Editrice [II.1, II.2, II.3, II.4, II.10, II.11, III.1]

Mitidieri, E., 2000, A Simple Approach to Hardy Inequalities, Math. Notes, 67,479-486 [Notes for II]

Miyakawa, T., 1982, On Non-Stationary Solutions of the Navier-Stokes Equationsin an Exterior Domain, Hiroshima Math. J., 12, 115-140 [III.2]

Miyakawa, T., 1994, The Helmholtz Decomposition of Vector Fields in Some Un-bounded Domains, Math. J. Toyama Univ., 17, 115-149 [Notes for III]

Miyakawa, T., 1995, On Uniqueness of Steady Navier-Stokes Flows in an ExteriorDomain, Adv. Math. Sci. Appl., 5, 411-420 [X.3, Notes for X]

Miyakawa, T., 1999, On Stationary Incompressible Navier-Stokes Flows with FastDecay and the Vanishing Flux Condition, Topics in nonlinear analysis, Hey-wood J.G., Masuda, K., Escher, J., & Simonett, G., Eds., Progress in Nonlin-ear Differential Equations and their Applications, Vol. 35, Birkhauser Verlag,Basel, 535-552 [Notes for X]

Miyakawa, T., and Sohr, H., 1988, On Energy inequality, Smoothness and LargeTime Behavior in L2 for Weak Solutions of the Navier-Stokes Equations in Ex-terior Domains, Math. Z., 199, 455-478 [Notes for II]

Mizumachi, R., 1984, On the Asymptotic Behavior of Incompressible ViscousFluid Motions Past Bodies, J. Math. Soc. Japan, 36, 497-522 [VIII.3, Notesfor VIII]

Mizuta, Y., 1989, Integral Representation of Beppo Levi Functions and the Exis-tence of Limits at Infinity, Hiroshima Math. J., 19, 259-279 [Notes for II]

Morimoto, H., 1992 , A Solution to the Stationary Navier-Stokes Equations Un-der the Boundary Condition with Non-vanishing Outflow, Memoirs of the In-stitute of Science and Technology, Meiji Univ., 31, 7-12 [IX.4, Notes for IX]

Morimoto, H., 1995, Stationary Navier-Stokes Equations with Non-Vanishing Out-flow Condition, Hokkaido Math. J., 24, 641-648 [Notes for IX]

Morimoto, H., 2007, Stationary Navier-Stokes Flow in 2-D Channels Involvingthe General Outflow Condition Handbook of Differential Equations, StationaryPartial Differential Equations, Vol. 4, Chipot, M., Ed., Elsevier Science, 299–353, [Notes for XIII]

Morimoto, H., and Fujita, H., 2002, A Remark on the Existence of Steady Na-vier-Stokes Flows in a Certain Two-Dimensional Infinite Channel, Tokyo J.Math., 25, 307-321 [Notes for XIII]

Morimoto, H., Ukai, S., 1996, Perturbation of the Navier-Stokes Flow in an An-nular Domain with the Non-vanishing Outflow Condition, J. Math. Sci. Univ.Tokyo, 3, 73-82 [IX.4, Notes for IX]

Morrey, C.B., Jr., 1958, On the Analyticity of the Solutions of Analytic Non-Linear Elliptic Systems of Partial Differential Equations. I. Analyticity in theinterior, Amer. J. Math., 80, 198-218 [Notes for IX]

Mozzi, G. 1763, Discorso Matematico sopra il Rotamento Momentaneo dei Corpi, Stamperia di Donato Campo, Napoli [Introduction to VIII]

Mucha, P.B., 2003, Asymptotic Behavior of a Steady Flow in a Two-DimensionalPipe, Studia Math., 158, 39-58 [Notes for I]

Page 23: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 997

Naumann, J., 1988, On a Maximum Principle for Weak Solutions of the StationaryStokes System, Ann. Scuola Norm. Pisa, (4) 15, 149-168 [Notes for IV]

Navier, C.L.M.H., 1827, Memoire sur les Lois du Mouvement des Fluides, Mem.Acad. Sci. Inst. de France, (2) 6, 389-440 [Introduction to I]

Nazarov, S.A., 1996, On the two-dimensional aperture problem for Navier-Stokesequations, C. R. Acad. Sci. Paris, 323, 699-703 [XIII.9, Notes for XIII]

Nazarov, S.A., and Pileckas, K.I., 1983, On the Behavior of Solutions of theStokes and Navier-Stokes Systems in Domains with a Periodically VaryingSection, Trudy Mat. Inst. Steklov, 159, 137-149; English Transl.: Proc. SteklovMath Inst., 159, 1984, 97-104 [Notes for VI, Notes for XIII]

Nazarov, S.A., and Pileckas, K.I., 1990, The Reynolds Flow of a Fluid in aThree-Dimensional Channel, Liet. Mat. Rink., 30, 772-783 (in Russian) [Notesfor XIII]

Nazarov, S.A., and Pileckas, K.I., 1998, Asymptotic of Solutions to Stokes andNavier-Stokes Equations in Domains with Paraboloidal Outlets to Infinity,Rend. Sem. Mat. Padova, 99, 1-43 [I.3, Notes for XIII]

Nazarov, S.A., and Pileckas, K., 1999a, The Asymptotic Properties of the So-lution to the Stokes Problem in Domains that are Layer-like at Infinity, J.Math. Fluid Mech., 1, 131-167 [Notes for VI]

Nazarov, S.A., and Pileckas, K., 1999b, On the Solvability of the Stokes andNavier–Stokes Problems in the Domains that are Layer-like at Infinity, J. Math.Fluid Mech., 1, 78-116 [Notes for XIII]

Nazarov, S.A., and Pileckas, K., 2000, On Steady Stokes and Navier-StokesProblems with Zero Velocity at Infinity in a Three-Dimensional Exterior Do-main, J. Math. Kyoto Univ., 40, 475-492 [X.9]

Nazarov, S.A., and Pileckas, K., 2001, On the Fredholm Property of the StokesOperator in a Layer-like Domain., Z. Anal. Anwendungen, 20, 155-182 [Notesfor VI]

Nazarov, S.A., and Plamenevskii, B.A., 1994, Elliptic Problems in Domainswith Piecewise Smooth Boundaries, De Gruyter Expositions in Mathematics,Vol. 13, W. De Gruyter, Berlin-New York [Notes for IV]

Nazarov, S.A., Sequeira, A., and Videman, J.H., 2001, Steady Flows of Jeffe-ry-Hamel Type from the Half-Plane into an Infinite Channel. I. Linearizationon an Antisymmetric Solution, J. Math. Pures Appl, 80, 1069-1098 [Notes forXIII]

Nazarov, S.A., Sequeira, A., and Videman, J.H., 2002, Steady Flows of Jeffe-ry-Hamel Type from the Half-Plane into an Infinite Channel. II. Linearizationon a Symmetric Solution, J. Math. Pures Appl., 81, 781-810 [Notes for XIII]

Necas, J., 1967, Les Methodes Directes en Theorie des Equations Elliptiques,Masson et Cie [II.1, II.4, Notes for II, Notes for III]

Newton, I., 1686, Philosophiae Naturalis Principia Mathematica, S. Pepys, Reg.Soc. Praeses, Londini [Introduction to I]

Nikol’skii, S.M., 1958, An Embedding Theorem for Functions with Partial Deriva-tives Considered in Different Metrics, Izv. Akad. Nauk SSSR Ser. Mat., 22,321-336 (in Russian) [II.3]

Nirenberg, L., 1959, On Elliptic Partial Differential Equations, Ann. Scuola Norm.Pisa, (3) 13, 115-162 [II.3]

Noether, F., 1911, Uber den Gultigkeitsbereich der Stokesschen Widerstands-formel, Z. Math. Phys., 62, 1-39 [Introduction to VIII]

Page 24: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

998 Bibliography

Novotny, A., and Padula, M., 1995, Note on Decay of Solutions of Steady Na-vier-Stokes Equations in 3-D Exterior Domains, Differential and Integral Equa-tions, 8, 1833-1842 [Notes for V, Notes for X]

Oden, J.T. and Reddy, J.N., 1976, Variational Methods in Theoretical Mechan-ics, Springer-Verlag, Berlin-Heidelberg-New York [Notes for III]

Odqvist, F.K.G., 1930, Uber die Randwertaufgaben der Hydrodynamik ZaherFlussigkeiten, Math. Z., 32, 329-375 [IV.3, IV.8, V.6, Notes for V, Notes forVII, Introduction to IX, IX.6]

Oleinik, O.A., 1952, On Properties of Some Boundary Problems for Equations ofElliptic Type, Mat. Sb., 30, 695-702 (in Russian) [XII.3]

Olmstead, W.E., 1968, Force Relationships and Integral Representations for theViscous Hydrodynamical Equations, Arch. Rational Mech. Anal., 31, 380-389[Introduction to VII]

Olmstead, W.E., and Gautesen, A.K., 1968, A New Paradox in Viscous Hy-drodynamics, Arch. Rational Mech. Anal., 29, 380-389 [Introduction to VII,VII.6]

Olmstead, W.E., and Gautesen, A.K., 1976, Integral Representations and theOseen Flow Problem, Mechanics Today, Vol 3., Nemat-Nasser, S., Ed., Perga-mon Press Inc., 125-189 [Notes to VII]

Olmstead, W.E., and Hector, D.L., 1966, On the Uniqueness of Oseen FlowPast a Half Plane, J. Math. and Phys., 45, 408-417 [Introduction to VII]

Oseen, C.W., 1910, Uber die Stokessche Formel und Uber eine Verwandte Auf-gabe in der Hydrodynamik, Ark. Mat. Astron. Fys., 6 (29), 1-20 [Introductionto VII]

Oseen, C.W., 1927, Neuere Methoden und Ergebnisse in der Hydrodynamik, Leipz-ig, Akad. Verlagsgesellschaft M.B.H. [Notes for IV, Notes for V, Introductionto VII, VII.3, VIII.3, X.10]

Padula, M., 1984, On the Traction Problem for Linear Elastostatics in ExteriorDomains, Proc. Roy. Soc. Edinburgh, 96 A, 55-64 [Notes for II]

Padula, M., 1986, Existence of Global Solutions for 2-Dimensional Viscous Com-pressible Flows, J. Funct. Anal., 69, 1-20 [Notes for II]

Padula, M., 1990, On the Existence and Uniqueness of Non-homogeneous Mo-tions in Exterior Domains, Math. Z., 203, 581-604 [II.9]

Padula, M., 1992, Stability Properties of Regular Flows of Heat-conducting Com-pressible Fluids, J. Math. Kyoto University, 32, 401-442 [III.3]

Padula, M., and Pileckas, K., 1992, Steady Flow of a Viscous Ideal Gas in Do-mains with Noncompact Boundaries: Existence and Asymptotic Behavior in aPipe, Ann. Mat. Pura Appl., 172, 191-218 [Notes for VI]

Pares, C., 1992, Existence, Uniqueness and Regularity of Solution of the Equa-tions of a Turbulence Model for Incompressible Fluids, Appl. Anal., 43, 245-296[Notes for I]

Payne, L.E., 1965, Uniqueness Criteria for Steady State Solutions of the Navier-Stokes Equations, Simposio InternazionaleApplicaz. Analisi alla Fisica Matem-atica, Ed. Cremonese, Roma, 130-165 [Notes for IX]

Payne, L.E., and Weinberger, H.F., 1957, Note on a Lemma of Finn and Gilb-arg, Acta Math., 98, 297-299 [Notes for II]

Pepe, L., 1978, Sulla Diseguaglianza di Sobolev-Poincare, Ann. Univ. Ferrara Sez.VII, 24, 1-9 [Notes for II]

Perucca, E., 1963, Fisica Generale e Sperimentale, Voll. I, II, Unione Tipografi-co-Editrice Torinese (UTET) [I.1, Introduction to V]

Page 25: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 999

Picone, M., 1946, Lezioni di Analisi Funzionale,Dispense Universitarie Tumminel-li, Citta Universitaria Roma [II.5]

Pileckas, K., 1980a, On Unique Solvability of Boundary Value Problems for theStokes System of Equations in Domains with Noncompact Boundaries, TrudyMat. Inst. Steklov, 147, 115-123; English Transl.: Proc. Steklov Math Inst.,147, 1981, 117-126 [Notes for VI]

Pileckas, K., 1980b, Three-Dimensional Solenoidal Vectors, Zap. Nauch. Sem.Len. Otdel. Mat. Inst. Steklov (LOMI), 96, 237-239; English Transl.: J. SovietMath. 21, 1983, 821-823 [Notes for III]

Pileckas, K., 1981, Existence of Solutions for the Navier-Stokes Equations, Hav-ing an Infinite Dissipation of Energy, in a Class of Domains with NoncompactBoundaries, Zap. Nauch. Sem. Len. Otdel. Mat. Inst. Steklov (LOMI), 110,180-202; English Transl.: J. Soviet Math. 25, 932-948 [ Notes for VI, Notes forXIII]

Pileckas, K., 1983, On Spaces of Solenoidal Vectors, Trudy Mat. Inst. Steklov,159, 137-149; English Transl.: Proc. Steklov Math Inst., 159, 1984, 141-154[III.4, III.5, Notes for III]

Pileckas, K., 1984, Existence of Axisymmetric Solutions of the Stationary Sys-tem of Navier-Stokes Equations in a Class of Domains with NoncompactBoundary, Liet. Mat. Rink., 24, 145-154 (in Russian) [Notes for XIII]

Pileckas, K., 1996a, Weighted Lq-Solvability for the Steady Stokes System in Do-mains with Noncompact Boundaries, Math. Models and Methods in Appl. Sci.,6, 97-136 [Introduction to VI, VI.3, VI.4, Notes for XIII]

Pileckas, K., 1996b, Classical Solvability and Uniform Estimates for the SteadyStokes System in Domains with Noncompact Boundaries, Math. Models andMethods in Appl. Sci., 6, 151-186 [Introduction to VI, VI.3, VI.4, Notes forXIII]

Pileckas, K., 1996c, Recent Advances in the Theory of Stokes and Navier-StokesEquations in Domains with Non-compact Boundaries, Mathematical Theory inFluid Mechanics, Galdi, G.P., Malek J., and Necas, J., Eds., Pitman ResearchNotes in Mathematics Series, Longman Scientific & Technical, Vol. 354, 30-85[I.3, Introduction to VI, VI.3, VI.4, Notes for XIII]

Pileckas, K., 1997, Strong Solutions of the Steady Nonlinear Navier-Stokes Sys-tem in Domains with Exits at Infinity, Rend. Sem. Mat. Padova, 97, 235-267[Introduction to I, Notes for XIII]

Pileckas, K., 2002, On the Asymptotic Behavior of Solutions of a Stationary Sys-tem of Navier–Stokes Equations in a Domain of Layer Type, Mat. Sb. , 193,69-104; English Transl.: Sb. Math., 193, 2002, 1801-1836 [Notes for XIII]

Poincare, H., 1894, Sur les Equations de la Physique Mathematique, Rend. Circ.Mat. Palermo, 8, 57-155 [II.5]

Poisson, S.D., 1831, Memoire sur les Equations Generales de l’Equilibre et duMouvement des Corps Solides Elastiques et des Fluides, J. Ecole Polytechnique,13, 1-174 [Introduction to I]

Preiss, D., 1997, Additional Regularity for Lipschitz Solutions of PDE, J. ReineAngew. Math., 485, 197-207 [III.3]

Preston, J.H., 1950, The Steady Circulatory Flow about a Circular Cylinder withUniformly Distributed Suction at the Surface, Aeronaut. Quart., 1, 319-338[XII.2]

Page 26: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

1000 Bibliography

Prodi, G., 1959, Rassegna di Ricerche Intorno alle Equazioni di Navier-Stokes,Quaderni dell’Istituto di Matematica dell’Universita degli Studi di Trieste, 2

[Notes for III]

Protter, M.H., and Weinberger, H. F., 1967, Maximum Principles in Differ-ential Equations, Prentice Hall [XII.3, XII.4]

Pukhnacev, V.V., 1989, Asymptotics of a Velocity Field at Considerable Dis-tances From a Self Propelled Body, Z. Prikl. Mekh. Tekh. Fiz., 30, 52-60;English Transl.: J. Appl. Mech. Tech. Phys., 30, 215-222 [X.6]

Pukhnacev, V.V., 1990a , Stokes Approximation in a Problem of the Flow Arounda Self-Propelled Body, Boundary Value Problems in Mathematical Physics,Naukova Dumka, Kiev, 65-73 (in Russian) [V.7, Notes for V]

Pukhnacev, V.V., 1990b , The Problem of Momentumless Flow for the Navier-Stokes Equations, Navier-Stokes Equations: Theory and Numerical Methods,Heywood, J.G., Masuda, K., Rautmann, R., & Solonnikov V.A., Eds., LectureNotes in Mathematics, Vol. 1431, Springer-Verlag, 87-94 [V.7, Notes for V]

Pukhnacev, V.V., and Solonnikov, V.A., 1982, On the Problem of DynamicContact Angle, Prikl. Matem. Mekhan., 46, 961-971; English Transl.: PMMU.S.S.R., 46, 1983, 771-779 [I.1]

Pulidori, R., 1993, Esistenza ed Unicita per il Problema di Stokes in Domini Es-terni in Spazi di Sobolev con Peso, Tesi di Laurea, Dipartimento di Matematicadell’Universita di Ferrara [Notes for IV, Notes for V]

Pulidori, R., and Specovius-Neugebauer, M., 1995, Generalized Solutionsfor the Stokes Equation in Exterior Domains, Navier-Stokes Equations andRelated Nonlinear Problems, Sequeira, A., Ed., Plenum Press, New York andLondon, 53-62 [Notes for V]

Rabier, P.J., 2002a, Invertibility of the Poiseuille Linearization for StationaryTwo-Dimensional Channel Flows: Symmetric Case, J. Math. Fluid Mech., 4,327-350 [Notes for XIII]

Rabier, P.J., 2002b, Invertibility of the Poiseuille Linearization for StationaryTwo-Dimensional Channel Flows: Nonsymmetric Case, J. Math. Fluid Mech.,4, 351-373 [Notes for XIII]

Reed, M., and Simon, B., 1975, Methods of Modern Mathematical Physics, Vol.II: Fourier Analysis, Self-Adjointness, Academic Press, New York [VII.4]

Rellich, F., 1930, Ein Satz Uber Mittlere Konvergenz, Nachr. Akad. Wiss. Gott-ingen Math. Phys. Kl., 52, 30-35 [II.5]

Rionero, S., and Galdi, G.P., 1979, The Weight Function Approach to Unique-ness of Viscous Flows in Unbounded Domains, Arch. Rational Mech. Anal., 69,37-52 [VII.4]

Rosenhead, L., 1940 , The Steady Two-dimensional Radial Flow of Viscous FluidBetween Two Inclined Plane Walls, Proc. Roy. Soc. London, Ser. A, 175, 436-467 [I.3, XIII.9]

Rudin, W., 1973, Functional Analysis, McGraw-Hill Series in Higher Math. [II.8,III.1]

Rudin, W., 1987, Real and Complex Analysis, Tata McGraw-Hill Publ. Co. Ltd.,New Delhi [II.1, II.3]

Russo, A., 2009, A Note on the Exterior Two-Dimensional Steady-State Navier-Stokes Problem, J. Math. Fluid Mech., 11, 407-414 [X.4]

Russo, A., 2010a, On the Asymptotic Behavior of D-Solutions of the Plane Steady-State NavierStokes Equations, Pacific. J. Math., 246, 253-256 [Notes for XII]

Page 27: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 1001

Russo, A., 2010b, On the Existence of D-Solutions of the Steady-State Navier-Stokes Equations in Plane Exterior Domains, published as arXiv:1101.1243[X.4]

Russo, A., and Starita, G., 2008, On the Existence of Steady-State Solutions tothe Navier-Stokes System for Large Fluxes, Ann. Sc. Norm. Super. Pisa Cl.Sci., 7, 171-180 [Notes for IX]

Russo, R., 2003, On the Existence of Solutions to the Stationary Navier-StokesEquations, Ricerche Mat., 52, 285-348 [Notes for IX]

Russo, R., 2011, On the Maximum Modulus Theorem for the Steady-State Navier-Stokes Equations in Lipschitz Bounded Domains, Applicable Analysis, 90, 193-200 [Notes for IX]

Saffman, P.G., 1965, The Lift on a Small Sphere in a Slow Shear Flow, J. FluidMech., 22, 385-400 [Notes for I]

Salvi, R., 1982, Some Problems of Vector Analysis in Solenoidal Spaces, Portu-galiae Math., 41, 199-209 [Notes for III]

Salvi, R., 1991, The Exterior Problem for the Stationary Navier-Stokes Equations:On the Existence and Regularity, Navier-Stokes Equations: Theory and Numer-ical Methods, Heywood, J.G., Masuda, K., Rautmann, R., & Solonnikov V.A.,Eds., Lecture Notes in Mathematics, Vol. 1530, Springer-Verlag, 132-145 [Notesfor VII]

Sather, J., 1963, The Initial Boundary Value Problem for the Navier-StokesEquations in Regions with Moving Boundaries, Ph.D. thesis, University ofMinnesota [X.3]

Sazonov, L.I., 1993, On the Existence of a Stationary Symmetric Solution of aTwo-dimensional Fluid Flow Problem, Math. Zametki, 54 (6), 138-141 (in Rus-sian) [Notes for IX]

Sazonov, L.I., 1999, Asymptotic Behavior of the Solution to the Two-DimensionalStationary Problem of Flow Past a Body Far from it, Mat. Zametki, 65, 246-253; English Transl.: Math. Notes, 65, 202-207 [Introduction to XII, XII.4,XII.7, Notes for XII]

Schechter, M., 1963a, On Lp Estimates and Regularity, I., Amer. J. Math., 85,1-13 [III.1, Notes for II]

Schechter, M., 1963b, Coerciveness in Lp, Trans. Amer. Math. Soc., 107, 10-29[III.1]

Schechter, M., 1971, Principles of Functional Analysis, Academic Press, NewYork [II.1, V.4]

Sedov, V.N., 1966 On Functions Becoming a Polynomial at ∞, Proc. All-UnionSymposium on Embedding Theorems, Baku (in Russian) [Notes for II]

Sennitskii, V.L., 1978, Liquid Flow Around a Self-Propelled Body, Z. Prikl.Mekh. Tekh. Fiz., 19, 76-83; English Transl.: J. Appl. Mech. Tech. Phys., 19,342-347 [X.6]

Sennitskii, V.L., 1984, An Example of Axisymmetric Fluid Flow Around a Self-Propelled Body, Z. Prikl. Mekh. Tekh. Fiz., 25, 31-36; English Transl.: J. Appl.Mech. Tech. Phys., 25, 526-530 [X.6]

Sequeira, A., 1981, Couplage Entre le Methode des Elements Finis et la Methodedes Equations Integrales.Application au Probleme Exterieur de Stokes Station-naire dans le Plane, These de Doctorat de 3-me Cycle. Universite P. et M. Curie[Notes for V]

Page 28: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

1002 Bibliography

Sequeira, A., 1983, The Coupling of Boundary Integral and Finite Element Meth-ods for the Bidimensional Exterior Steady Stokes Problem, Math. Meth. inAppl. Sci., 5, 356-375 [Notes for V]

Sequeira, A., 1986, On the Computer Implementation of a Coupled Bound-ary and Finite Element Method for the Bidimensional Exterior Steady StokesProblem, Math. Meth. in Appl. Sci., 8, 117-133 [Notes for V]

Serre, D., 1983, Equations de Navier-Stokes Stationnaires avec Donnees peu Re-gulieres, Ann. Sc. Norm. Sup. Pisa (4) 10, 543-559 [Notes for IX]

Serre, D., 1987, Chute Libre d’un Solide dans un Fluide Visqueux Incompressible.Existence, Jap. J. Appl. Math., 4, 99-110 [Notes for VIII, Introduction to X,Notes for XI]

Serrin, J.B., 1959a, On the Stability of Viscous Fluid Motions, Arch. RationalMech. Anal., 3, 1-13 [Notes for IX]

Serrin, J.B., 1959b, Mathematical Principles of Classical Fluid Mechanics, Hand-buch der Physik, Bd VIII/1, Springer-Verlag, Berlin-Heidelberg-NewYork, 125-263 [I.1]

Serrin, J.B., 1962, Strong Convergence in a Product Space, Proc. Am. Math.Soc., 13, 651-655 [Notes for II]

Serrin, J.B., 1963, The initial Value Problem for the Navier-Stokes Equations,, from Nonlinear Problems,Langer R.E., Ed., University of Wisconsin Press,69-98 [Notes for II, X.3]

Shapiro, V.L., 1974, Isolated Singularities for Solutions of the Nonlinear Station-ary NavierStokes Equations, Trans. Amer. Math. Soc., 187, 335-363 [Notes forIX]

Shapiro, V.L., 1976a, Generalized and Classical Solutions of the Nonlinear Sta-tionary Navier-Stokes Equations, Trans. Amer. Math. Soc., 216, 61-79 [Notesfor IX]

Shapiro, V.L., 1976b, Isolated Singularities in Steady State Fluid Flow, SIAM J.Math. Anal., 7, 577-601 [Notes for IX]

Shapiro, V.L., 1976c, A Counterexample in the Theory of Planar Viscous Incom-pressible Flow, J. Differential Equations, 22, 164-179 [Notes for IX]

Shen, Z-W., 1995, A Note on the Dirichlet Problem for the Stokes System in Lip-schitz Domains, Proc. Amer. Math. Soc., 123, 801-811 [Notes for IV]

Shibata, Y., 1999, On an Exterior Initial Boundary Value Problem for Navier-Stokes Equation, Quart. Appl. Math., 57, 117155 [Notes for VII]

Shinbrot, M., 1964, A Fixed Point Theorem and Some Applications, Arch. Ra-tional Mech. Anal., 17, 255-271 [Notes for IX]

Silvestre, A. L., 2004, On the Existence of Steady Flows of a Navier-Stokes Liq-uid Around a Moving Rigid Body, Math. Methods Appl. Sci., 27, 1399-1409[Notes for VIII, Notes for XI]

Silvestre, A.L., 2009, Steady Solutions with Finite Kinetic Energy for a Per-turbed Navier-Stokes System in R

3, J. Differential Equations, 247, 2124-2139[X.6]

Simader, C.G., 1972, On Dirichlets Boundary Value Problem. An Lp-TheoryBased on a Generalization of Gardings Inequality., Lecture Notes in Mathe-matics, Springer-Verlag, Berlin-New York Vol. 268 [II.2, Notes for II]

Simader, C.G., 1990, The Weak Dirichlet and Neumann Problem for the Lapla-cian in Lq for Bounded and Exterior Domains. Applications, Nonlinear Analy-sis, Function Spaces and Applications Vol. 4, Krbec, M., Kufner, A., Opic, B.,& Rakosnık, J. Eds., Teubner-Texte Math., Vol. 119, 180-223 [III.1]

Page 29: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 1003

Simader, C.G., 1992, Mean Value Formulas, Weyl’s Lemma and Liouville Theo-rem for ∆2 and Stokes’ System, Resultate der Mathematik, 22, 761-780 [II.11,Notes for IV]

Simader, C.G., and Sohr, H., 1992, The Helmholtz Decomposition in Lq andRelated Topics, Mathematical Problems Related to the Navier-Stokes Equation,Galdi, G.P., Ed., Advances in Mathematics for Applied Science, 11, World Sci-entific, 1-35 [III.1, Notes for III]

Simader, C.G., and Sohr, H., 1997, The Dirichlet Problem for the Laplacianin Bounded and Unbounded Domains, Pitman Research Notes in MathematicsSeries, Longman Scientific & Technical, Vol. 360 [II.6, Notes for II]

Simader, C.G., and Ziegler, T., 1998, The Weak Dirichlet and Neumann Prob-lems in Lq for the Laplacian and the Helmholtz Decomposition in InfiniteCylinders and Layers, Recent Developments in Partial Differential Equations,Solonnikov V.A. Ed., Quad. Mat., Vol.2, Dip. Mat., Seconda Univ. Napoli,Caserta, 37-161 [Notes for III]

Simon, J., 1991, Primitives de Distributions et Applications, Seminaire d’Analyse,Universite B. Pascal, 20, 1-17 [Notes for III]

Simon, J., 1993, Demonstration Constructive d’un Theoreme de G. de Rham, C.R.Acad. Sci. Paris, 316, 1167-1172 [Notes for III]

Smirnov, V.I., 1964, A Course of Higher Mathematics, Vol. V, Pergamon Press[II.1, II.3, II.5, III.2]

Smith, D.R., 1965, Estimates at Infinity for Stationary Solutions of the Navier-Stokes Equations in Two Dimensions, Arch. Rational Mech. Anal., 20, 341-372[I.2, X.5, X.6, Notes for X, XII.7, XII.8, Notes for XII]

Sobolev, S.L., 1938, On a Theorem of Functional Analysis, Mat. Sbornik, 4, 471-497; English Transl.: Amer. Math. Soc. Transl., 34, 39-68 [II.3, II.11, Notes forII]

Sobolev, S.L., 1963a, Applications of Functional Analysis to Mathematical Physics,Amer. Math. Soc. Transl. of Monographs, Vol. 7 [II.4, II.5, Notes for II, Notesfor III]

Sobolev, S.L., 1963b, Denseness of Finite Fields in the Space Lmq (En), Sib. Mat.

Zh., 3, 673-682 (in Russian) [II.6, Notes for II]Sobolev, S.L., 1964, Partial Differential Equations of Mathematical Physics, In-

tern. Ser. of Monographs in Pure and Appl. Math., Vol. 50, Pergamon Press,Oxford [II.11]

Sobolevski, P.E., 1960, On the Smoothness of Generalized Solutions of the Navier-Stokes Equations, Dokl. Akad. Nauk SSSR, 128, 45-48; English Transl.: SovietMath. Dokl., 1, 1960, 341-343 [Notes for IV]

Sohr, H., and Thater, G., 1998, Imaginary Powers of Second Order DifferentialOperators and Lq-Helmholtz Decomposition in the Infinite Cylinder, Math.Ann., 311, 577-602 [Notes for III]

Sohr, H., and Varnhorn, W., 1990, On Decay Properties of the Stokes Equa-tions in Exterior Domains, Navier-Stokes Equations: Theory and NumericalMethods, Heywood, J.G., Masuda, K., Rautmann, R., & Solonnikov V.A. Eds.,Lecture Notes in Mathematics, Vol. 1431, Springer-Verlag, 134-151 [Notes forV]

Sohr, H., and von Wahl, W., 1984, On the Singular Set and the Uniqueness ofWeak Solutions of the Navier-Stokes Equations, Manuscripta Math., 49, 27-59[Notes for IX]

Page 30: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

1004 Bibliography

Solonnikov, V.A., 1960, On the Estimates of the Tensor Green’s Function forSome Boundary-Value Problems, Dokl. Akad. Nauk SSSR, 130, 988-991; En-glish Transl.: Soviet Math. Dokl., 1, 1960, 128-131 [Notes for IV]

Solonnikov, V.A., 1964, Estimates of the Solutions of a Nonsationary LinearizedSystem of Navier-Stokes Equations, Trudy Mat. Inst. Steklov, 70, 213-317;English Transl.: Amer. Math. Soc. Transl.,75, 1968, 1-116 [VIII.3]

Solonnikov, V.A., 1966, General Boundary Value Problems for Douglis-Niren-berg Elliptic Systems II, Trudy Mat. Inst. Steklov, 92, 233-297; English Transl.:Proc. Steklov Inst. Math, 92, 1966, 212-272 [Introduction to IV, IV.7]

Solonnikov, V.A., 1970, On Green’s Matrices for Elliptic Boundary Value Prob-lems I, Trudy Mat. Inst. Steklov, 110, 107-145; English Transl.: Proc. SteklovInst. Math, 110, 1970, 123-170 [IV.8]

Solonnikov, V.A., 1973, Estimates for Solutions of Nonstationary Navier-StokesEquations, Zap. Nauch. Sem. Len. Otdel. Mat. Inst. Steklov (LOMI), 38, 153-231; English Transl.: J. Soviet Math. 8, 1977, 467-528 [Notes for III, Notes forV]

Solonnikov, V.A., 1981, On the Solvability of Boundary and Initial-BoundaryValue Problems for the Navier-Stokes System in Domains with NoncompactBoundaries, Pacific. J. Math., 93, 443-458 [III.3, III.4, Introduction to VI,XIII.5, XIII.7]

Solonnikov, V.A., 1982, On the Stokes Equations in Domains with Non-SmoothBoundaries and on Viscous Incompressible Flow with a Free Surface, Collegede France Seminar, Vol. III, Pitman Research Notes in Mathematics Series,Longman Scientific & Technical , Vol. 70, 340-423 [Notes for I, Notes for IV]

Solonnikov, V.A., 1983, Stokes and Navier-Stokes Equations in Domains withNon-Compact Boundaries, College de France Seminar, Vol. IV, Pitman Re-search Notes in Mathematics Series, Longman Scientific & Technical , Vol. 84,240-349 [III.4, Notes for III, Introduction to VI, VI.3, XIII.5, XIII.7, Notes forXIII]

Solonnikov V.A., 1988, Solvability of the Problem of the Effluence of a ViscousIncompressible Fluid in to an Infinite Open Basin, Trudy Mat. Inst. Steklov,179, 193-225; English Transl.: Proc. Steklov Inst. Mat., (2) (1989) 193-225[VI.5]

Solonnikov V.A., 1996, Estimates of Oseen’s Potentials in Weighted Holder Spac-es, Math. Nachr., 177, 307-321 [Notes for VII]

Solonnikov, V.A., and Pileckas, K., 1977, On Certain Spaces of SolenoidalVectors and on the Solvability of a Boundary-Value Problem for the Systemof Navier-Stokes Equations in Domains with Noncompact Boundaries, Zap.Nauch. Sem. Len. Otdel. Mat. Inst. Steklov (LOMI), 73, 136-151; EnglishTransl.: J. Soviet Math., 34, 1986, 2101-2111 [III.4, Notes for VI, XIII.7]

Solonnikov, V.A., and Scadilov, V.E., 1973, On a Boundary Value Problemfor Stationary Navier-Stokes Equations, Tr. Math. Inst. Steklov, 125, 196-210;English Transl.: Proc. Steklov Math. Inst. 125, 1973, 186-199 [I.1, Notes for I,Notes for III, Notes for IV, Notes for V, Notes for IX, Notes for X]

Specovius-Neugebauer, M., 1986, Exterior Stokes Problems and Decay at Infin-ity, Math. Meth. in the Appl. Sci., 8, 351-367 [Notes for III, Notes for V]

Specovius-Neugebauer, M., 1990, The Helmholtz Decomposition of WeightedLr-Spaces, Commun. in Partial Diff. Equations, 15, 273-288 [Notes for III]

Specovius-Neugebauer, M., 1995 The Weak Neuman Problem and the Helmho-ltz Decomposition of Two-dimensional Vector Fields in Weighted Lr-Spaces,

Page 31: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 1005

Navier-Stokes Equations and Related Nonlinear Problems, Sequeira, M., Ed.,Plenum Press, New York and London, 105-116 [Notes for III]

Specovius-Neugebauer, M., 1996, The Two-Dimensional Exterior Stokes Prob-lem, Existence, Regularity and Decay Properties, Math. Methods Appl. Sci.,19, 507-528 [Notes for V]

Squire, H.B., 1951, The Round Laminar Jet, Quart. J. Mech. Appl. Math., 4,321-329 [X.9]

Stein, E.M., 1957, Note on Singular Integrals, Proc. Amer. Math. Soc., 8, 250-254[II.11]

Stein, E.M., 1970, Singular Integrals and Differentiability Properties of Functions,Princeton University Press, Princeton [II.3, II.11, III.6, Notes for III]

Stein, E.M., and Weiss, G., 1958, Functional Integrals on n-dimensional Euclid-ean Space, J. Math. Mech., 7, 503-514 [Notes for II]

Stokes, G.G., 1845, On the Theories of the Internal Friction of Fluids in Motion,Trans. Cambridge Phil. Soc., 8, 287-319 [Introduction to I, Introduction to IV]

Stokes, G.G., 1851, On the Effect of the Internal Friction of Fluids on the Motionof Pendulums, Trans. Cambridge Phil. Soc., 9, 8-106 [I.2, Introduction to V]

Stoyan, G., 2001, Iterative Stokes Solvers in the Harmonic Velte Subspace, Com-puting, 67, 13-33 [Notes for III]

Struwe, M., 1988, On Partial Regularity Results for the Navier-Stokes Equations,Comm. Pure Appl. Math., 41, 437-458 [Introduction to IX]

Struwe, M., 1995, Regular Solutions of the Stationary Navier-Stokes Equationson R

5, Math. Annalen, 302, 719-741 [Introduction to IX, IX.2, IX.5]Sverak, V., 1993, On Optimal shape Design, J. Math. Pures Appl., 72, 1-15

[Notes for III]Sverak, V., 2006, On Landau’s Solutions of the Navier-Stokes Equations, pub-

lished as arXiv:math/0604550 [X.9]Sverak, V., and Tsai, T-P., 2000, On the Spatial Decay of 3-D Steady-State

Navier-Stokes Flows, Comm. Partial Differential Equations, 25, 2107-2117[Notes for IV, X.9]

Takeshita, A., 1993, A Remark on Leray’s Inequality, Pacific J. Math., 157, 151-158 [I.1, IX.4]

Tanaka, N., 1995, On the Boundary Value Problem for the Stationary Stokes Sys-tem in the Half-Space, J. Differential Equations, 115, 70-74 [Notes for IV]

Thater, G., 1995, Lq Helmholtz Decomposition and Regularity Results for theInfinite Cylinder, Dissertation zur Erlangung des Doktorgrades der Natur-wissenschaften im Fachbereich Mathematik-Informatik, Universitat Paderborn[Notes for III]

Temam, R., 1973, On the Theory and Numerical Analysis of the Navier-StokesEquations, Lecture Note #9, Dept. of Mathematics, University of Maryland[Notes for III, Notes for IV]

Temam, R., 1977, Navier-Stokes Equations, North-Holland Pub. Co., Amsterdam-New York-Tokyo [III.2, Notes for III, Introduction to IV, Notes for IV, Notesfor IX]

Thomann, E.A. and Guenther, R.B., 2006, The Fundamental Solution of theLinearized Navier-Stokes Equations for Spinning Bodies in Three Spatial Di-mensions. Time Dependent Case, J. Math. Fluid Mech., 8, 77-98 [VIII.3]

Tian, G., and Xin, Z., 1999, Gradient Estimation on Navier-Stokes Equations,Comm. Anal. Geom., 7, 221-257 [Introduction to IX]

Page 32: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

1006 Bibliography

Triebel, H., 1978, Interpolation Theory, Function Spaces, Differential Operators,North-Holland Pub. Co., Amsterdam-New York-Tokyo [II.3]

Troisi, M., 1969, Teoremi di Inclusione per Spazi di Sobolev non Isotropi, RicercheMat., 18, 3-24 [II.3]

Trudinger, N.S., 1967, On Embeddings in Orlicz Spaces and Some Applications,J. Math. Mech., 17, 473-483 [II.3]

Truesdell, C.A., 1952, The Mathematical Foundations of Elasticity and FluidMechanics, J. Rational Mech. Anal., 1, 125-300; corrected reprint, 1966, Con-tinuum Mechanics I, Int. Science Review Series, Vol 8, Gordon & Breach, NewYork [I.1]

Truesdell, C.A., 1953, Notes on the History of the General Equations of Hydro-dynamics, Am. Math. Monthly, 60, 445-458 [Introduction to I]

Udeschini, P., 1941, Incompatibilita dell’Adesione Completa al Contorno con laRegolarita e le Condizioni Asintotiche Euleriane per Correnti Viscose Stazionarie,Reale Accad. Lincei, Cl. Sci. Fis. Mat. Nat., (VII) 2, 957-963 [X.8]

Uspenskii, S.V., 1961, Imbedding Theorems for Weighted Classes, Trudy Mat.Inst. Steklov, 60, 282-303; English Transl.: Amer. Math. Soc. Transl., 87, 1980,121-145 [Notes for II]

Valli, A., 1985, On the Integral Representation of the Solution to the Stokes Sys-tem, Rend. Sem. Mat. Padova, 74, 85-114 [Notes for IV]

Varnhorn, W., 1994, The Stokes Equations, Mathematical Research, Vol. 76,Akademie Verlag [Notes for IV]

Vasil’ev, M.M., 1973, On the Asymptotic Behaviour of the Velocity and ForcesExerted on a Body, in a Stationary Viscous Fluid Flow, Preprint Institute ofAppl. Math., 50, Moscow University [X.8]

Velte, W., 1990, On Optimal Constants in Some Inequalities, Navier-StokesEqua-tions: Theory and Numerical Methods, Heywood, J.G., Masuda, K., Rautmann,R., & Solonnikov, V.A. Eds., Lecture Notes in Mathematics, Springer-Verlag,Vol. 1431, 158-168 [Notes for III]

Voldrich, J., 1984, On Solvability of the Stokes Problem in Sobolev Power WeightSpaces, Comm. Math. Univ. Carolinae, 25, 325-336 [Notes for III, Notes forIV]

von Wahl, W., 1978, Die Stationaren Gleichungen von Navier-Stokes und Semi-lineare Elliptische Systeme, American J. Math., 100, 1173-1184 [Notes for IX]

von Wahl, W., 1986, Regularity of Weak Solutions of the Navier-Stokes Equa-tions, Proc Symp. Pure Math., 45, Part 2, 497-503 [Notes for IX]

von Wahl, W., 1989, Das Außenraumproblem fur die Instationaren Navier-StokesGleichungen, Sonderforschungsbereich, 256, Rudolf-Lipschitz-Vorlesung [Notesfor III]

von Wahl, W., 1990a, On Necessary and Sufficient Conditions for the Solvabilityof the Equations rotu = γ and div u = ε with u vanishing on the Bound-ary, Navier-Stokes Equations: Theory and Numerical Methods, Heywood, J.G.,Masuda, K., Rautmann, R., & Solonnikov, V.A., Eds., Lecture Notes in Math-ematics, Vol. 1431, Springer-Verlag, 152-157 [Notes for III]

von Wahl, W., 1990b, Abschatzungen fur das Neumann-Problem und die Helmholtz-Zerlegung von Lp, Nach. Akad. Wiss. Gottingen, Math. Phys. Klasse (2) [Notesfor III]

Youdovich, V.I., 1967, An Example of Loss of Stability and Generation of Sec-ondary Flow in a Closed Vessel, Mat. Sb., 74, 306-329; English Transl.: Math.USSR Sbornik, 3, 1967, 519-533 [Introduction to IX, IX.2]

Page 33: Bibliography - Springer978-0-387-09620-9/1.pdfBibliography Numbers in square brackets denote the chapter and section where the reference is quoted. Acosta, G., Duran, R.G., and Muschietti

Bibliography 1007

Vorovich, I.I., and Youdovich, V.I., 1959, Stationary Flow of a Viscous Fluid,Dokl. Akad. Nauk SSSR, 124 542-545 (in Russian) [IX.3]

Vorovich, I.I., and Youdovich, V.I., 1961, Stationary Flow of a Viscous In-compressible Fluid, Mat. Sb., 53, 393-428 (in Russian) [Notes for IV, Intro-duction to IX, IX.3, Notes for IX]

Wang, G., and Yang, D., 2008, Decomposition of Vector-Valued Divergence FreeSobolev Functions and Shape Optimization for Stationary Navier-Stokes equa-tions, Comm. Partial Differential Equations, 33, 429-449 [Notes for III]

Watson, G.N., 1962, A Treatise on the Theory of Bessel Functions, CambridgeUniv. Press [VII.3]

Weinberger, H.F., 1965, A First Course in Partial Differential Equations withComplex Variables and Transform Methods, Blaisdell Publ. Co. Waltham-Toronto-London [XII.3]

Weinberger, H.F., 1972, Variational Properties of Steady fall in Stokes Flow, J.Fluid Mech., 52, 321-344 [Introduction to X]

Weinberger, H.F., 1973, On the Steady Fall of a Body in a Navier-Stokes Fluid,Proc. Symp. Pure Mathematics, 23, 421-440 [Notes for VIII, Introduction toX, Notes for XI]

Weinberger, H.F., 1974, Variational Principles for a Body Falling in SteadyStokes Flow, Proc. Symp. on Continuum Mechanics and Related Problems ofAnalysis, 2, Mecniereba, 330-339 [Introduction to X]

Weyl, H., 1940, The Method of Orthogonal Projection in Potential Theory, DukeMath. J., 7, 411-444 [II.11, III.1, IV.2, VIII.2]

Whitehead, A.N., 1888, Second Approximation to Viscous Fluid Motion, Quart.J. Math., 23, 143-152 [Introduction to VII, X.10]

Xiao, Y., and Xin, Z., 2007, On the Vanishing Viscosity Limit for the 3-D Navier-Stokes Equations with a Slip Boundary Condition, Comm. Pure Appl. Math.,60, 1027-1055 [Notes for I]

Yamazaki, M., 2009, The Stationary Navier-Stokes Equation on the Whole Planewith External Force with Antisymmetry, Ann. Univ. Ferrara Sez. VII, 55,407-423 [Notes for XII]

Zeidler, E., 1997, Nonlinear Functional Analysis and Applications, Vol. IV. Cor-rected Second Printing, Springer-Verlag, New York Inc. [Introduction to IX]

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Index

Adherence condition, 4–5Anisotropic Sobolev spaces, 55Annihilator, 143Aperture domain, 20, 199, 368, 901,

928Navier–Stokes flow in, 928ff

existence and uniqueness for, 937asymptotic structure of, 965–966,

970see also Navier–Stokes flow in

aperture domainsStokes flow in, 393, 407

existence and uniqueness for, 407asymptotic behavior of, 414–415

see also Stokes flow in aperturedomains

Approximating solutionsin bounded domains, 599, 617in domains with noncompact

boundaries, 915, 945in exterior domains, 682, 767

Approximation of functionsin D1,q

0 ∩ Lr, 218, 225in H1

q ∩ Lr , 218, 222in Lq ∩D−1,q

0 , 456in Wm,q , 50, 51

Aronszajn-Gagliardo theorem, 742Asymptotic behavior, see Behavior,

Decay, Oseen flow, Stokes flow,Navier–Stokes flow

Behavior at large distances of functionsfrom D1,q , 86, 88–89

pointwise, 117

Bessel functions, 431, 887Body force, 2Bogovskiı formula, 163Boundary

bounded, 4, 37unbounded, 4Stokes flow, estimates near the, 271ff

Boundary inequalities, 61ffBoundary portion

of class Ck, 37of class Ck,λ, 37

Bounded domainNavier–Stokes flow in a, 583ffsee also Navier–Stokes flow in a

bounded domainOseen flow in a, 469Stokes flow in a, 231ffsee also Stokes flow in a bounded

domainBounded regions

flow in, 4Brouwer theorem, 597

Calderon-Zygmund theorem, 130Canonical basis in R

n, 26Carnot theorem, 127Cauchy inequality, 42Cauchy sequence, 30

weak, 32Cauchy stress tensor, 2Coincidence of H1

q and bH1q

in bounded domains, 196in domains with a noncompact

boundary, 198

1009

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1010 Index

in exterior domains, 197Coincidence of D1,q

0 and bD1,q0 , 214ff

Compactness criterion, 73Compatibility condition for the exis-

tence of Navier-Stokes flow in abounded domain, 584

Compatibility condition for the exis-tence of q-generalized solutionsfor Navier-Stokes flow in exteriordomains, 743

Compatibility condition for the exis-tence of Stokes flow in exteriordomains, 338

Cone property, 170Constants, 27Continuity properties of trilinear form,

588, 592, 661, 908Convective term, 1Convergent sequence, 29

weakly, 32Convexity inequality, 42Convolution, 125Counterexample to solvability of

∇ · v = f , 173, 606Cross section, 10, 20, 192, 193, 370, 902

unbounded, 387Curl operator, 28Cut-off function

anisotropic, 93Sobolev, 102

D-solutions, 650two-dimensional, 803

asymptotic behavior of, 806ffpressure field associated to, 818ff,886derivatives of, 812, 828

total head pressure field associatedto, 831ffvelocity field, 824, 876derivatives of, 812, 827–828, 885

vorticity field, 826ff, 887D-solutions, three-dimensional,

see generalized solutionsDecay, pointwise

for functions from D1,q , 117for Navier–Stokes flow

in aperture domains,three-dimensional case, 960ff

two-dimensional case, 321in semi-infinite straight channels,

918ffin three-dimensional exterior

domains; irrotational casewith v∞ 6= 0, 688ff, 709ffwith v∞ = 0, 724ff

in three-dimensional exteriordomains; rotational casewith v0 · ω 6= 0, 777ffwith v0 · ω = 0, 795ff

in two-dimensional exteriordomains,with v∞ 6= 0, 876ffwith v∞ = 0, 857

see also Navier-Stokes flow in exteriordomains, and in domains with anunbounded boundary

for Oseen flow, 471, 472–473for Stokes flow in exterior domains,

313, 314for Stokes flow in a semi-infinite

straight channel, 379fffor Stokes flow in channels with

unbounded cross sections, 393fffor Stokes flow in aperture domains,

414–415Decomposition of Lq , 112ff, 141ffDerivative

generalized (or weak), 48normal, 67

Diameter of a set, 27Difference quotient, 59Differential inequality, 381Dirichlet integral

bounded, 7, 8, 13, 15unbounded, 20, 367–368

Dirichlet problem for the Poissonequation in a half-space

existence, 133uniqueness, 134

Dirichlet problem for the Poissonequation in exterior domains

existence and uniqueness, 349Distance, 26

regularized, 219Distorted channel, 902Divergence operator, 2, 28

generalized, 155

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Index 1011

Domain, 27

of class Ck, 37

Ck-smooth, 37of class Ck,λ, 37

Ck,λ-smooth, 37

locally Lipschitz, 37

star-shaped, 38star-like, 38

with cylindrical ends, 370

Double-layer potentials for Stokes flowin a half-space, 247

Duality pairing

〈 , 〉, 60

[ , ], 112Dyadic product, 26

Ehrling inequality, 77

Embedding theorems, 57, 59Energy equation, 11, 602, 651, 660,

727, 757, 932

generalized, 662, 667Energy inequality, 602, 726, 761, 766

generalized, 639, 663, 669, 682

Erenhaft–Millikan experiment, 300

Estimatesfor generalized Oseen flow, 560ff

for Oseen flow, exterior domains,481ff

for Stokes flow, interior, 263fffor Stokes flow, near the boundary,

271ff

for Stokes flow, exterior domains,320ff, 337ff

for Stokes flow, in Holder spaces,287ff

for Stokes flow in a semi-infinitestraight channel, in Wm,q , 375ff

Exceptional solutions, 337

Existence see generalized Oseen flow,Navier-Stokes flow, Oseen flow,Stokes flow, generalized solutions

Extension, 57–58Leray–Hopf, 604

solenoidal 176, 181, 603, 616, 678

Extension Condition (EC), 603

counterexample to, 605–606Exterior domain, 37

Generalized Oseen flow in an, 485ff

Navier–Stokes flow in a three-dimensional,

irrotational case, 649ffrotational case, 747ff

Navier–Stokes flow in a two-dimensional, 799ff

see also Navier–Stokes flow in exteriordomains

Oseen flow in an, 417ffsee also Oseen flow in exterior

domainsStokes flow in an, 299ffsee also Stokes flow in exterior

domainsExterior regions, 4

flow in, 8

Fluid, 1plane flow of a, 14

Fluxthrough the boundary

of a bounded domain, 5, 604ffof an exterior domain, 678

three-dimensional, 13, 680, 686two-dimensional, 680–681, 687

through the aperture, 929through the cross section, 18, 900

Flux carrier, 900, 914, 938Force exerted by the liquid on the

boundary, 703, 741, 743, 854, 893Friedrichs inequality, 73Function spaces

of hydrodynamics, 155ff, 193ff, 214ffsee also SpacesFundamental solution

for the biharmonic equation, 239for the Laplace equation, 115for the Oseen equation, 429ff

three-dimensional, 434–435estimates, 436ff

two-dimensional, 439–440estimates, 440ff

n-dimensional, estimates, 443for the Stokes equation, 238ff

estimates at large distances, 240for the time-dependent Oseen

equation, 516asymptotic estimates, 517ff

truncated Oseen-Fujita, 470

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1012 Index

truncated Stokes-Fujita, 310

Gagliardo theorem, 64Galerkin method, 424, 463, 497, 597,

682, 766, 914, 945Gauss divergence theorem, 61, 68

generalized, 159Generalized derivative, 48Generalized Oseen flow

generalized solutions for,existence of, 501uniqueness of, 505ff

q-generalized solutions for, 499asymptotic behavior of, 547, 555,

558–559existence, uniqueness and estimates

in R3 of, 547

existence, uniqueness and estimatesin exterior domains of, 555

pressure associated to, 500regularity of, 500–501representation of, 498

Generalized solutionssee Generalized Oseen flow, Navier–

Stokes flow, Oseen flow, Stokesflow

Gradient operator, 2, 28generalized, 175

Green’s identityfor the Laplace operator, 115for the Oseen system, 467for the Stokes system, 290for the time-dependent Oseen system,

531–532Green’s tensor for the Stokes problem

in bounded domains, 288ffestimates for, 289

in exterior domains, 349ffestimates for, 350–351

in half-space, 261ffestimates for, 263

Hagen–Poiseuille flow, 18, 366Hamel solution, 805Hardy inequality, 66Helmholtz–Weyl decomposition of Lq,

141ffHeywood’s problem, 20, 928

see also aperture domain

Holder continuous, 36Holder inequality, 41

generalized, 42Homogeneous Sobolev spaces, 80

Incompressibility condition, 2Inequality

Cauchy, 42convexity, 42differential 381Ehrling, 77Friedrichs, 73Hardy, 66Holder, 41

generalized, 42integro-differential 381–382Ladyzhenskaya, 55Minkowski, 42

generalized, 42Nirenberg, 51

generalized, 54Poincare, 69, 71, 72, 75

generalized, 175Poincare-Sobolev, 75Schwarz, 42Sobolev, 54trace 62–64, 68, 122Troisi, 55for vector functions with normal

component vanishing at theboundary, 71, 77

weighted, 85–86, 98, 135Wirtinger, 76Young, for convolutions, 125Young, for numbers, 42

Integral transform, 125ffIntegro-differential inequality, 381–382

Jeffery–Hamel solution, 971

Kernel, 125weakly singular, 126singular, 129

Kinematic viscosity coefficient, 4Kinetic energy,

finite, 703infinite, 698, 702ff, 750, 774

Ladyzhenskaya inequality, 55

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Index 1013

Ladyzhenskaya’s variant of Leray’smethod, 643

Laminar motion, 584

Landau solution, 12, 728–729, 793, 797

Laplace operator, 2Laplace equation, fundamental solution

for, 115

Lebesgue spaces, 40ffLeray’s contradiction argument, 645ff

Leray–Hopf extension, 604

Leray’s problem, 18, 370ff, 903ffNavier–Stokes flow for,

generalized solutions for, 903

asymptotic behavior of, 927

existence of, 914regularity of, 904

uniqueness of, 911, 918

pressure associated to, 904Stokes flow for,

generalized solutions to, 371

asymptotic decay of, 379ffexistence and uniqueness of, 378

pressure associated to, 371

regularity of, 372Leray–Schauder theorem, 584, 644

Limit of vanishing Reynolds number,

for Oseen flow, 487fffor Navier–Stokes flow

in bounded domains, 640ff

in three-dimensional exteriordomains, 731ff

in two-dimensional exteriordomains, 887ff

Liouville theorem for generalizedsolutions

in R3, 12, 705, 729

in Rn, n > 3, 731

in R2, 808

Liquid, 1Lizorkin theorem, 446

Locally Lipschitz domain, 37

Minkowski inequality, 42

generalized, 54Multi-index, 28–29

Mollifier, 43–44

Mollifying kernel, 43Mozzi–Chasles transformation, 13, 496

Navier–Stokes equations, 2steady-state, 4

Navier–Stokes flow in aperture domains,see also Heywood’s problem, 928ff

generalized solutions for, 929asymptotic structure of, 965–966derivative of, 966pressure associated to, 970

existence of, 945global summability properties of,

951pressure field associated to, 929–930regularity of, 931uniqueness of, 951

Navier–Stokes flow in bounded domainsgeneralized solutions for, 587

existence of,with homogeneous boundary

data, 596ffwith nonhomogeneous boundary

data, 602ffnon-uniqueness of, 596pressure field associated to, 590regularity of, 621ff, 636–639uniqueness of, 592

with homogeneous boundarydata, 602

with nonhomogeneous boundarydata, 619

Navier–Stokes flow in distortedchannels, see Leray’s problem

Navier–Stokes flow in three-dimensionalexterior domains; irrotational case

generalized solutions for, 654asymptotic structure of,

with v∞ 6= 0, 709ffderivative of, 717pressure associated to, 719vorticity of, 720

asymptotic structure of,with v∞ = 0, 726derivative of, 724, 726pressure associated to, 724, 726

existence of, 681global summability properties of,

701pressure field associated to, 655regularity of, 658–659uniqueness of, 668ff

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1014 Index

with v∞ 6= 0, 709with v∞ = 0, 727

Navier–Stokes flow in three-dimensionalexterior domains; rotational case

generalized solutions for, 763–764asymptotic structure of,

with v0 · ω 6= 0, 777ffderivative of, 787pressure associated to, 790

asymptotic structure of,with v0 · ω = 0, 795ffderivative of, 796, 797pressure associated to, 796, 797

existence of, 765global summability properties of,

772pressure field associated to, 752, 756regularity of, 757uniqueness of, 760ff

with v0 · ω 6= 0, 777with v0 · ω = 0, 796

Navier–Stokes flow in two-dimensionalexterior domains, 799ff

generalized solutions for, 654asymptotic structure of,

with v∞ 6= 0, 876derivative of, 885pressure associated to, 886vorticity of, 828–829, 887

asymptotic structure of,with v∞ = 0, 857

existence of, 687, 838ffglobal summability properties of,

857ffnon-existence for large data of, 17,

856non-uniqueness of, 805–806pressure field associated to, 655regularity of, 658–659uniqueness of, 803ff

Non-existence for large data forNavier–Stokes flow, 17, 856

Non-uniqueness for Navier–Stokes flowin bounded domains, 596in exterior domains, 805–806

Neumann problem, generalized, 146ffNirenberg inequality, 51

generalized 54Norm, 29

‖ ‖q , 40‖ ‖m,q , 50| |m,q , 83‖ ‖(q,r),A,t , 526

Notation 6, 26ff

Olmstead–Gautesen drag paradox, 474Orthogonal complement, 143Oseen flow

generalized, see generalized Oseenflow

q-generalized solutions for, 420asymptotic behavior of, 471, 472asymptotic behavior of the vorticity

of, 475existence, uniqueness and estimates

in Rn of, 452, 459

existence, uniqueness and estimatesin exterior domains of, 481, 484

local representation of, 471pressure associated to, 421regularity of, 422representation of, 470

generalized solutions for,three-dimensional; existence of, 425two-dimensional; existence of, 465uniqueness of, 422ff

limit of vanishing Reynolds number,487ff

time-dependent, see time-dependentOseen flow

Oseen fundamental solution, 429ffthree-dimensional, 434–435

estimates of, 436fftwo-dimensional, 439–440

estimates of, 440ffn-dimensional; estimates of, 443paraboloidal wake region exhibited

by the,three-dimensional, 436two-dimensional, 440

Oseen volume potentials, 444time-dependent, 532

Oseen-Fujita truncated fundamentalsolution, 470

Paraboloidal wake regionthree-dimensional, 436, 714two-dimensional, 440, 881–882

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Index 1015

Paradoxof Olmstead–Gautesen, 474of Stokes, 302, 309, 318, 319, 351ff,

887ffof Whitehead 417, 732within the Oseen approximation,

419, 474Partition of unity, 40Perturbation series around Stokes flow,

723–724Physically Reasonable (PR) solutions,

651,Plane flow in exterior domains, 14,

799ffPoincare constant, 70Poincare inequality, 69, 70, 71, 75

generalized, 175, 191Poincare–Sobolev inequality, 75Poiseuille solution, 18, 366, 900Poiseuille constant, 369Poisson integral, 133PR solutions, see Physically Reasonable

solutionsPressure field associated to a q-gene-

ralized solution, 235, 305, 371,390, 421, 500, 590, 655, 752, 904,929–930

Problem ∇ · v = fin bounded domains, 161ffin domains with noncompact

boundary, 191ffin exterior domains, 188ffin a half-space, 261

Projection operator Pq, 142

q-generalized solutionsfor generalized Oseen flow, 499for Navier–Stokes flow 587, 644for Oseen flow 420for Stokes flow in bounded domains,

234for Stokes flow, interior estimates,

266, 270for Stokes flow, estimates near the

boundary, 278for Stokes flow in R

n; existence anduniqueness of, 244

for Stokes flow in a half-space;existence and uniqueness of, 257

for Stokes flow in exterior domains,341

asymptotic behavior of, 313, 314regularity of, 265, 266, 276, 306, 372

q-weak solutions, see q-generalizedsolutions

Regularity of generalized solutionssee generalized Oseen flow,Navier–Stokes flow, Oseen flow,Stokes flow

Regularized distance, 219Regularizer, 44Representation formulas

for Navier–Stokes flow in exteriordomains

three-dimensional case 692–693two-dimensional case 867–868

for Navier–Stokes flow in aperturedomains 965–966

for Oseen flow, 472–473local, 471

for Stokes flow in aperture domains,411, 414

for Stokes flow in bounded domains,292, 294

for Stokes flow in exterior domains,315

local, 312for time-dependent Oseen flow, 532

Reynolds number, 231, 420, 496effective, 497

see also limit of vanishing Reynoldsnumber

Riesz potential, 126

Scalar potential, 141Schauder estimates, 287Schmidt orthogonalization procedure,

425Schwarz inequality, 42Segment property, 51Self-propelled body 352, 364, 653, 703Semi-infinite straight channel

Stokes flow in; estimates in Wm,q ,375

Stokes flow in; asymptotic decay, 379Sequence

Cauchy, 30

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1016 Index

weak, 32

convergent, 29weakly convergent 32

Singular kernel, 129

Sobolev “cut-off” function, 102Sobolev theorem, 128

Sobolev space, 50Space

Banach, 30Ck(Ω), 35

C∞(Ω), 35Ck

0 (Ω), 35

C∞0 (Ω), 35

Ck,λ(Ω), 36Dm,q(Ω), 80

Dm,q(Ω), 83Dm,q

0 (Ω), 84

D−m,q0 (Ω) 109

D(Ω), 142

D1,q0 (Ω), 214bD1,q

0 (Ω), 214Gq(Ω), 142Hq(Ω), 142

H1q (Ω), 193bH1

q (Ω), 193Lq(Ω) 40

Lqloc(Ω),Lq

loc(Ω), 43Lq(At), 526

Lr,q(At), 526Wm,q(Ω), 49Wm,q

0 (Ω), 50

W−m,q′ (Ω), W−m,q′0 (Ω), 60

Wm−1/q,q (∂Ω), 64, 67Wm,q

loc (Ω),Wm,qloc (Ω), 81

Sobolev, 50anisotropic, 55

homogeneous, 80negative, 60

trace, 64, 67–68Star-shaped or star-like 38

Steady-state Navier-Stokesequations, 4

Steady fall of a body, 653Stein theorem

on extension maps, 58on singular transforms in weighted

spaces, 131on regularized distance, 219

Stokes flow in an aperture domain, seealso Heywood’s problem, 407

generalized solutions for, 388asymptotic behavior of, 414–415existence and uniqueness of, 392,

407pressure associated to, 390representation of, 411, 414

existence and uniqueness in D1,q,407

Stokes flow in bounded domains, 231ffgeneralized solutions for, 234

existence and uniqueness of, 237regularity of, 267, 277

q-generalized solutions for, 234estimates of

in Holder spaces,in Wm,q , 227 234interior, 263ffnear the boundary, 271ff

existence and uniqueness ofin Holder spaces, 287ffin Wm,q , 279ff

pressure field associated to, 186uniqueness of, 228

maximum modulus theorem for, 298Green’s tensor, 288

estimates for 289Stokes flow in channels with unbounded

cross sections 387ffgeneralized solutions for, 388

asymptotic behavior of, 393ffexistence and uniqueness of, 390pressure associated to, 390

Stokes flow in exterior domains, 299ffexistence and uniqueness in Dm,q ,

334–335generalized solutions, 304

asymptotic behavior of, 313, 314pressure field associated to, 305regularity of, 306

q-generalized solutions for, 304existence and uniqueness of, 341

Green’s tensor, 349estimates for, 350–351

representation of linear functionals,345

Stokes flow in Rn, 238ff

existence and uniqueness

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Index 1017

in Dm,q, 243of q-generalized solutions, 244

Stokes flow in Rn+, 247ff

existence and uniquenessin Dm,q, 256of q-generalized solutions, 257

Green’s tensor, 261–262estimates for, 263

Stokes flow in a semi-infinite straightchannel

asymptotic behavior of, 379ffestimates in Wm,q , 374–375

Stokes flow in an unbounded distortedchannel, see Leray’s problem

Stokes flow, transition tofrom Navier-Stokes flow

in bounded domains, 640ffin three-dimensional exterior

domains, 731ffin two-dimensional exterior

domains, 887fffrom Oseen flow, 487ff

Stokes fundamental solution, 239–240Stokes paradox, 302, 309, 318, 319,

351ff, 839, 854, 894for generalized solutions, 309

Stokes potentialvolume, 240double-layer in a half-space, 247

Stokes solutionpast a sphere, 300past a cylinder, 302

Stokes-Fujita truncated fundamentalsolution, 310

Stream function, 301Stretching tensor, 2Stress tensor, 2Support of a function, 28Symmetric flow, 826, 855

Tensorstretching, 2Cauchy stress, 2

Time-dependent Oseen flowCauchy problem for

existence of solutions to, 527, 538,540–541

uniqueness of solutions to, 527, 536,539, 540–541

representation of solutions to, 532Time-dependent Oseen fundamental

solution, 515–516estimates of

integral, 517ffpointwise, 517

Total head pressure, 819Trace inequalities, 62–64, 68, 122Trace of a function, 63–64

on a bounded boundary, 61ffon a bounded portion of the

boundary, 68defined in a half-space, 121ff

Trace operator, 64, 67Trace space of functions

from Wm,q , 64, 67from D1,q(Rn

+), 122, 125Transition to the Stokes flow

from Oseen flow, 487fffrom Navier–Stokes flow,

in bounded domains, 640ffin three-dimensional exterior

domains, 731ffin two-dimensional exterior

domains, 887ffTrilinear form, 588

continuity of, 588, 592, 661, 908Troisi inequality, 55Truncated fundamental solution

Oseen-Fujita, 470Stokes-Fujita, 310

Unbounded Dirichlet integral, 20, 367Unbounded regions with unbounded

boundaryflow in, 17Stokes flow in, 365ffsee also Stokes flow in a half

space, in semi-infinite channels,in channels with unbounded crosssection, in aperture domains

Navier–Stokes flow in, 899ffsee also Heywood’s problem, Leray’s

problemUniqueness

see generalized Oseen flow, Navier–Stokes flow, Oseen flow, Stokesflow

Unsteady flow, 4

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1018 Index

Unsteady Oseen flow, 514ff

Variational formulation, 233, 304, 420,586, 653, 751, 903

Vector potential, 141Very weak solution, 297, 644–645Viscosity

infinite limit, see limit of vanishingReynolds number

kinematic, 4Vorticity, 474, 806

asymptotic behavior of, 475, 720,828–829, 887

see also Navier–Stokes flow, Oseenflow

Wake regiongeneralized Oseen flow, 554, 555Navier–Stokes flow

irrotational, three-dimensional 714

rotational, 749, 777two-dimensional, 881–882

Oseen flowthree-dimensional, 436two-dimensional, 440

Weak compactness, 32Weak convergence, 32Weak derivative, 48Weak solution, see generalized solutionWeakly complete, 32Weakly divergence free, 155Weakly singular kernels, 126Weierstrass kernel, 515Weighted inequalities, 85–86, 98, 135Whitehead paradox, 417, 432Wirtinger inequality, 76

Young inequalityfor convolutions, 125for numbers, 42


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