Parabolic and Navier-Stokes Equations
Bedlewo, September 2-8, 2012
BOOK OF ABSTRACTS
Scientific Committee: Herbert Amann, Piotr Biler, Bogdan Bojarski, Eduard Feireisl,Antonin Novotny, Gregory Seregin, Vsevolod A. Solonnikov and Gerhard Strohmer
Organizing Committee: Reinhard Farwig, Piotr B. Mucha, Jiri Neustupa, JoannaRencławowicz, Yoshihiro Shibata and Wojciech Zajaczkowski
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CONTENT
HELMUT ABELS, On a phase-field model for two-phase flows of viscous incompress-ible fluids with degenerate mobilityHERBERT AMANN, Parabolic equations on singular manifoldsCHÉRIF AMROUCHE, New results for the Stokes and Navier-Stokes equations withnon standard boundary conditionsHYEONG-OHK BAE, Strong solutions for the interaction of particles and fluidHUGO BEIRÃO DA VEIGA, On some sharp results concerning vorticity and regularityof solutionsGALINA BIZHANOVA, On the solutions of the singularly perturbed problem for theparabolic equationJAN BURCZAK, Almost everywhere Hölder continuity of gradients of parabolic sym-metric p-LaplacianDONGHO CHAE, On the blow-up problem for the Euler equations and the Liouvilletype results in the fluid equationsMICHEL CHIPOT, Asymptotic behavior for nonlocal p-Laplace type problemsTOMASZ CIESLAK, An alternative proof of the Yudovich theoremFRANCESCA CRISPO, Regularity results for p-laplacian parabolic systemsRAPHAËL DANCHIN, The Oberbeck-Boussinesq approximation in critical spacesIRINA VLAD. DENISOVA, Global solvability of a problem governing the motion oftwo immiscible capillary fluidsREINHARD FARWIG, Concentration-diffusion phenomena for the Boussinesq systemEDUARD FEIREISL, Relative entropies and weak-strong uniquenessMETHIAS GEISSERT, Weak Neuman implies H∞-calculus for the Stokes operatorCARLO ROMANO GRISANTI, Time periodic Leray’s problem for a shear-thinning fluidPIOTR GWIAZDA, Renormalized solutions to the parabolic equation in Musielak-OrliczspacesTOSHIAKI HISHIDA, Decay estimates of the Oseen flow in the planeTSUKASA IWABUCHI, Global solutions for the Navier-Stokes equations in the rota-tional frameworkPIOTR KACPRZYK, Long time existence of solutions to the Navier-Stokes equationswith inflow-outflow and heat convectionPETR KAPLICKY, On generalized Kelvin-Voigt modelKRISTINA KAULAKYTE, On the Navier-Stokes equations with nonhomogeneous bound-ary conditions in a system of connected layersHYUNSEOK KIM, Very weak solutions of the stationary Stokes equations on exteriordomainsNERINGA KLOVIENE, Non-stationary Poiseuille type solution for the second gradefluidsTAKAYUKI KOBAYASHI, L2 boundedness for the solutions to the 2D Navies-StokesequationsHIDEO KOZONO, Uniqueness criterion of weak solutions to the Navier-Stokesequations in general unbounded domainsADAM KUBICA, Regularity criteria for axially symmetric weak solutions to the Navier-Stokes equationsTAKAYUKI KUBO, Weighted Lp − Lq estimates of Stokes semigroup in exterior do-mains
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PETR KUCERA, Perturbations of initial conditions of strong solutions of the Navier-Stokes equations in L3-norms.DANIEL LENGELER, Global existence for the interaction of a Navier-Stokes fluid witha linearly elastic shellYASUNORI MAEKAWA, On the inviscid limit for the viscous incompressible flows inthe half planeALEX MAHALOV, Stochastic 3D rotating Navier-Stokes equations: averaging, conver-gence and regularityMARCIN MAŁOGROSZ, A model of morphogen transport in the presence of glypicansPAOLO MAREMONTI, On the Stokes problem in exterior domains: a functional analy-sis approach to the maximum modulus theoremANCA-VOICHITA MATIOC, Analysis of a mathematical model describing water-mudinteractionBOGDAN-VASILE MATIOC, Global non-negative weak solutions for the thin film ap-proximation of the Muskat problemNORIKO MIZOGUCHI, Refined asymptotics of blowup solutions to a simplified chemo-taxis systemJOACHIM NAUMANN, On Prandtl’s model of turbulence: existence of weak solutionsto the equations of unsteady turbulent pipe-flowSARKA NECASOVÁ, Weak solutions to the barotropic Navier-Stokes system with slipboundary conditions in time dependent domainsJIRI NEUSTUPA, Regularity of a weak solution to the Navier-Stokes equations via onecomponent of a spectral projection of vorticity.TAKAAKI NISHIDA, Heat convection problems of compressible viscous fluidsBERNARD NOWAKOWSKI, Navier-Stokes equations in thin domainsTAKAHIRO OKABE, Initial profile for the slow decay of the Navier-Stokes flow in thehalf-spaceIRENA PAWŁOW, The global solvability of a sixth order Cahn-Hilliard type equationvia the Bäcklund transformationPATRICK PENEL, On the conditional regularity of Leray-Hopf weak solutions to theNavier-Stokes equations one half century after Prodi-SerrinMILAN POKORNÝ, Time-periodic solutions to the full Navier–Stokes–Fourier systemDALIBOR PRAŽÁK, Regularity and uniqueness of non-Newtonian binary fluid mix-turesREIMUND RAUTMANN, On vorticity transport & diffusion in bounded 3D-domainsWITOLD SADOWSKI, On the local existence in L3 for the three-dimensional Navier–Stokes equationsOKIHIRO SAWADA, Ill-posedness theory and norm-inflation argument of the 3-D Navier-Stokes equations in the critical spaceYOSHIHIRO SHIBATA, Some decay properties of compressible viscous fluid flow in 2dimensional exterior domains.SENJO SHIMIZU, On R-sectoriality of the Stokes equations with first order boundarycondition in a general domainANDREY SHISHKOV, Very singular and large solutions of semi-linear parabolic equa-tions with degenerate absorption potentialMIKOŁAJ SIERZEGA, Classical solutions for reaction-diffusion equations with singu-lar initial dataANA SILVESTRE, Steady solutions with finite kinetic energy for the Navier-Stokesequations in a three-dimensional exterior domainZDENEK SKALÁK, Asymptotic behavior of solutions to the Navier-Stokes equations
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EMMA SKOPIN, The boundary value problems for the scalar Oseen equationMINDAUGAS SKUJUS, On the correct asymptotic conditions at infinity for the time-periodic Stokes problem in a system of semi-infinite pipesVSEVOLOD A.SOLONNIKOV, On free boundary problems of magnetohydrodynamicsGERHARD STRÖHMER, Existence and stability of solutions close to equilibria for gasballs under the influence of gravityMASAHIRO SUZUKI, Stability of stationary solutions to the Euler-Poisson equationsarising in plasma physicsYOSHIAKI TERAMOTO, Free surface flow of viscous incompressible fluidKYOKO TOMOEDA, Optimal Korn’s inequality for solenoidal vector fields on a peri-odic slabYOSHIO TSUTSUMI, Gibbs measure for isothermal Falk modelYOSHIHIRO UEDA, Decay structure of the regularity-loss type and the asymptotic sta-bility for the Euler-Maxwell systemMORIMICHI UMEHARA, On a spherically symmetric motion of a self-gravitating vis-cous gasERIKA USHIKOSHI, Hadamard variational formula for the velocity and pressure of theStokes equations of the perturbation of domainsWERNER VARNHORN, On extensions of Serrin’s condition for the Navier-Stokes equa-tionsJÖRG WOLF, A regularity criterion of Serrin-type for the Navier-Stokes equations interms of one component of the velocityANETA WRÓBLEWSKA-KAMINSKA, The motion of rigid bodies in non-Newtonianfluids with nonstandard rheologyNORIKAZU YAMAGUCHI, A mathematical justification of the penalty method for theStokes and Navier-Stokes equationsWOJCIECH ZAJACZKOWSKI, Regularity of axially symmetric solutions to the Navier-Stokes equations.FLORIAN ZANGER, The fractional step theta method in fluid dynamicsEWELINA ZATORSKA, On a model of reactive flow with multicomponent diffusionNIKOLAI CHEMETOV, Solvability of a generalized Buckley-Leverett model
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ABSTRACTS:
On a phase-field model for two-phase flows of viscous incompressiblefluids with degenerate mobility
Helmut Abelsa
aUniversity of Regensburg, Faculty for Mathematics93040 Regensburg, Germany
We discuss a recent model for the two-phase flow of two immiscible, incom-pressible fluids in the case when the densities of the fluids are different. Such modelswere introduced to describe the flow when singularities in the interface, which sepa-rates the fluids, (droplet formation/coalescence) occur. The fluids are assumed to bemacroscopically immiscible, but a partial mixing in a small interfacial region is as-sumed. We will present recent results on existence of weak solutions for this model inthe case of degenerate and non-degenerate mobility. Here the case of degenerate mo-bility is of special interest since there is no diffusion in the pure phases and the effectof Ostwald ripening does not occur for the sharp interface limit. This is a joint-workwith Harald Garcke and Daniel Depner from Regensburg.
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Parabolic equations on singular manifolds
Herbert Amann
University of ZurichZurich, Switzerland
We shall describe a general theory for parabolic equations on non-smooth do-mains. In particular, we shall be interested in transmission-boundary value problemsin situations where the interface meets the boundary.
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New results for the Stokes and Navier-Stokes equations with nonstandard boundary conditions
Chérif Amrouchea and Nour El Houda Seloulab
aUniversity de Pau, [email protected]
bUniversité de Caen,
We consider here elliptical systems as Stokes and Navier-Stokes problems ina bounded domain, eventually multiply connected, whose boundary consists of multi-connected components. We investigate the solvability in Lp theory, with 1 < p < ∞,under the non standard boundary conditions
u · n = g, curlu× n = h or u× n = g, π = π∗ onΓ.
The main ingredients for this solvability are given by the Inf-Sup conditions, someSobolev’s inequalities for vector fields and the theory of vector potentials satisfying
ψ · n = 0, or ψ × n = 0 onΓ.
Those inequalities play a fundamental key and are obtained thanks to Calderon-Zygmundinequalities and integral representations. In the study of elliptical problems, we con-sider both generalized solutions and strong solutions that very weak solutions.
In a second part, we will consider the nonstationary case for the Stokes equa-tions. This last part is in collaboration with S. Necasova.
References[1] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector Potentials In Three-
dimensional Non-smooth Domains, Math. Meth. Applied Sc., Vol. 21, pp. 823–864, 1998.
[2] C. Amrouche, N.E.H. Seloula, Theory for Vector Potentials and Sobolev’s In-equalities for Vector Fields. Application to the Stokes Equations With Pressureboundary conditions, to appear in Math. Mod. Meth. Appl. Sciences (M3AS)
[3] C. Amrouche, N.E.H. Seloula, On the Stokes equations with the Navier-typeboundary conditions, Differential Equations and Applications, 3-4, 581- 607,(2011)
[4] C. Amrouche, N.E.H. Seloula, Navier-Stokes Equations with pressure boundaryconditions. Lp -theory for the Navier-Stokes equations with pressure boundaryconditions, to appear in Discrete Contin. Dyn. Syst, Ser S
[5] J. Bolik, W. Von. Wahl, Estimating ∇u in terms of div u, curlu either (ν, u) and(ν × u) and the topology, Math. Meth. Appl. Sci.,
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Strong solutions for the interaction of particles and fluid
Hyeong-Ohk Baea, Young-Pil Choib, Seung-Yeal Hab and Moon-Jin Kangb
a Department of Financial EngineeringAjou University, Suwon 443-749, Korea
b Department of Mathematical SciencesSeoul National University, Seoul 151-747, Korea
We presented a coupled kinetic-fluid model for the interactions between Cucker-Smale(C-S) flocking particles and incompressible fluid on the periodic spatial domainTd. Our coupled system consists of the kinetic Cucker-Smale equation and the in-compressible Navier-Stokes equations, and these two systems are coupled through theadvection term and drag force. For the proposed model, we provide a strong solutionsin the periodic domain and bounded domains.
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On some old results on non-linear elliptic equations
Hugo Beirão da Veiga
In their famous 1993 paper, Constantin and Fefferman consider the evolutionNavier-Stokes equations in the whole space R3 and prove, essentially, that if the di-rection of the vorticity is Lipschitz continuous in the space variables, during a giventime-interval, then the corresponding solution is regular. In this talk we consider theinitial-boundary value problem for the Navier-Stokes equations in a regular, bounded,domain under a slip boundary condition, and describe some recent results concerningregularity of solutions under a significant, non-uniform, Lipschitz-continuity assump-tion on the direction of the vorticity. The interest of our result highly follows from thefact that it is at the same level of strength as that reached by the classical "Prodi-Serrin"type conditions. In addition, our proof is elementary. We also state other related results.
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On the solutions of the singularly perturbed problem for the parabolicequation
G. Bizhanova
Institute of MathematicsAlmaty
There is considered the problem for a parabolic equation with a small parameterat a time derivative (principle term) in a boundary condition. It is a linearized one phaseStefan type free boundary problem.
The properties of the solution of the perturbed problem are studied.
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Almost everywhere Hölder continuity of gradients of parabolicsymmetric p-Laplacian
Jan Burczaka
aInstitute of Mathematics, Polish Academy of SciencesWarsaw
There is classical by now theory, which gives local Hölder continuity of gradi-ents to parabolic p-Laplacian, summed up by school around DiBenedetto. For physicalapplications, however, one should replace gradient with its symmetrical part. Sur-prisingly, this causes fundamental problems, as quasi-subharmonicity breaks down.Nevertheless, one can still use caloric approximation approach as in [2] to obtain a.e.regularity results, which is presented.
References[1] Burczak, P.. Almost everywhere Hölder continuity of gradients for parabolic symmetric
p-systems, preprint
[2] Duzaar, F., Mingione, G., Steffen K. Parabolic systems with polynomial growth and regu-larity., Mem. Am. Math. Soc. 214, 2011
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On the blow-up problem for the Euler equations and the Liouville typeresults in the fluid equations
Dongho Chae
Chung-Ang University, Korea
In this talk we discuss some observations connected with the blow-up problemin the 3D Euler equations. We first consider the scenarios of the self-similar blow-upand its generalizations. For the associated self-similar Euler equations we prove a Liou-ville type theorem by a simpler argument than the previous one, which shows that fastdecaying vorticity at spatial infinity implies the triviality of solution. For an extremecase of the self-similar Euler equations, which corresponds to the Euler equations withdamping, we show that any velocity decaying solution at spatial infinity(independentof the decay rate) is trivial. For the axisymmetric Euler equations we observe that thecomplex Riccati structure exists excluding the pressure term. In this case show thatsome uniformity condition for the pressure is not consistent with the global regularity.In the second part of talk we present Liouville type theorems for the steady Navier-Stokes equations for both of the incompressible and the compressible cases.
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Asymptotic behavior for nonlocal p-Laplace type problems
M. Chipot
Institute for MathematicsUniversity of Zurich
Winterthurerstrasse 190CH-8057 Zurich
We would like to report on the asymptotic behaviour of problems of the form
ut −∇ · a(∫Ω
u(t, x)dx)|∇u|p−2∇u = f in (0,+∞)× Ω
together with boundary and initial conditions.
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An alternative proof of the Yudovich theorem
T. Cieslak
Institute of Mathematics, Polish Academy of [email protected]
In my talk I will review a new proof of the famous Yudovich uniqueness theoremof vortex patches. The proof is of a completely different nature than the original proofof Yudovich. In particular the detailed dependence of a constant in Marcinkiewiczinterpolation inequality on the exponents of the Lp spaces is not required.
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Regularity results for p-laplacian parabolic systems1
F. Crispo
Seconda Università degli Studi di NapoliCaserta (Italy)
We consider a singular parabolic system of p-laplacian type, p ∈ (1, 2). Theanalysis of this kind of system is strictly related to the study of non-stationary non-Newtonian fluids such as shear-thinning fluids. We address different aspects of theregularity theory for solutions of such systems, in particular higher space-time regu-larity in Sobolev spaces and Hölder continuity of the gradient of the solution. We geta global integrability result for second order derivatives for p ∈ (p0, 2) with p0 =max 2N
N+2 ,32. It is interesting to note that the proof of such results does not rely on
the difference quotient method, so that for the regularity up to the boundary we avoidany kind of localization. Higher integrability results of second order derivatives, andthe consequent space-time global Hölder continuity of the gradient of weak solutions,need some further restrictions on the range of p.
1The results are part of a work in progress with P. Maremonti
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The Oberbeck-Boussinesq approximation in critical spaces
R. Danchina and L. Heb
aUniversité Paris-Est, LAMA, UMR 8050,61 avenue du Général de Gaulle, 94010 Créteil Cedex, France
bDepartment of Mathematical Sciences, Tsinghua UniversityBeijing 100084, P. R. China
This talk is devoted to the study of the so called Oberbeck-Boussinesq approx-imation for compressible viscous perfect gases in the whole three-dimensional space.Both the cases of fluids with positive heat conductivity and zero conductivity are con-sidered. For small perturbations of a constant equilibrium, we establish the global ex-istence of unique strong solutions in a critical regularity functional framework. Next,taking advantage of Strichartz estimates for the associated system of acoustic waves,and of uniform estimates with respect to the Mach number, we obtain all-time conver-gence to the Boussinesq system.
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Global solvability of a problem governing the motion of twoimmiscible capillary fluids
I. Denisova
Institute for Problems in Mechanical Engineering , Russian Academy of SciencesSt. Petersburg, Russia
We deal with the motion of two immiscible fluids in a container. The liquids areseparated by a close unknown interface on which surface tension is taken into account.We prove that this problem is uniquely solvable in an infinite time interval providedthat the initial velocity of the liquids is small and the initial configuration of the innerfluid is close to a ball. Moreover, we show that the velocity decays exponentially atinfinity with respect to time and that the interface between the fluids tends to a sphereof the certain radius. The proof is based on an exponential estimate of a generalizedenergy and on a local existence theorem of the problem.
References[1] Denisova I. V., Solonnikov V. A., Global solvability of a problem governing the motion
of two incompressible fluids in a container, Zapiski nauchn. semin. POMI 397/2011, pp.20–52 (in Russian).
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Concentration-diffusion phenomena for the Boussinesq system
Reinhard Farwiga, Raphael Schulza, and Masao Yamazakib
a Technische Universität Darmstadt ,b Waseda University, Tokyo
In the whole space Rn we study the asymptotic behaviour of solutions to theBoussinesq equations with a suitable gravity field. Therefore, we investigate the solv-ability of these equations in weighted L∞-spaces using semigroup techniques in ho-mogeneous Besov spaces and determine the asymptotic profile for sufficiently fast de-caying initial data.
For n = 2, 3 we are able to construct initial data such that the velocity exhibitsan interesting concentration-diffusion phenomenon. To be more precise, let 0 =: t0 <t1 < ... < tN < tN+1 := T , N ∈ N, be an arbitrary finite sequence and let the initialvelocity u0 satisfy some symmetry properties.
Then there exists an initial temperature θ0 and for each i = 1, . . . , N there areinstants t′i, t
∗i arbitrarily close to ti such that the corresponding unique strong solution
u, θ of the Boussinesq system with initial data (ηu0, ηθ0) and η > 0 sufficiently smallsatisfies, for all i = 1, ..., N and all |x| large enough, (more or less) the pointwiseestimate
|u(x, t∗i )| ≤ C|x|−n−2, but |u(x, t′i )| ≥ c|x|−n.
References[1] Farwig, R., Schulz, R., Yamazaki, M.: Concentration-diffusion phenomena of heat convec-
tion in an incompressible fluid. FB Mathematik, TU Darmstadt, Preprint no. 2649 (2012)
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Relative entropies and weak-strong uniqueness
E. Feireisla
aInstitute of Mathematics AVCRPraha
We discuss applications of the concept of relative entropy in the dynamics ofviscous, compressible and heat conducting fluids described by means of the Navier-Stokes-Fourier system. A new relative entropy is identified based on the thermody-namic potential termed ballistic free energy. Applications are given to the problem ofweak-strong uniqueness of solutions to the Navier-Stokes-Fourier system.
References[1] Feireisl, E., Novotný, A., Weak-strong uniqueness property for the full Navier-Stokes-
Fourier system., Arch. Rational Mech. Anal., to appear.
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Weak Neuman implies H∞-calculus for the Stokes operator
M. Geisserta and P. C. Kunstmannb
aUniversity of HannoverHannover
bKarlsruhe Institute of TechnologyKarlrsruhe
We show that the Stokes operator admits an H∞-calculus on Lqσ(Ω) provided
the Helmholtz decomposition exists in Lq(Ω) and the boundary of Ω ⊂ Rn is smoothenough. The proof is based on properties of the Dirichlet-Laplacian and an abstractresult by Kalton, Kunstmann and Weis (see [1]). We also discuss some related results.
References[1] N.J. Kalton, P.C. Kunstmann, L. Weis, Perturbation and interpolation theorems for the
H∞-calculus with applications to differential operators, Math. Ann. 336, 747–801 (2006).
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Time periodic Leray’s problem for a shear-thinning fluid
C.R. Grisantia & G.P. Galdib
aUniversity of PisaPisa
bUniversity of PittsburghPittsburgh
We study the Leray’s Problem for a shear-thinning fluid in the framework oftime periodic flows. We consider a system of pipes connected to a reservoir. We usethe unbounded approach, hence the pipes are supposed to be of infinite length. The no-slip boundary condition is completed with the request that the velocity field approachesat infinity a Womersley-type fully developed flow in every pipe. The system object ofour study is the following:
∂V
∂t+ V · ∇V = µ0∆V + S(DV )−∇p in Ω× R
∇ · V = 0 in Ω× R∫Σ
V · ndσ = α(t) T - periodic
V |∂Ω = 0lim
|x|→∞, x∈Ωj
V (x, t)−Wj(x, t) = 0 t ∈ R
V and p are T - periodic in t
where Ωj is the j-th pipe, Wj is the corresponding fully developed flow, Σ is thegeneric cross-section of the system and α(t) is the prescribed flux. The system abovedescribes an inverse problem, since the only datum is the assigned flux α(t) and wehave to find both the velocity field and the pressure gradient.
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Renormalized solutions to the parabolic equation in Musielak-Orliczspaces
Piotr Gwiazdaa
aInstytut Matematyki Stosowanej i MechanikiUniwersytet Warszawski
We study a general class of nonlinear parabolic problems associated with thedifferential inclusion ∂tβ(x, u) − div (a(x,∇u) + F (u)) ∋ f , where f ∈ L1(Ω).The vector field a(·, ·) is monotone in the second variable and satisfies a non-standardgrowth condition described by an x-dependent convex function that generalizes bothLp(x) and classical Orlicz settings. Using truncation techniques and a generalizedMinty method in the functional setting of non reflexive spaces we prove existence ofrenormalized solutions for general L1-data. Under an additional strict monotonicity as-sumption uniqueness of the renormalized solution is established. Sufficient conditionsare specified which guarantee that the renormalized solution is already a weak solutionto the problem.
References[1] Gwiazda, P. , Wittbold, P., Wróblewska, A., Zimmermann, A., Renormalized solutions of
nonlinear elliptic problems in generalized Orlicz spaces, J. Differ. Equations 253, (2) pp.635–666 (2012)
[2] Gwiazda, P. , Wittbold, P., Wróblewska, A., Zimmermann, A., Renormalized solutions ofnonlinear parabolic problems in generalized Orlicz spaces, in preperation
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Decay estimates of the Oseen flow in the plane
Toshiaki Hishida
Nagoya UniversityNagoya, Japan
We consider the initial value problem for the Oseen system in 2D exterior do-mains and study the local energy decay of solutions. For 3D case the theory was welldeveloped by Kobayashi and Shibata, while 2D case has remained open. The result isapplied to deduction of Lp-Lq estimates of the Oseen semigroup. The dependence ofestimates on the Oseen parameter is also discussed.
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Global solutions for the Navier-Stokes equationsin the rotational framework
Tsukasa Iwabuchi
Department of Mathematics, Faculty of Science and Engineering, Chuo UniversityTokyo, JAPAN
In this talk, we consider the initial value problem for the Navier-Stokes equa-tions with the Coriolis force
∂u
∂t−∆u+Ωe3 × u+ (u · ∇)u+∇p = 0 in R3 × (0,∞),
div u = 0 in R3 × (0,∞),
u(x, 0) = u0(x) in R3,
(NSC)
where u = u(x, t) = (u1(x, t), u2(x, t), u3(x, t)) and p = p(x, t) denote the unknownvelocity field and the unknown pressure of the fluid at the point (x, t) ∈ R3 × (0,∞),respectively. The purpose of this talk is to show the existence and the uniqueness ofthe global solutions to (NSC) in the homogeneous Sobolev spaces Hs(R3) (s ≥ 1/2).In particular, we consider global solutions for large initial data u0 if the speed of therotation is sufficiently fast.
References[1] H. Brezis, Remarks on the preceding paper by M. Ben-Artzi, “Global solutions of two-
dimensional Navier-Stokes and Euler equations”, Arch. Rational Mech. Anal., 128 (1994),no. 4, 359–360.
[2] Y. Giga, K. Inui, A. Mahalov, J. Saal, Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data, Indiana Univ. Math. J., 57 (2008), 2775–2791.
[3] M. Hieber, Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in therotational framework, Math. Z., 265 (2010), 481–491.
[4] T. Kato, Strong Lp-solutions of the Navier-Stokes equation in Rm, with applications toweak solutions, Math. Z., 187 (1984), 471–480.
[5] H. Kozono, On well-posedness of the Navier-Stokes equations, in: J. Neustupa, P. Penel(eds.) Mathematical Fluid Mechanics, Recent Results and Open Questions, Birkhaüser,Basel, (2001) 207–236.
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On generalized Kelvin-Voigt model
Petr Kaplicky
In the talk the existence and uniqueness of solutions to generalized Kelvin-Voigtmodel describing the motion of a compressible viscoelastic body will be discussed. Inparticular we establish the existence of a unique classical solution to such a model inthe spatially periodic 2D setting.
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L2–asymptotic stability of mild solutionsto Navier-Stokes systemx
Grzegorz Karch
University of Wrocł[email protected]
Abstract
We consider the following initial value problem for the Navier–Stokes systemfor an incompressible fluid in the whole three dimensional space
ut −∆u+∇ · (u⊗ u) +∇p = F, (x, t) ∈ R3 × (0,∞)
div u = 0,
u(x, 0) = u0(x).
It is well-known that this problem has a unique global-in-time mild solution for a suf-ficiently small initial condition u0 and for a small external force F in suitable scalinginvariant spaces. We show that these global-in-time mild solutions are asymptoticallystable under every (arbitrary large) L2-perturbation of their initial conditions.
This is a joint work with Dominika Pilarczyk and Maria Elena Schonbek.
References[1] Karch, G.; Pilarczyk, D., Asymptotic stability of Landau solutions to Navier-Stokes system.
Arch. Ration. Mech. Anal. 202 (2011), no. 1, 115âAS131.
[1] Karch, G.; Pilarczyk, D., & Schonbek M.E. L2–asymptotic stability of mild solutions toNavier-Stokes system. Work in preparation.
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On the Navier-Stokes equations with nonhomogeneous boundaryconditions in a system of connected layers
K. Kaulakytea and K. Pileckasb
aVilnius UniversityVilnius
bVilnius UniversityVilnius
The stationary Navier–Stokes equations with nonhomogeneous boundary con-ditions are studied in a system of connected layers. The boundary of the domain ismultiply connected and consists of finite number of infinite connected components,which form the outer boundary, and finite number of compact connected components,forming the inner boundary. The boundary value is assumed to have a compact supportand it is supposed that the fluxes of the boundary value over the components of theinner boundary are sufficiently small. We do not pose any restrictions on fluxes of theboundary value over the infinite components of outer boundary. The existence of atleast one weak solution to the mentioned Navier–Stokes problem is proved. The solu-tion may have finite or infinite Dirichlet integral depending on geometrical propertiesof the domain.
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Very weak solutions of the stationary Stokes equations on exteriordomains
D. Kima, H. Kimb and S. Parkc
a Sogang UniversitySeoul
b Sogang UniversitySeoul
cTohoku UniversitySendai
We study the nonhomogeneous Dirichlet problem for the stationary Stokes equa-tions on exterior smooth domains Ω in Rn, n ≥ 3. Our main result is the existence anduniqueness of very weak solutions in the Lorentz space Lp,q(Ω)n, where (p, q) satisfieseither (p, q) = (n/(n− 2),∞) or n/(n− 2) < p < ∞, 1 ≤ q ≤ ∞. This is deducedby a duality argument from our new solvability results on strong solutions in homoge-neous Sobolev-Lorentz spaces. Homogeneous Sobolev-Lorentz spaces are also studiedin some details: particularly, we establish basic interpolation and density results.
References[1] Kim, D., Kim, H., Park, S., Very weak solutions of the stationary Stokes equations on
exterior domains., Preprint(June 2012), 45 pages.
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Non-stationary Poiseuille type solution for the second grade fluids
Neringa Kloviene and Konstantin PileckasVilnius University
Vilnius UniversityVilnius
We consider the incompressible non–Newtonian second grade fluid flow in athree–dimensional pipe Π = x = (x′, x3) ∈ R3 : x′ ∈ σ, x3 ∈ R:
∂t(u − α∆u)− ν∆u + curl(u − α∆u)× u +∇p = f,div u = 0,u|ST = 0, u(x, 0) = u0(x),∫σu3(x
′, x3, t)dx′ = F (t),
(1)
here ST = ∂Π × (0, T ), u and p are the velocity and the pressure of the fluid, f is theexternal body force, u0 is the initial velocity, α is the normal stress module, ν is thekinematic viscosity (α and ν are positive constants) and condition (14) prescribes theflux F (t) over an arbitrary bounded two–dimensional cross–section σ of class C4.
In the two-dimensional channel and in three-dimensional pipe with rotationalsymmetry problem (1) has the unidirectional Poiseuille type solution. However, in ageneral three–dimensional pipe secondary flows appear and the velocity field has allthree components.
We look for the solution (u(x, t), p(x, t)) of system (1) in the formu(x, t) = (u1(x
′, t), u2(x′, t), u3(x
′, t)),p(x, t) = p(x′, t)− q(t)x3 + p0(t),
where p0(t) is an arbitrary function and q(t) is unknown and has to be found.We prove the existence and uniqueness for the solution. The problem is uniquely
solvable for small data by using special basis.
23
L2 boundedness for the solutions to the 2D Navies-Stokes equations
T. Kobayashi
Saga UniversitySaga City, Japan
We consider the initial boundary value problems for the semilinear heat equa-tions and semilinear dissipative wave equations in two dimensional exterior domains.Also, we consider the Cauchy problems for the linear wave equations with strongdamping terms and Navier-Stokes equations in R2. We will give the L2 bounded-ness of the solutions for the initial data in Hardy space. The results in this talk wereobtained in a joint work with M. Misawa (Kumamoto University, Japan).
References[1] M. Misawa, S. Okamura and T. Kobayashi., Decay property for the linear wave equations
in two dimensional exterior domains., Differential and Integral Equations Vol. 24, No. 9-10(2011), 941-964
24
Uniqueness criterion of weak solutions to the Navier-Stokesequations in general unbounded domains
Hideo Kozonoa, Takahiro Okabeb and Hideori Takahashic
aWaseda UniversityTokyo, Japan
[email protected] University
Hirosaki, [email protected]
cTohoku UniversitySendai, Japan
In any domain Ω ⊂ Rn, possibly unbounded with non-compact boundary ∂Ω,Masuda [1] had proved that if u is a weak solution of the Navier-Stokes equations inL∞(0, T ;Ln(Ω)) and if u(·, t) is continuous from the right in the Ln(Ω)-norm forevery t ∈ [0, T ), then u is the only weak solution on Ω × [0, T ). Later on, Kozono-Sohr [2] succeeded to remove such a restriction on u as right continuity in Ln(Ω)and showed that the class L∞(0, T ;Ln(Ω)) guarantees uniqueness of weak solutionsprovided Ω is the whole space R, the half-space Rn
+, and interior-exterior domains withthe compact boundary. In this article, we consider general unbounded domains Ω withuniformly C2-boundary ∂Ω and prove the uniqueness of weak solutions in the classL∞(0, T ;Ln(Ω)). The new function space Lr(Ω) = Lr(Ω) + L2(Ω) for 1 < r ≤ 2,Lr(Ω) = Lr(Ω) ∩ L2(Ω) for 2 ≤ r < ∞ plays an important role for our proof. Ourresult reads as follows:
Theorem. Let Ω be a uniformly C2 domain in Rn. Suppose that u and v are two weaksolutions of the the Navier-Stokes equations in Ω× (0, T ) in the Leray-Hopf class withsame initial data a ∈ L2
σ(Ω). Assume that
u ∈ L∞(0, T ;Ln(Ω))
and that
∥v(t)∥2L2(Ω) + 2
∫ t
0
∥∇u(τ)∥2L2(Ω)dτ ≤ ∥a∥2L2(Ω), 0 < t < T.
Then it holds that u ≡ v in Ω× [0, T ).
References[1] Masuda, K., Weak solutions of the Navier-Stokes equations. Tohoku Math. J. 36, 623–646
(1984).
[2] Kozono, H., Sohr, H., Remark on uniqueness of weak solutions to the Navier-Stokes equa-tions. Analysis 16, 255–271 (1996).
25
Regularity criteria for axially symmetric weak solutions to theNavier-Stokes equations
A. Kubica
Warsaw University of TechnologyWarsaw
We will consider an axially symmetric solutions of the Navier-Stokes equations.In papers [1] and [2] the authors proved the regularity of solution under the assumptionthat the radial or angular component of velocity satisfy Serrin-type condition. I willpresent analogous results ([3]), which hold under weighted version of Serrin-type con-dition, where the weight is a power of the distance from the axis of symmetry of thesolution. This is a joint work with M. Pokorny and W. Zajaczkowski.
References[1] J. Neustupa, M. Pokorny, J. Math. Fluid Mech.2 (2000), no. 4, pp. 381-399.
[2] O. Kreml, M. Pokorny, Elec. J. Diff. Eq., Vol 2007, No. 08, pp. 1-10.
[3] A. Kubica, M. Pokorny, W. Zajaczkowski, „Remarks on regularity criteria for axially sym-metric weak solutions to the Navier-Stokes equations", Math. M. Appl. Sc. Vol. 35, Issue3, 2012, pp. 360-371.
26
Weighted Lp− Lq estimates of Stokes semigroup in exterior domains
Takayuki Kobayashia and Takayuki Kubob
aSaga UniversitySaga, Japan
bUniversity of TsukubaIbaraki, Japan
We consider the Navier-Stokes equations in exterior domains and in the weightedLp space. For this purpose, we consider the Lp − Lq estimates of Stokes semigroupwith weight ⟨x⟩s type. Our proof is based on the cut-off technique with local energydecay estimate proved by Dan, Kobayashi and Shibata [1] and the weighted Lp − Lq
estimates of Stokes semigroup in the whole space proved by Kobayashi and Kubo [2].Finally, as the application of the weighted Lp − Lq estimates to the Navier-Stokesequations, we obtain the weighted asymptotic behavior of global solution as t → ∞.
References[1] W. Dan, T. Kobayashi and Y. Shibata, On the local energy decay approach to some fluid
flow in exterior domain, Recent Topics on Mathematical Theory of Viscous IncompressibleFluid, 1–51, Lecture Notes Numer. Appl. Math. 16, Kinokuniya, Tokyo, 1998.
[2] T. Kobayashi and T. Kubo, Weighted estimates for Navier-Stokes folw in some unboundeddomains, (preprint).
*****
Perturbations of initial conditions of strong solutions of theNavier-Stokes equations in L3-norms.
P. Kucera
Czech Technical UniversityPrague
We solve a system of the Navier-Stokes equations with slip boundary conditionson a bounded domain. We deal with perturbations of initial conditions of strong solu-tions of our system. We prove that if these perturbations are sufficiently small in L3 -norm then corresponding solutions are strong too.
27
Global existence for the interaction of a Navier-Stokes fluid with alinearly elastic shell
Daniel Lengeler
University of Regensburg
I will present a result from my PhD thesis, namely the existence of global-in-time weak solutions for a Navier-Stokes fluid interacting with a linearly elastic shell ofKoiter type. This generalizes [1, 2]. A key step of the proof consists in the introductionof a new method for showing the compactness of bounded sequences of approximateweak solutions. This method might be of general interest in the study of fluid dynami-cal problems involving a free boundary. There is no damping term involved in the shellequations.
References[1] Grandmont, Céline, Existence of weak solutions for the unsteady interaction of
a viscous fluid with an elastic plate., SIAM J. Math. Anal. 40 (2008), no. 2,716âAS737.
[2] Chambolle, Antonin; Desjardins, BenoÃot; Esteban, Maria J.; Grandmont, Cé-line, Existence of weak solutions for the unsteady interaction of a viscous fluidwith an elastic plate., J. Math. Fluid Mech. 7 (2005), no. 3, 368âAS404.
28
On the inviscid limit for the viscous incompressible flows in the halfplane
Y. Maekawaa
aKobe University1-1 Rokkodai Nada-ku, Kobe, [email protected]
We consider the Navier-Stokes equations for viscous incompressible flows inthe half plane under the no-slip boundary condition. By using the vorticity formulationwe prove the (local in time) convergence of the Navier-Stokes flows to the Euler flowsoutside a boundary layer and to the Prandtl flows in the boundary layer at the inviscidlimit when the initial vorticity is located away from the boundary.
References[1] Maekawa, Y.; On the inviscid limit problem of the vorticity equations for vis-
cous incompressible flows in the half plane, Hokkaido university preprint series# 1005: available at http://eprints3.math.sci.hokudai.ac.jp/2195/
*****
Stochastic 3D rotating Navier-Stokes equations: averaging,convergence and regularity
Alex Mahalov
Arizona State [email protected]
Stochastic 3D rotating Navier-Stokes equations are considered. Averaging the-orems for the stochastic problems are proven in the case of strong rotation. Regularityresults are established by bootstrapping from global regularity of the limit stochasticequations and convergence theorems (joint work with Franco Flandoli).
Reference: Archive for Rational Mechanics and Analysis, doi: 10.1007/s00205-012-0507-6, vol. 205, Issue 1, p. 195-237, 2012.
29
A model of morphogen transport in the presence of glypicans
Marcin Małogrosza
aUniversity of WarsawWarsaw
We analyze a one dimensional version of a model of morphogen transport, a biologicalprocess governing cell differentiation. The model was proposed by Hufnagel et al [1].to describe the forming of morphogen gradient in the wing imaginal disc of the fruitfly. In mathematical terms the model is a system of reaction-diffusion equations whichconsists of two parabolic PDE’s and three ODE’s. The source of ligands is modelledby a Dirac Delta. Using semigroup approach and L1 techniques we prove that thesystem is well-posed and possesses a unique steady state. All results are proved withoutimposing any artificial restrictions on the range of parameters.
References[1] L. Hufnagel, J. Kreuger, S. M. Cohen, B. I. Shraiman, On the role of glypicans in the
process of morphogen gradient formation, Developmental Biology 300, (2006) pp 512-522.
[2] MM A model of morphogen transport in the presence of glypicans I, submitted.
30
On the Stokes problem in exterior domains: a functional analysisapproach to the maximum modulus theorem
P. Maremonti
Seconda Università degli Studi di NapoliCaserta (Italy)
In this talk we show a result concerning the maximum modulus theorem forsolutions to the Stokes initial boundary value problem in exterior domains. It is wellknown that a result of this kind has been given by Solonnikov by means of the potentialtheory. Subsequently, as announced at Luminy conference (2011), Y. Giga, in a paperjointly with K. Abe, has given a new result concerning the maximum modulus theorem.More precisely, if Ω is bounded they prove that the Stokes operator forms an analyticsemigroup continuous in C|0(Ω) and not continuous in the subset of solenoidal fields (inweak form) in L∞(Ω). Moreover, locally in time, they prove that the Stokes operatoris continuous in C|0(Ω), Ω exterior domains. However, the distinctive feature of theirpaper is the fact that they use a functional analysis approach.
In this talk we show that a functional analysis approach for the maximum mod-ulus theorem is also possible in the case of the exterior domain. Starting from the resultby Abe & Giga of a continuous analytic semigroup in C|0(Ω), Ω bounded, we obtainthe result for the initial boundary value problem in exterior domains with data in thesubset of L∞(Ω) characterized by the divergence free property.
31
Analysis of a mathematical model describing water-mud interaction
A.-V. Matioc
University of ViennaFaculty of Mathematics
Nordbergstraße 15, 1090 Vienna, [email protected]
We consider a mathematical model describing the two-phase interaction be-tween water and mud in a water canal when the width of the canal is small comparedto its depth. The mud is treated as a non-Newtonian fluid and the interface between themud and fluid is allowed to move under the influence of gravity and surface tension.
We reduce the mathematical formulation, for small boundary and initial data,to a fully nonlocal and nonlinear problem and prove its local well-posedness by us-ing abstract parabolic theory [1]. This is a joint-work with Joachim Escher (LeibnizUniversity Hanover).
References[1] J. ESCHER & A.-V. MATIOC: On the well-posedness of a mathematical model
describing water-mud interaction, submitted.
*****
32
Global non-negative weak solutions for the thin film approximation ofthe Muskat problem
Bogdan-Vasile Matioc
University of ViennaVienna
In this talk we present some recent results on the global existence of non-negative weak solutions for the lubrication approximation of the two-phase Muskatproblem. For thin fluid layers the Muskat problem is approximated by a second orderstrongly coupled degenerate system of parabolic type. When the constitutive equationsof this system are posed on a bounded interval, we construct regularized problemswhich possess global strong solutions. The non-negative weak solutions of the thinfilm approximation are then found as limit points of these global strong solutions. Inthe case when the system is defined on the real line we interpret it as a gradient flowfor the 2-Wasserstein distance in the space of probability measures with finite secondmoment. The weak solutions are constructed in this case by using a variational scheme.The talk is based on joint works with Joachim Escher, Philippe Laurençot, and Anca-Voichita Matioc.
*****
Refined asymptotics of blowup solutions to a simplified chemotaxissystem
Noriko Mizoguchi
Department of Mathematics, Tokyo Gakugei University,Koganei, Tokyo 184-8501, Japan
(e-mail: [email protected]),and
Precursory Research for Embryonic Science and Technology (PRESTO),Japan Science and Technology Agency (JST),
4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan
This is a joint work with T. Senba. We treat the Cauchy problem of a parabolic-elliptic system in two dimensional domains, which is a model for several biologicalproblems and physical problems. In this paper, we consider the blowup of radial so-lutions to the system. If the total mass is larger than the critical mass 8π, any radialsolution blows up in finite time and has a delta function singularity at the blowup time.However, the blowup rate and the asymptotic profile of blowup solutions have not beeninvestigated. Exceptionally, Herrero and Velázquez constructed a radial blowup solu-tion with explicit blowup rate and blowup profile of the solution in 1996 (Math. Ann.306, 583–623). We show that all radial blowup solutions behave as same as the solutionobtained by Herrero and Velázquez.
33
On Prandtl’s model of turbulence: existence of weak solutionsto the equations of unsteady turbulent pipe-flow
J. Naumann
Department of MathematicsHumboldt University of [email protected]
Let Ω ⊂ R2 be a bounded domain, let 0 < T < +∞; define Q = Ω×]0, T [.We consider the following system of PDEs:
∂u
∂t− div((ν + ℓ
√k)∇u) = f in Q, (1)
∂k
∂t− div((µ+ ℓ
√k)∇k) = ℓ
√k|∇u|2 − k
√k
ℓin Q (2)
(ν, µ = const ≥ 0, f given, ℓ = ℓ(x) mixing length, ℓ(x) > 0 for all x ∈ Ω, ℓ(x) → 0as x → ∂Ω). As a model for the fully developed turbulence, Prandtl postulated thissystem with ν = µ = 0 for general flow profiles. System (1), (2) characterizes theunidirectional flow through a pipe with cross section Ω.
In this talk we consider the special case ℓ ≡ 1. We complete (1), (2) by bound-ary conditions on ∂Ω×(0, T ) and initial conditions in Ω×0. We prove the existenceof a weak solution (u, k) to this boundary-initial value problem such that
k ≥ 0 a. e. in Q, ∃Q∗ ⊂ Q : mes(Q∗) > 0, k > 0 a. e. in Q∗.
34
Weak solutions to the barotropic Navier-Stokes system with slipboundary conditions in time dependent domains
E. Feireisla O. Kremlb, Š. Necasovác J. Neustupa d, J. Stebel e
aMathematical Institute, Academy of Sciences of the Czech RepublicPrague, Czech Republic
bMathematical Institute, Academy of Sciences of the Czech RepublicPrague, Czech [email protected]
cMathematical Institute, Academy of Sciences of the Czech RepublicPrague, Czech [email protected]
dMathematical Institute, Academy of Sciences of the Czech RepublicPrague, Czech Republic
eMathematical Institute, Academy of Sciences of the Czech RepublicPrague, Czech Republic
We consider the compressible (barotropic) Navier-Stokes system on time-dependentdomains, supplemented with slip boundary conditions. Our approach is based on penal-ization of the boundary behaviour, viscosity, and the pressure in the weak formulation.Global-in-time weak solutions are obtained see [1].
References[1] E. Feireisl, O. Kreml, Š. Necasová, J. Neustupa and J. Stebel. Weak solutions to the
barotropic Navier-Stokes system with slip boundary conditions in time dependent domains,Submitted
*****
Regularity of a weak solution to the Navier-Stokes equations via onecomponent of a spectral projection of vorticity.
Jiri Neustupa
The lecture is motivated by the regularity criteria, showing that some particu-lar quantities (one or two components of velocity, two components of vorticity, somechosen components of the gradient of velocity, etc) control the regularity of a weaksolution to the Navier-Stokes equations. We show that a certain spectral projection ofvorticity, or even its one component, also belong to such quantities.
35
Heat convection problems of compressible viscous fluids
T.Nishidaa , M.Padulab and Y.Teramotoc
aKyoto UniversityKyoto City
bUniversity of FerraraFerrara City
bSetsunan UniversityNeyagawa City
Steady solutions are considered for heat convection problems of compressibleviscous fluids. They include steady solutions for the Oberbeck-Boussinesq system as alimit.
References[1] Spiegel, E. A., Convective instability in a compressible atomosphere. I, Astro-
phys. J., Vol.141, 1965
[2] Nishida, T., Padula, M., and Teramoto, Y., Heat convection problems of com-pressible viscous fluids. I, Preprint, pp. 1–10.
[3] Nishida, T., Padula, M., and Teramoto, Y., Heat convection problems of com-pressible viscous fluids. II, In preparation.
36
Navier-Stokes equations in thin domains
B. Nowakowski
MIM UWWarszawa
We investigate the existence of regular solutions to the Navier-Stokes equationsin bounded thin domains in R3 under different boundary conditions. First results ofthis kind were obtained by G. Raugel and R. Sell in the middle of 90’ (see [2], [3]).They were based on the mean value operator which was extensively used in furtherstudies on the subject (for comprehensive summary see [1]).
In our talk we present a substantially different approach, which uses energyestimates and the Poincaré inequality. The price we pay for much shorter proof isslightly stronger assumption on the class of the initial data.
References[1] Kukavica I., Mohammed Z., On the regularity of the Navier-Stokes equation in a thin
periodic domain, J. Differential Equations 234 (2007), 485–506.
[2] Raugel R., Sell G., Navier-Stokes equations on thin 3D domains. I. Global attractors andglobal regularity of solutions, J. Amer. Math. Soc. 6 (1993), 503–568.
[3] Raugel R., Sell G., Navier-Stokes equations on thin 3D domains. II. Global regularityof spatially periodic solutions, Proc. Collège de France Sem., Vol XI, Paris, 1989–1991,Pitman Res. Notes Math. Ser. Vol. 299, Longman Sci. Tech., Harlow, 1994, 205–247.
37
Initial profile for the slow decay of the Navier-Stokes flow in thehalf-space
T. Okabea
aHirosaki UniversityHirosaki, Aomori Japan
We consider the asymptotic behavior of the incompressible viscous fluid in thehalf-space governed by the usual Navier-Stokes equations. Especially, we considerthe energy-decay problem of the weak solution to the Navier-Stokes equations. Forthis direction, there are many results on the upper bound for the energy-decay. Forthe whole space case, by the Fourier splitting method for the linear Stokes flow, wealso have the lower bound of the energy decay. However, for the half-space case, itseems that there are few result on the lower bound of the decay problem. Our previousresult [1] derived the lower bound of the slow decay of the linear Stokes flow and[1] characterized the initial profile which causes the lower bound. However, since weconsider specific initial data, tangential flow, we do not include two dimensional case.Our aim is to consider more general initial data and to derive the lower bound for thetwo-dimensional flow.
References[1] T. Okabe, Lower bound of L2 decay of the Navier-Stokes flow in the half-space Rn
+ andits asymptotic behavior in the frequency space, submitted.
38
The global solvability of a sixth order Cahn-Hilliard type equation viathe Bäcklund transformation
I. Pawłowa and W. M. Zajaczkowskib
aSystems Research Institute, Polish Academy of Sciences,and Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University
of Technology, Warsaw,[email protected]
bInstitute of Mathematics, Polish Academy of Sciences, and Institute of Mathematicsand Cryptology, Cybernetics Faculty, Military University of Technology Warsaw,
We reconsider the sixth order Cahn-Hilliard type equation with a nonlinear dif-fusion, addressed previously in [1]. Such PDE arises as a model of oil-water-surfactantmixtures. Applying the approach based on the Bäcklund transformation and the Leray-Schauder fixed point theorem we generalize the existence result of [1] by imposingessentially weaker assumptions on the data.
References[1] Pawłow, I., Zajaczkowski, W. M., A sixth order Cahn-Hilliard type equation arising in
oil-water-surfactant mixtures., Comm. Pure Appl. Anal. 10/2011, pp. 1823–1847.
39
On the conditional regularity of Leray-Hopf weak solutions to theNavier-Stokes equations one half century after Prodi-Serrin
Patrick Penel
We consider the incompressible 3D Navier-Stokes equations in the whole space.Even if the data are more regular than necessarily to justify the existence of weaksolutions, the problem of their regularity remains unsolved. Among all weak solutionsu of the Leray-Hopf class, u := (u1, u2, u3) is known as globally in time regular,strong and unique provided a posteriori regularity information [*] in the form of α-scaled criteria for all three components
uj ; j = 1, 2, 3;uj ∈ Lr(0, T ;Ls(R3));
2
r+
3
s= α;
s in certain interval Iα(s), for all T > 0. The so-called Prodi-Serrin conditions (1959-63) correspond to the 1-scaled criterion with I1(s) = (3,∞], later extended to [3,∞].A ’natural’ question (initiated by Neustupa-P. in 1999) is to get weaker a posterioricriteria, playing with the key parameter α and the interval Iα(s), for a minimum quan-tity of information related to u, i.e. involving only some components of u, or just one,or some components of the gradient ∇u, or some components of the correspondingvorticity, or looking for the role of the pressure. From 1999 ..., many positive answershave been obtained by many authors, leading to a large collection of α-scaled criteriaand to a better understanding of this fact that the only possible epochs of singularityof solutions concern simultaneously all three components. A survey is of interest. Wewill add our last regularity criteria (see [1]) in dependence on integrability propertiesof spectral projections of vorticity. The question, whether the regularity of u can becontrolled by only one vorticity component, is open. The question, whether the regu-larity of u can be controlled by a 1-scaled criterion for only one component, is openbut a reasonable perspective (the best known result is closed : a α-scaled criterion foru3 with α = 3/4 + 1/2s and Iα(s) = (10/3,∞], see [2]).
[*] We choose to discuss Prodi-Serrin-type conditions, but to infer regularity ofu one can use another a posteriori regularity information in terms of the direction ofvorticity [following P. Constantin and C. Fefferman].
References[1] J. Neustupa, P. Penel., Regularity criteria for weak solutions to the Navier-Stokes equa-
tions based on spectral projections of vorticity, Note to CRAS Paris Š2012 to appear.
[2] Y. Zhou, M. Pokorny: On the regularity of the solutions to the Navier-Stokes equations viaone velocity component. Nonlinearity, 23-5 (2010), p.1097-1107.
40
Time-periodic solutions to the full Navier–Stokes–Fourier system
Milan Pokorný
Praha, Czech Republicjoint work with: E. Feireisl, P.B. Mucha and A. Novotný
We consider the system of partial differential equations describing the flow ofa Newtonian compressible heat conducting fluid in a bounded three-dimensional do-main. We consider homogeneous Dirichlet condition for the velocity and Newton-type boundary condition for the temperature. Under certain assumptions on the model(which still include e.g. the monoatomic gas) we show that for arbitrarily large time-periodic force and arbitrarily large data there exists a time periodic weak solution toour problem. The presentation is based on the recent paper: Feireisl, E., Mucha, P.B.,Novotný, A., Pokorný, M.: Time-periodic solutions to the full Navier-Stokes-Fouriersystem, Arch. Ration. Mech. Anal. 204 (2012), No. 3, 745–786.
41
Regularity and uniqueness of non-Newtonian binary fluid mixtures
M. Grassellia and D. Pražákb
aPolitecnico di MilanoMilano
bCharles UniversityPrague
We consider a binary fluid mixture, described by a Ladyzhenskaya type modelfor the velocity, coupled with a Cahn-Hilliard equation for the order parameter:
∂tu+ (u · ∇)u−∇ · T (ϕ,D(u))) +∇π = kµ∇ϕ+ f(t), ∇ · u = 0
∂tϕ+ (u · ∇)ϕ−∆µ = 0, µ = −ε∆ϕ+ αF ′(ϕ)
with x ∈ Ω a bounded three-dimensional domain, t ∈ (0, T ). The system is comple-mented with some reasonable boundary condition, e.g. u = 0 and ϕ = ∆ϕ = 0 on ∂Ω.The point is that the analysis will focus exclusively on time regularity of solutions, sothat the boundary condition is not relevant.
It is well-known that for p ≥ 11/5 (the parameter of the growth of T w.r. toD(u)) weak solutions to Ladyzhenskaya model exist globally, and satisfy the energyequality. Recently, it has been shown in [1] that for p > 11/5, any weak solution isregular enough to ensure uniqueness; moreover, there exists a finite-dimensional expo-nential attractor. The argument uses iterative scheme of improving the time regularityof solutions in fractional Nikolskii spaces.
Adapting and partly simplifying the presentation of [1], we extend the result toa model of binary fluid mixture described above.
References[1] Bulícek, M., Ettwein, F., Kaplický, P., Pražák, D. On uniqueness and time regularity of
flows of power-law like non-Newtonian fluids, Math. Meth. Appl. Sci 33/2010, pp. 1995–2010.
42
On vorticity transport & diffusion in bounded 3D-domains.
R.Rautmanna
a University of Paderborn
The condition of adherence for the velocity of a viscous flow implies a nonlocalboundary condition for its vorticity. In suitable function spaces, in which this conditionholds, a Hopf-Galerkin method leads to unique solutions of the initial- boundary valueproblem of the vorticity transport diffusion equation locally in time.
*****
On the local existence in L3 for the three-dimensional Navier–Stokesequations
Witold Sadowski
Warsaw UniversityBanacha 2, 02-097 Warsaw
We consider the 3D Navier–Stokes equations. We present an elementary proofof local existence in L3 based only on energy estimates and regularisation of the initialdata with the heat semigroup. We also derive a condition on u0 that prevents a putativeblowup on the time interval [0, T ] and we give a priori estimates that provide a rate ofblowup in Lp, p > 3. This is a joint work with James Robinson.
Ill-posedness theory and norm-inflation argument of the 3-DNavier-Stokes equations in the critical space
O. Sawada
Gifu UniversityYanagido 1-1, Gifu City, Japan
The local well-posedness theory of the 3-dimensional Navier-Stokes equationsin BMO−1 = F−1
∞,2 was proved by Koch-Tataru, using bilinear estimates and fixed-point arguments (successive approximation) due to the notion of mild solutions as Katoor Giga-Miyakawa. Besides, the ill-posedness theory in the critical space B−1
∞,∞ =
F−1∞,∞ was established by Bourgain-Pavlovic. They showed a lack of equicontinuity
of mild solutions in C([0, T ]; B−1∞,∞(R3)), using the norm-inflation arguments. The
precise proofs are given in this talk. Moreover, it is seen that some subsequence of thesuccessive approximation of mild solutions diverges in finite time.
43
Some decay properties of compressible viscous fluid flow in 2dimensional exterior domains.
Yoshihiro Shibataa
aWaseda University Tokyo, [email protected]
Many years ago, Matsumura and Nishida proved a global in time existence the-orem with initial data close to positive constant density and zero velocity field in theexterior domain case. They used so called the energy method, so that the decay prop-erties of solutions were obtained only for the H4 initial data. In this talk, I will talkabout some decay properties with L1∩H4 initial data. The key estimate is some Lp-Lq
decay estimates of the Stokes semigroup for the compressible viscous fluid flow. Thisis a joint work with Y. Enomoto and M. Suzuki.
*****
On R-sectoriality of the Stokes equations with first order boundarycondition in a general domain
Y. Shibataa and S. Shimizub
aWaseda UniversityTokyo, Japan
bShizuoka UniversityShizuoka, Japan
In this talk, we discuss R-sectoriality of the Stokes equations with Neumanntype boundary condition in a general domain, which implies the maximal Lp-Lq reg-ularity for the initial boundary value problem of the Stokes equations. Our essentialassumption of a general domain is the unique existence of solutions to the weak Dirich-let problem, and it is satisfied by the whole space, the half-space, a layer, a boundeddomain, an exterior domain, a perturbed half-space, a perturbed layer. As an appli-cation of this result, we obtain local well-posedness of free surface problems for theNavier-Stokes equations in general domains.
44
Very singular and large solutions of semi-linear parabolic equationswith degenerate absorption potential
A.E. Shishkov
Inst. of Appl. Math. and Mech. of NAS of UkraineDonetsk, Ukraine
E-MAIL: [email protected]
It is well known that semilinear heat equation
ut −∆u+ hup = 0 in Q := RN × (0,∞), N ≥ 1, p > 1,
with strongly positive continuous absorption potential h = h(x, t) ≥ h0 = const > 0admit both large ( i.e. solutions u(x, t) ≥ 0 : u(x, 0) = ∞ ∀x ∈ RN ) and verysingular (i.e. solutions u(x, t) ≥ 0 : u(x, 0) = 0 ∀x ∈ RN \0, ∥u(·, t)∥L1 → ∞as t → 0) solutions. Such a property remains true for wide class of potentials h,degenerating on some manifolds Γ ⊂ Q : h(x, t) = 0 on Γ.
Essentially new phenomenon happens if potential h is very flat near to Γ. In thissituation solutions uk(x, t), k = 1, 2, . . . , approximating corresponding large or v.s.solution u∞(x, t), may elaborate singularity which propagates along all Γ as k → ∞.As result, limiting function u∞ is not large or v.s. solution. It turns into some solutionof equation in the domain Q \ Γ only (for example, into "raizor blade" solution). Forsome class of manifolds Γ we obtain sharp necessary and sufficient (almost criterium)conditions on the flatness of h near to Γ, guaranteeing existence or nonexistence oflarge and v.s. solution. Analogous analysis we provide for solutions of correspondingelliptic semilinear equations with degenerate potential. Some of mentioned results arepublished in [1]–[5].
Results of joint investigations with Laurent Veron and Moshe Marcus.
References[1] Shishkov A., Veron L. The balance between diffusion and absorption on semilinear
parabolic equations. Atti Accad. Naz. Lincei. Cl. Sci. Fis. Math. Natur. Rend Lincei (9)Mat. Appl. – 2007. – 18, z 1. – P. 59–96.
[2] Shishkov A., Veron L. Singular solutions of some nonlinear parabolic equations withspatially inhomogeneous absorption. Calc. Var. Part. Differ. Equat. – 2008. – 33,z 3. – P. 343–375.
[3] Shishkov A., Veron L. Diffusion versus absorption in semilinear elliptic equations.J. Math. Anal. Appl. – 2009. – 352, z 1. – P. 206–217.
[4] Shishkov A., Veron L. Propagation of singularities of nonlinear heat Flow in FissuredMedia, arXiv: 1103.5893v1 (to appear in Commun. in Pure Appl. Anal.).
[5] Murcus M., Shishkov A. Fading absorption in non-linear elliptic equations, arXiv:1201.5325v1 (to appear in Ann. of Inst. H. Poincare).
45
Classical solutions for reaction-diffusion equationswith singular initial data
Mikołaj Sierzega
Warwick University
Let Ω ⊂ Rn be a smooth bounded domain. We consider the equation
ut = ∆u+ f(u) in Ω, t > 0,
with Dirichlet boundary conditions, nondecreasing source term f ∈ C1([0,∞)) andnonnegative initial data ϕ ∈ L1
loc(Ω) (supplemented by additional requirements de-pendent on f and Ω). We will present a theory of local existence of solutions forunbounded initial data that are classical for positive times. The technique involvesintegral reformulation of the PDE, known as the variation-of-constants formula. Theargument is based on positivity and monotonicity of an operator associated with theintegral formulation. An application of the method to rapidly growing nonlinearitieswill be given with emphasis on exponential nonlinearities.
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Steady solutions with finite kinetic energy for the Navier-Stokesequations in a three-dimensional exterior domain
Ana L. Silvestrea
aInstituto Superior TécnicoT. U. Lisbon
We consider incompressible, three-dimensional Navier-Stokes flow, around amoving rigid body. The equations of motion are written with respect to a referenceframe attached to the body, where the domain becomes time-independent, but such achange of frame produces new terms in the equations, related with the rotation, whichdifficult the analysis in exterior domains. We begin by presenting a result on existenceand uniqueness of steady solutions with finite kinetic energy to this problem. Thesquare-integrability of the velocity field is shown without resorting to methods basedon the (complicated) fundamental solution of the underlying linear operator. Then,we show, for sufficiently small data, the global existence of a transient solution whichconverges, when time goes to infinity, to the steady solution with finite kinetic energy.Finally, we prove, under small data assumption, the stability of such a steady solution.
46
Asymptotic behavior of solutions to the Navier-Stokes equations
Z. Skalak
aThe Institute of Hydrodynamics of the Czech Academy of SciencesPrague
We study the large time behavior of solutions to the Navier-Stokes equations.The emphasis is put on the rate of the decay of the solutions and their behavior in thefrequency space (see [1] and [2]).
References[1] Skalak, Z., Solutions to the Navier-Stokes equations with the large time energy
concentration in the low frequencies, ZAMM Z. Angew. Math. Mech. 91 (2011),no. 9, pp. 733–742.
[2] Miyakawa, T., On upper and lower bounds of rates of decay for nonstationaryNavier-Stokes flows in the whole space, Hiroshima Math. J. 32 (2002), no. 3, pp.431–462.
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The boundary value problems for the scalar Oseen equation
E. Skopin
University of [email protected]
The scalar Oseen equation represents a linearized form of the Navier Stokesequations, well-known in hydrodynamics. We present an explicit potential theory forthis equation and solve the interior and exterior Oseen Dirichlet and Oseen Neumannboundary value problems via a boundary integral equation method in spaces of con-tinuous functions on a C2-boundary, extending the classical approach for the isotropicselfadjoint Laplace operator to the anisotropic non-selfadjoint scalar Oseen operator. Itturns out that the solution to all boundary value problems can be presented by bound-ary potentials with source densities constructed as uniquely determined solutions ofboundary integral equations.
References[1] Skopin, E., Zur Potentialtheorie der skalaren Oseen-Gleichung in R3., Diplomar-
beit. University of Kassel, 2011
[2] Medkova, D., Skopin, E., Varnhorn, W., The boundary value problems for thescalar Oseen equation., Math. Nachr., to appear
47
On the correct asymptotic conditions at infinity for the time-periodicStokes problem in a system of semi-infinite pipes
M. Skujusa and K. Pileckasb
a,bVilnius UniversityVilnius
[email protected]@MIF.VU.LT
We consider the time-periodic Stokes problem in domains with cylindrical out-lets to infinity. It is well known that this problem may have infinitely many solutions.Prescribing the time-periodic flow rates through the cross-sections of the outlets onecan obtain a solution which is unique up to a time-dependent function in its pressureterm. We constructed a basis in the set of solutions to the homogeneous problem. Thebasis possesses several properties that are essential for the setting of asymptotic condi-tions at infinity (see [1]). We use the generalized Green’s formula in order to specifysome asymptotic conditions (different from the prescription of fluxes) which ensure thecorrect formulation of the problem.
References[1] Skujus M. On the timeâAS-periodic Stokes problem in domains with cylindrical outlets to
infinity, to appear in "Asymptotic analysis".
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On free boundary problems of magnetohydrodynamics.
Vsevolod Alekseevich Solonnikov
Abstract The communication is concerned with well-posedness and differentia-bility properties of solutions of free boundary problems in magnetohydrodynamics ofa viscous incompressible fluid. Main arttention is given to the case where the domainoccupied with the fluid is not simply connected.
48
Existence and stability of solutions close to equilibria for gas ballsunder the influence of gravity
G. Strohmer
Department of MathematicsUniversity of Iowa
We consider configurations of n gas balls (n=1, 2,...) moving in equilibriumunder the influence of gravity and discuss the existence and stability of solutions forall time that begin close to these equilibria. These gas balls consist of barotropic vis-cous compressible fluids assumed to be chemically and physically homogeneous andbounded by a free boundary moving with the flow without surface tension. We studythe question of existence of such equilibrium states, and, in case the equilibria are en-ergy stable, prove that solutions for initial values close to the equilibria exist for alltime.
*****
49
Stability of stationary solutions to the Euler-Poisson equations arisingin plasma physics
Masahiro Suzuki
Tokyo Institute of TechnologyTokyo, Japan
The main concern of this talk is to analyze the behavior of a plasma boundarylayer, called a sheath. It occurs on the surface of materials with which plasma contacts.For the formation of a sheath, the well-known Bohm criterion claims the velocity ofpositive ions should be faster than a certain constant. The behavior of positive ions isgoverned by the Euler-Poisson equations. We discuss that the sheath is understood asa special stationary solution to the equations. We first show that the Bohm criterionis a sufficient condition for the existence of the stationary solution. Then it is alsoshown that the stationary solution is time asymptotically stable provided that an initialperturbation is sufficiently small in the weighted Sobolev space. Moreover we obtainthe convergence rate of the time global solution towards the stationary solution.Acknowledgment. The present result is obtained through the joint research with Prof.Shinya Nishibata at Tokyo Institute of Technology and Dr. Masashi Ohnawa at WasedaUniversity.
References[1] S. Nishibata, M. Ohnawa and M. Suzuki, Asymptotic stability of boundary layers to the
Euler-Poisson equation arising in plasma physics., SIAM Math. Anal, Vol.44 (2012),pp.761-790.
[2] M. Suzuki, Asymptotic stability of stationary solutions to the Euler-Poisson equations aris-ing in plasma physics., Kinetic and Related Models, Vol.4 (2010), pp.569-588.
*****
50
Free surface flow of viscous incompressible fluid
T. Nishida, Y. Teramotoa and K. Tomoedaa
aSetsunan UniversityNeyagawa City
[email protected], [email protected]
The motion of viscous incompressible fluid flowing down an inclined plane un-der the effect of gravity will be discussed. We explain how to formulate in a mathemat-ical way the free boundary value problem which describes this physical phenomena.To show existence results as well as qualitative behaviors of solutions to this problem itis essential to get certain estimates for the resolvent operators arising in the linearizedproblem. Based on this we show asymptotic behaviours of solutions.
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Optimal Korn’s inequality inequality for solenoidal vector fields on aperiodic slab
Y. Teramotoa and K. Tomoedab
aSetsunan [email protected]
bSetsunan [email protected]
In this talk I will talk about the best constant of Korn’s inequality for solenoidalvector fields on a periodic slab in the two dimension.
The Korn’s inequalities are essential in establishing coercive estimate for bound-ary value problems of viscous incompressible fluid dynamics. The best constant ofKorn’s inequality is called Korn constant. Korn constant was investigated for varioussituations. However in the case of a periodic slab, this is not investigated yet. We ob-tain Korn constant for solenoidal vector fields on a periodic slab [1]. The proof of thisresult was based on Ito’s works (1994, 1999).
References[1] Y. Teramoto, K. Tomoeda, Optimal Korn’s inequality for solenoidal vector fields on a
periodic slab, submitted in Proc. Japan Acad.
51
Gibbs measure for isothermal Falk model
Y. Tsutsumia and S. Yoshikawab
aKyoto UniversityKyoto, JAPAN
bEhime UniversityMatsuyama, JAPAN
We consider the Gibbs measure for the isothermal Falk model (IFM) in onespace dimension under periodic boundary condition. (IFM) is the following equationmodeling the shape memory alloy:
∂2t u+ ∂4
xu = ∂x(f(∂xu)
), t ∈ R, x ∈ T = R/2πZ,
f(v) = v5 − v3 + av, a ∈ R.
(IFM) is regarded as a Hamiltonian system in infinite dimensions and the Gibbs mea-sure is an invariant Gaussian measure under the flow of (IFM) with the following en-ergy functional (or Hamiltonian) as weight:
H(v, w) =1
2∥w∥2L2 +
1
2∥∂2
xv∥2L2 +
∫T
F(∂xv(x)
)dx, F (v) =
∫ v
0
f(s) ds.
We show not only the existence of the Gibbs measure with its support in ∩s<3/2Hs ×
Hs−2 but also the global existence of solutions for almost sure initial data in its support.We note that the support of the Gibbs measure is below the energy space H2 ×L2. Asa result, we also have the recurrence property of the system by the Poincaré recurrencetheorem.
References[1] Bourgain, J., “Global Slutions of Nonlinear Schrödinger Equations", Colloquium Publi-
cations Vol. 46, AMS, 1999.
[2] Brokate, M., Sprekels, J., “Hysteresis and Phase Transitions", Applied Math. Sci. Vol.121, Springer, 1996.
[3] Yoshikawa, S., Weak solutions for the Falk model system of shape memory alloys in energyclass, Math. Methods Appl. Sci., 28 (2005), 1423-1443.
[4] Zhidkov, P.E., “Korteweg-de Vries and Nonlinear Schrödinger Equations: QualitativeTheory", Lect. Notes Math., Vol. 1756, Springer, 2001.
52
Decay structure of the regularity-loss typeand the asymptotic stability for the Euler-Maxwell system
Yoshihiro Uedaa
aKobe UniversityKobe City, JAPAN
This talk is based on a joint work with Shuichi Kawashima (Kyushu University)and Shu Wang (Beijing University of Technology). In this talk, we consider the Cauchyproblem of the Euler–Maxwell system in R3 (see, for example, [1, 2]).
The Euler-Maxwell system describes the dynamics of compressible electrons inplasma physics under the interaction of the magnetic and electric fields via the Lorentzforce.
Our purpose is to study the large-time behavior of solutions to the initial valueproblem for the Euler-Maxwell system in R3. This system verifies the decay propertyof the regularity-loss type. Under smallness condition on the initial perturbation, weshow that the solution to the problem exists globally in time and converges to the equi-librium state. Moreover we derive the corresponding convergence rate of the solutions.The key to the proof of our main theorems are to derive a priori estimates of solutionsby using the energy method.
References[1] F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press,
New York, 1984.
[2] Y. Peng, S. Wang and Q. L. Gu, Relaxation limit and global existence of smooth solutionsof compressible Euler-Maxwell equations, SIAM J. Math. Anal. 43 (2011), no. 2, 944–970.
53
On a spherically symmetric motion of a self-gravitating viscous gas
M. Umehara
Ibaraki UniversityMito, Japan
We consider a system of equations describing a spherically symmetric motionof a viscous and ideal gas. The gas is bounded by a free-surface from above, and on acentral rigid sphere. The motion of the gas is driven by the gravitation due to both therigid core and the gas itself (self-gravitation). Equations describing the motion aboveare, in the Lagrangian-mass coordinate, for (x, t) ∈ Ω× (0,∞), Ω := (0,M)
vt = (r2u)x,
ut = r2(−R
θ
v+ ζ
(r2u)xv
)x
−GMc + x
r2,
cvθt =
(−R
θ
v+ ζ
(r2u)xv
)(r2u)x − 4µ(ru2)x +
(r4κ
vθx
)x
.
Imposed initial and boundary conditions are: (v, u, θ)|t=0 = (v0, u0, θ0),u|x=0 = 0,
(−R
θ
v+ ζ
(r2u)xv
− 4µu
r
) ∣∣∣∣∣x=M
= −pe,
(r4κ
vθx − κc(θ − θc)
) ∣∣∣∣∣x=0
=
(r4κ
vθx − κe(θ − θe)
) ∣∣∣∣∣x=M
= 0.
Unknown functions are the density ρ = ρ(x, t), the velocity v = v(x, t) and the abso-lute temperature θ = θ(x, t). Here r satisfies r = r(x, t) =
(Rc
3 + 3∫ x
0v(s, t) ds
)1/3;
M is the total mass of the gas; Rc, Mc, G, R are positive constants; µ, ζ, cv, κ, κc, κe,θc, θe and pe are all assumed to be constants with µ, cv, κ, θc, θe > 0 and 3ζ − 4µ, κc,κe, pe ≥ 0.
We shall obtain a large-time behaviour of the flow under a certain restricted con-dition similar to the one obtained in [1] of a one-dimensional problem. Correspondingstationary problem is also discussed in a similar manner found in [2].
References[1] Umeahra, M., Tani, A., Free-boundary problem of the one-dimensional equations for a
viscous and heat-conductive gaseous flow under the self-gravitation, submitted.
[2] Zlotnik, A., Ducomet, B., Stabilization rate and stability for viscous compressiblebarotropic symmetric flows with free boundary for a general mass force, Sb. Math., 196(2005), pp. 1745–1799.
54
Hadamard variational formula for the velocity and pressure of theStokes equations of the perturbation of domains
Erika Ushikoshi
Tohoku UniversitySendai City
We are interested in examing the variation of the Green function under the per-turbations of its domain. More specifically, we intend to derive so-called the Hadamardvariational formula, which represents the first coefficient term in the expansion of theGreen function in power series of the small parameter describing an infinitesimal dis-placement of the boundary of the domain. Hadamard [4] proved this formula for theGreen function of the Laplace equation under a certain perturbation of its domain.The proof of such a formula for general perturbations was established, for example,in Garabedian-Schiffer [3]. Moreover, Fujiwara-Ozawa [2] and Aomoto [1] proveda variational formula for the Green functions of some normal elliptic boundary valueproblems.
The aim of this article is to prove the Hadamard variational formula for theGreen function for the velocity and pressure of the Stokes equations, describing themotion of viscous incompressible fluid motion by a new approach.
References[1] K.Aomoto, Formule variationnelle d’Hadamard et modèle euclidien des variétés différen-
tiables plongées 34(1979), 493-523.
[2] D.Fujiwara and S.Ozawa, The Hadamard variational formula for the Green functions ofsome normal elliptic boundary value problems, Proc. Japan Acad., 54(1978), 215-220.
[3] P.R.Garabedian and M.Schiffer, Convexity of domain functionals, J.Anal.Math., 2(1952-53), 281-368.
[4] J.Hadamard, Mémoire sur le probleme d’analyse relatif à l’equilibre des plaques élastiquesencastrées, Oeuvres., 2(1908), 515-631.
[5] J.Peetre, On Hadamard’s variational formula, J.Differential Equations, 36(1980), no.3,335-346.
55
On extensions of Serrin’s condition for the Navier-Stokes equations
R. Farwiga, H. Sohrb and W. Varnhornc
aTechnical University of [email protected]
bUniversity of [email protected]
cUniversity of [email protected]
We consider a smoothly bounded domain Ω ⊆ R3, some time interval [0, T ),0 < T ≤ ∞, and a weak solution u of the Navier-Stokes system in [0, T )×Ω with ini-tial value u0 ∈ L2
σ(Ω) and conservative external forces, for simplicity. We assume thatu is weakly continuous and satisfies the so-called strong energy inequality. Our generaluniqueness result strictly extends Serrin’s uniqueness condition u ∈ Ls(0, T ;Lq(Ω)),2 < s ≤ ∞, 3 ≤ q < ∞, 2
s+3q = 1. In particular, we are interested in extensions of the
condition u ∈ L∞(0, T ;L3(Ω)) since there are several recent results in this context.The following condition only contains local in time properties formulated with help of
the dual Besov space B−2s
q,s (Ω) which is the optimal initial value space (see [1]) for localstrong solutions in Ls(0, T ;Lq(Ω)): The given weak solution u is uniquely determined
within the class of weak solutions with initial value u0 if (i) u(t) ∈ B−2s
q,s (Ω) for eacht ∈ [0, T ), and if (ii) ∥u∥2 : [0, T ) → R is left-side continuous at each t ∈ (0, T ).
Using the strict embedding L3σ(Ω) ⊂ B
−2s
q,s (Ω) if q ≤ s, this extends, in particular, theuniqueness condition u ∈ L∞(0, T ;L3
σ(Ω)).
References[1] Farwig, R., Sohr, H., Varnhorn, W., On optimal initial value conditions for local strong
solutions of the Navier-Stokes equations., Ann. Univ. Ferrara 55, pp. 89–110, 2009.
56
On the boundary regularity of the pressureof unsteady Stokes flow in exterior domains
J. Wolf
Department of MathematicsUniversity of Magdeburg
Let Ω ⊂ Rn be a domain with compact smooth boundary, let 0 < T < +∞;define Q = Ω×]0, T [. We consider the following system of PDEs:
div u = 0 in Q, (1)∂u
∂t−∆u = −∇p+ f in Q, (2)
together with the boundary-initial value condition
u(0) = 0 on ∂Ω×]0, T [∪Ω× 0.
By the well-known Lp-theory of the Stokes equation the gradient of pressure ∇p be-longs to Ls(0, T ;Lq(Ω)) if f ∈ Ls(0, T ;Lq(Ω)) (1 < s, q < +∞). In our talk wepresent an extension of this result which says that ∇p belongs to Ls(0, T ;W k, p(Ω))(k ∈ N) if f belongs to this space. In particular, we prove that ∇p is smooth withrespect to x if f is smooth with respect to x.
57
The motion of rigid bodies in non-Newtonian fluids with nonstandardrheology
A. Wróblewska-Kaminskaa
aUniversity of WarsawWarsaw
We prove the existence of a weak solution to the problem of the motion of oneor several nonhomogeneous rigid bodies immersed in an incompressible homogenousnon-Newtonian fluid of rheology more general than power-law-type. The nonlinearviscous term in the equation is described with help of general convex function defin-ing Orlicz spaces. The main ingredient of the proof is convergence of nonlinear termachieved with help of pressure localisation method. Therefore we provide the decom-position and local estimates for the pressure function in Orlicz spaces.
References[1] E. Feireisl, M. Hillairet and Š. Necasová. On the motion of several rigid bodies in an
incompressible non-Newtonian fluid. Nonlinearity. 21:1349-1366, 2008.
[2] A. Wróblewska-Kaminska. Existence results for unsteady flows of nonhomogeneous non-Newtonian incompressible fluids - monotonicity methods in generalized Orlicz spaces.Prepreprint PhD Programme: Mathematical Methods in Natural Sciences, Nr 2011 - 024,2012.
[3] A. Wróblewska-Kaminska. Local pressure methods in Orlicz spaces for the motionsof rigid bodies in an non-Newtonian fluid with general growth conditions. Accepted toDiscrete and Continuous Dynamical Systems - S, 2012.
58
A mathematical justification of the penalty method for the Stokes andNavier-Stokes equations
Norikazu Yamaguchi
University of Toyama3190 Gofuku, Toyama-shi, Toyama 930-8555, JAPAN
The motion of viscous incompressible fluid is governed by the Navier-Stokesequations. In the incompressible Navier-Stokes equations, the pressure term does nothave time evolutional structure. Therefore we need to eliminate the pressure from theequation, when we treat the Navier-Stokes equations as an evolution equation.
In numerical computation of the motion of fluid, it is better to eliminate thepressure in some ways. The penalty method is widely used in numerical computation ofthe motion of viscous incompressible fluid to remove the pressure from the equations.By the penalty method, the solenoidal condition is approximated by ∇ · u = −p/ηwith penalty parameter ϵ > 0, which is assumed to be very large. By such a relation,we get a approximated equations for motion. In this talk, I will talk about mathematicaljustification of the penalty method, that is, I will show that the solution of penalizedStokes/Navier-Stokes equations uη(t) converges to original one u(t) in some topologywhen the penalty parameter goes to infinity. Ingredients of the proof are the Helmholtzdecomposition of Lr-vector fields, Lr-Lq estimates for the heat kernel with diffusivityand semigroup theory.
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Regularity of axially symmetric solutions to the Navier-Stokesequations.
Wojciech Zajaczkowski.
Institute of Mathematics, Polish Academy of [email protected]
We consider axially symmetric solutions to the Navier-Stokes equations in aperiodic cylinder. We assume the slip boundary conditions on the lateral boundary ofthe cylinder. The proof is divided into the steps. Having initial velocity in H1 space wehave local existence in W 2,1
2 space. Next we prove global a priori estimate for norm ofvelocity in H1. Then the local solution can be extended step by step infinitely.
59
The fractional step theta method in fluid dynamics
F. Zanger
University of [email protected]
We consider the non-stationary incompressible linear Stokes equations in [0, T ]×Ω with bounded Ω ⊂ Rn. Its solution can be approximated with the fractional steptheta time stepping procedure. Using energy estimates and assuming a certain degree ofregularity for data and boundary ∂Ω we can show second order ℓ∞(L2)-convergence.We plan to carry over the result to the nonlinear case.
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On a model of reactive flow with multicomponent diffusion
E. Zatorska
Institute of Applied Mathematics and Mechanics,University of Warsaw,
ul Banacha 2, 02-097 Warszawa, [email protected]
We consider the Cauchy problem for the system of equations governing flowof reactive mixture of compressible gases [1]. We will present the proof of sequentialstability of weak solutions when the state equation essentially depends on the speciesconcentration and the viscosity coefficients vanish on vacuum (isothermal reaction) [2].We shall also discuss the main steps of the approximation procedure, under additionalassumption on the ”cold” component of the pressure in the regions of small density [2].
References[1] V. Giovangigli, Multicomponent flow modeling.Modeling and Simulation in Science, En-
gineering and Technology. Birkhäuser Boston Inc., Boston, MA, 1999.
[2] E. Zatorska, On the flow of chemically reacting gaseous mixture.http://mmns.mimuw.edu.pl/preprints.html, Preprint no. 2011 - 016, 2011.
60
Solvability of a generalized Buckley-Leverett model
N. V. Chemetova and W. Nevesb
a CMAF / University of LisbonLisbon, Portugal
b Federal University of Rio de JaneiroRio de Janeiro, Brasil
We propose a new mathematical modelling of the Buckley-Leverett system,which describes the two-phase flows in porous media. We prove the solvability ofthe initial-boundary value problem for a deduced model
∂tu+ div(v g(u)
)= 0, (1)
−ν∆v + h(u)v = −∇p, div(v) = 0, (2)
where u = u(t,x) and v = v(t,x) are the saturation and the total velocity of thetwo-phase flow. The equation (2) is a generalized Darcy Law (Darcy-Brinkman’s law).
In order to show the solvability result, we consider an approximated parabolic-elliptic system. Since the approximated solutions do not have ANY type compactnessproperty, the limit transition is justified by the kinetic method [1]-[3]. The main issue isto study a linear (kinetic) transport equation, instead of the nonlinear original system.
References[1] Chemetov N.V., Neves W., The generalized Buckley-Leverett System. Solvability, accepted
to be published in "Arch. Rational Mech. Anal.", http://arxiv.org/abs/1011.5461
[2] Chemetov N.V., Arruda L., L_p-Solvability of a Full Superconductive Model., NonlinearAnalysis: Real World Applications, 12(4)/2011, pp. 2118–2129.
[3] Chemetov N.V., Nonlinear Hyperbolic-Elliptic Systems in the Bounded Domain, Commu-nications on Pure and Applied Analysis, 10(4)/2011, 1079–1096.
61
