An entropy satisfying relaxation/HLLC solver for gas dynamics that computes vacuum without correction Fran¸ cois Bouchut CNRS & Ecole Normale Sup´ erieure, Paris, France [email protected]Several solvers proved entropy satisfying for the gas dynamics system have been derived these last years. I shall present an approach by relaxation which allows to derive such a solver, of HLLC type, and to analyze precisely its stablity for arbitrary large data. This allows an optimal choice of the wave velocities, which ensures a finite value in the neighbourhood of vacuum, without any particular treatment. The solver does not need any iterative procedure, and is valid for any convex pressure law. Reference F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conser- vation laws, and well-balanced schemes for sources, Frontiers in Mathematics series, Birkh¨ auser, 2004,
Transcript
An entropy satisfying relaxation/HLLC solver for gas dynamicsthat computes vacuum without correction
Francois BouchutCNRS & Ecole Normale Superieure, Paris, France
Several solvers proved entropy satisfying for the gas dynamics system have beenderived these last years. I shall present an approach by relaxation which allows toderive such a solver, of HLLC type, and to analyze precisely its stablity for arbitrarylarge data. This allows an optimal choice of the wave velocities, which ensures afinite value in the neighbourhood of vacuum, without any particular treatment. Thesolver does not need any iterative procedure, and is valid for any convex pressurelaw.
Reference
F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conser-
vation laws, and well-balanced schemes for sources, Frontiers in Mathematics series,Birkhauser, 2004,
Abstract: The limit of vanishing ratio of the electron mass to the ionmass in the isentropic transient Euler-Poisson equations with periodicboundary conditions is proved. The equations consist of the conservationlaws for the electron density and current density for given ion density,coupled to the Poisson equation for the electrostatic potential. The limitis related to the low-Mach-number limit of Klainerman and Majda. Inparticular, the limit velocity satisfies the incompressible Euler equationwith damping. The difference to the zero-Mach-number limit comes fromthe electrostatic potential which needs to be controlled. This is done bya reformulation of the equations in terms of the enthalpy, higher-orderenergy estimates and a careful use of the Poisson equation.
1
Some Reviews on Nonlinear System of Conservation Laws
Abstract: The persistence of sharp and flat interfaces under multi-dimensional perturbations is investigated from a viewpoint analogousto Majda’s treatment of classical shocks. The main novelty here isthat jump conditions for the free interfaces take surface tension intoaccount. This means that, unlike classical jump conditions, they arenon-homogeneous, containing first-order and second-order differentialsof the front. A normal modes analysis shows that neutral modes maypropagate along the front. In the standard setting, this would imply aweak stability result, involving energy estimates with loss of derivatives.In our case the lack of homogeneity of the underlying boundary valueproblem implies that neutral modes can only be of large enough wavelength. Suitable frequency cut-offs then yield energy estimates withoutloss of derivatives - for the constant-coefficients linearized problem, asin the case of uniformly stable classical shocks. (Joint work with S.Benzoni-Gavage.)
1
Collision potentials and L1-stability of Some kinetic equations
Seung-Yeal HaDepartment of Mathematical Sciences, Korea
In this talk, I will present collision potentials and L1-stability for the Boltzmannequation and Vlasov-Poisson system. Collision potentials measures the possiblefuture collisions between particles and are employed to the L1-scattering and stabilityanalysis of aforementioned equations.
Abstract: The contact discontinuity is one of the basic wave patternsin gas motions. The stability of contact discontinuities with general per-turbations is a long standing open problem. One of the reasons is thatcontact discontinuities are linearly degenerate waves in the nonlinearsettings, like the Navier-Stokes equations and the Boltzmann equation.The nonlinear diffusion waves generated by the perturbations in sound-wave families couple and interact with the contact discontinuity and thencause analytic difficulties. Another reason is that in contrast to the basicnonlinear waves, shock waves and rarefaction waves, for which the corre-sponding characteristic speeds are strictly monotone, the characteristicspeed is constant across a contact discontinuity, and the derivative ofcontact wave decays slower than the one for rarefaction wave. In thistalk, I will present some recent works on the time asymptotic stabil-ity of a damped contact wave pattern with an convergence rate for theNavier-Stokes equations and the Boltzmann equations. One of the keyobservations is that even though the energy estimate involving the lo werorder may grow at the rate (1+ t)
12 , it can be compensated by the decay
in the energy estimate for derivatives which is of the order of (1 + t)−12 .
Thus, these reciprocal order of decay rates for the time evolution of theperturbation are essential to close the priori estimate containing the uni-form bounds of the L∞ norm on the lower order estimate and then itgives the decay of the solution to the contact wave pattern.
Abstract: We will survey weak shock Mach reflection and the von Neu-mann triple point paradox. We will show numerical solutions (obtainedjointly with Allen Tesdall) of the transonic small disturbance equation,which provides an asymptotic description of weak shock Mach reflection,that contain an expansion fan at the triple point and a remarkably com-plex flow immediately behind it, thus resolving the triple point paradox.We will also show some very recent experimental results of Skews andAshworth that support these theoretical results.
1
Vanishing viscosity limit to rarefaction waves for the
Navier-Stokes equations of one-dimensional compressible
heat-conducting fluids
Song Jiang, Guoxi Ni, Wenjun Sun
Beijing Institute of Applied Physics and Computational Mathematics, P.R.China
Abstract: We prove the solution of the Navier-Stokes equations forone-dimensional compressible heat-conducting fluids with centered rar-efaction data of arbitrary strength exists globally in time, and moreover,as the viscosity and heat-conductivity coefficients tend to zero, the globalsolution converges to the center rarefaction wave solution of the corre-sponding Euler equations uniformly away from the initial discontinuity.
1
On Strong Convergence in Vortex Sheets Problem
for 3-D Axisymmetric Euler Equations
Quansen JIU
Department of Mathematics,Capital Normal University,
Abstract: In this report, we summarize some of the mainresults on the structures and convergence properties of thesequences of approximate solutions to the vortex-sheets prob-lem for axisymmetric flows without swirls, in both unsteadyand steady case. The main results include two aspects: First,we will show that if the approximate solutions of the 3-Dunsteady axisymmetric Euler equations converge strongly (inL2-space) over the region outside the symmetry axis, thenthey must converge strongly in the whole space. This im-plies if there would appear energy concentration in the pro-cess of limit for the approximate solutions, the set of energy-concentration must contain points in the region outside thesymmetry axis. There is no restriction on the signs of the ini-tial vorticity here. Second, we will give a strong convergencecriterion for the approximate solutions of the 3-D steady ax-isymmetric Euler equations.
1
Transonic problems for two dimensional self-similar potential
Abstract: In this talk, I will focus on studying the propagation ofshock waves away from the boundary for scalar viscous conservationlaw. We will introduce an iteration scheme to decouple two effects onsolution behavior: the compressibility of the shock and the presence ofthe boundary. Through this scheme we can obtain pointwise descriptionof the perturbation of the shock profile.
1
Large-Time Behavior of Solutions for the Boltzmann
Abstract: We study the solutions of the one-dimensional Boltzmannequation for hard potential models with Grad’s angular cutoff. Thecollision frequency ν(ξ) ≈ (1+ |ξ|)γ makes the spectrum of the operator−iξ1η +L non-analytic in η. We resolve this complication using the realanalytic method in the estimates of the fluidlike long waves. We devisea new energy method to account for the sub-exponential behavior ofwaves outside fluid region.
1
INTRODUCTION TONONCLASSICAL SHOCKS AND KINETIC RELATIONS
PHILIPPE G. LEFLOCHUNIVERSITY OF PARIS 6 & CNRS
Abstract. I will review results on undercompressive, nonclassical shockwaves and discuss various tools that have been introduced to studythe effect of small scales and the selection of physically meaning-ful, discontinuous solutions. I will explain how to characterize zerodiffusion-dispersion limits for hyperbolic systems that are strictlyhyperbolic but not globally genuinely nonlinear, and for systemsof mixed (hyperbolic-elliptic) type. Solutions typically contain un-dercompressive shocks or subsonic phase transitions. These wavesare fundamental in, for instance, phase transition dynamics (van derWaals fluids, martensitic materials) when both viscosity and capil-larity effects come into play.
While classical shocks are compressive, independent of small-scaleregularization mechanisms, and can be characterized by a single en-tropy inequality, by contrast nonclassical waves are undercompres-sive, are very sensitive to diffusive and dispersive mechanisms. Theselection of the latter is more delicate and requires an additional jumpcondition, referred to as a kinetic relation.
Among the issues of interest we can mention : the role of the kineticrelation for solving the Riemann problem, the derivation of kineticrelations from a traveling wave analysis, the existence of nonclas-sical entropy solutions via Glimm scheme and wave front tracking,the uniqueness of admissible solutions in the class of solutions withtame variation, the nucleation criterion for thin films, the design ofdifference schemes based on entropy conservative flux.
References
[1] P.G. LeFloch, Propagating phase boundaries: formulation of the problem and existence via theGlimm scheme, Arch. Rational Mech. Anal. 123 (1993), 153–197.
[2] P.G. LeFloch, Hyperbolic systems of conservation laws: The theory of classical and nonclassicalshock waves, Lectures in Mathematics, ETH Zurich, Birkauser, 2002.
[3] P.G. LeFloch and M. Shearer, Nonclassical Riemann solvers with nucleation, Proc. Royal Soc. Edin-burgh 134A (2004), 941–964.
Abstract: We review the recent progress on the quantum hydrodynam-ical model for charge transport.
1
RECENT DEVELOPMENT OF THE GENERALIZEDRIEMANN PROBLEM (GRP) SCHEME
JIEQUAN LI
We will talk about the recent development of the generalized Rie-mann problem scheme for hyperbolic balance laws. The concept ofRiemann invariants are extensively used in the analytic resolution of thegeneralized Riemann problem associated with high resolution Godunov-type schemes. The resulting scheme is independent of the Eulerian orLagrangian formulation of physical systems. The Delicate sonic caseare simply treated and the multidimensional extension are straightfor-ward.
1
On Traffic Instabilities
Tong LiDepartment of Mathematics, University of Iowa, USA
We will review recent results on traffic instabilities. An innovative approach totraffic dynamics is proposed. The self-organized oscillatory and chaotic behaviorof traffic flow are identified and formulated. The results agree with the empiricalfindings for freeway traffic and with the previous numerical simulations. Thus thework gives a justification for observed and simulated traffic instabilities and someinsight into their meanings.
The Green’s function for one-dimensional Broadwell
Abstract: The Broadwell model system is one type of the discreteBoltzmann equation which appears in the kinetic theory for rarifiedgases. In this talk, we plan to devise a Picard- type iteration and useFourier method to obtain pointwise estimate for the Green function forthe linearized Broadwell model system in one-dimensional space arounda local Maxwellian.
1
Some thoughts on shock wave theory and Boltzmann
equation
Tai-Ping Liu
Department of Mathematics, Stanford University, USA
Abstract: The understanding in conservation laws is helpful for thestudy of Boltzmann equation in two regards: First, the shock wavetheory naturally gives rise to corresponding problems for Boltzmannequation. Some of the techniques for shock wave theory can be appliedto Boltzmann equation. Secondly, the understanding in conservationlaws helps to highlight the striking differences between fluid mechanicsand kinetic theory. This is particularly so for the behavior near solidboundary. We plan to give some examples to exhibit these two regards.
Abstract: We present the analysis of QHD models arising in the physicsof semiconductor devices and regarding models of various condensationphenomena. The analysis regards dissipative and dispersive propertiesof nonlinear waves.References:[1]Hailiang Li, Marcati, Pierangelo, Existence and asymptotic behaviorof multi-dimensional quantum hydrodynamical model for semiconduc-tors Comm. Math. Phys. 245 (2004), no. 2, 215–247.[2]Donatelli, Donatella; Marcati, Pierangelo, in preparation
1
On a Shockley-Read-Hall Model forSemiconductors: Convergence to
Equilibrium
Vera Miljanovic, Christian Schmeiser
Abstract
We are considering a drift-diffusion and a kinetic model for the flow of electrons in asemiconductor crystal, incorporating the effects of recombination-generation via trapsdistributed in the forbidden band. In mathematical terms, model consists of a reaction-diffusion-convection equation for the electric field and an integro-differential equationfor the distribution of occupied traps. We derive formal and rigorous asymptotics, andshow convergence.
1
Conservation laws and completely conservativedifference schemes for the nonlinear kinetic
Landau-Fokker-Planck equation
I. F. Potapenko
Keldysh Institute for Applied Mathematics, RAS, Department of Kinetic Equations,
The kinetic Landau - Fokker - Planck (LFP) equation is widely used for the descrip-
tion of collision processes. As an intrinsic part of physical models, both analytical and
numerical, this equation has many applications in laboratory as well as in space plasma
physics. It should be remarked that for the nonlinear LFP equation a nontrivial situation
exists: at least two conservation laws (for particle and energy) are valid. If the differ-
ence scheme possesses only an approximate analog of the conservation laws, then this can
easily lead to the accumulation of errors in the analysis of the nonstationary problem.
The difference schemes which satisfy these two laws we call the completely conservative
schemes. They allow to carry out numerical calculations in a large time interval without
error accumulation what has utmost importance for a gas with light and heavy particles,
when the characteristic time scales differ by hundreds times.
Illustrative examples are given. The formation of a non equilibrium steady-state distri-
bution function of particles with the power law interaction potentials U = α/rs, where
1 ≤ β < 4, is studied numerically in the presence of particle (energy) sources. Results can
be useful in connection with the development of high-power particle and energy sources
and for the prediction of the semiconductor behavior under the action of particle fluxes or
electromagnetic radiation. Also the problems connected with wave-particle interactions
are considered.
The comparison of the numerical calculations with the analytical asymptotic results
proves the high accuracy of the exploited difference schemes.
Finally, the results obtained on the base of finite-difference schemes are compared
with the results used a collision simulation algorithm based on the time-explicit formula
derived from the Boltzmann equation.
1.I.F.Potapenko, C.A. de Azevedo. J.of Comp. and Appl. Math.,103(1999) 115
2.I.F.Potapenko, A.V.Bobylev, C.A. de Azevedo,A.S. de Assis. Phys.Rev. E 56 (1997)
7159
3. A.V.Bobylev, K.Nanbu Phys.Rev. E 61 (2000) 4576
Global Attractor for A Nonlinear Thermoviscoelasticity with ANon-convex Free Energy Density
Yuming Qin
Department of Applied Mathematics, College of Science, Donghua University, Shanghai200051,P.R. China. E-mail: [email protected]
Abstract
This paper is concerned with the existence of a global attractor for a semiflowgoverned by the weak solutions to a nonlinear one-dimensional thermoviscoelastic-ity with a non-convex free energy density. The constitutive assumptions for theHelmholtz free energy include the model for the study of martensitic phase tran-sitions in shape memory alloys. To describe physically phase transitions betweendifferent configurations of crystal lattices, we work in a framework in which thestrain u belongs to L∞. New approaches are introduced and more delicate esti-mates are derived to establish the crucial L∞-estimate of strain u in deriving thecompactness of the orbit of the semiflow and existence of an absorbing set.
1
Linear and nonlinear viscoelastic models with memory term
In the talk I will recall the nonlinear 3-D viscoelastic model with memory termand consider the Cauchy problem. The assumptions on the memory kernel includesintegrable singularities at zero and polynomial decay at infinity. I will give the resultsof existence and the time decay of the solutions both for the linear and nonlinearproblems. Furthermore, I will discuss a preliminary idea to obtain the stability ofplanar travelling waves.
Acceleration Waves and Weaker Kawashima Condition
Jie LouDepartment of Mathematics, Shanghai University, China
Research Center of Applied Mathematics (CIRAM), University of Bologna, [email protected]
We consider dissipative hyperbolic systems of balance laws in which a block ofequations are conservation laws which arise typically in the Extended Thermody-namics [1]. In this case, a coupling condition firstly introduced by Kawashima (K-condition) [2], play a fundamental role for the global existence of smooth solutionfor small initial data and for the stability of constant state [3]-[5]. Nevertheless thecounterexample by Zeng [6] prove that the K-condition is only a sufficient condition.
In this talk we propose a weaker K-condition in which the K-condition is re-quired only for the genuine non linear characteristic velocities and not for the lineardegenerate one.
We prove that this weaker condition (that is satisfied also by the Zeng example)is, together with the dissipative condition, a necessary and sufficient condition atleast for smooth solution in the class of discontinuity wave (C1-piecewise) [7].
References:
[1] I. Muller, T. Ruggeri, Rational Extended Thermodynamics. 2nd ed. SpringerTracts in Natural Philosophy 37, Springer-Verlag, New York, 1998.
[2] S. Kawashima, Proc. Roy. Soc. Edimburgh, 106A, 169 (1987).
[3] B. Hanouzet, R. Natalini, Arch. Rat. Mech. Anal. 169, 89 (2003).
[5] T. Ruggeri, D. Serre, Quarterly of Applied Math., 62(1), 163 (2003).
[6] Y. Zeng, Arch. Rat. Mech. Anal. 150 no. 3, 225 (1999).
[7] J. Lou, T. Ruggeri, Acceleration Waves and Weaker Kawashima Condition. Sub-mitted.
1 Presented by T. Ruggeri
Two dimensional transonic flows through a nozzle for the
steady full Euler equations
Kyungwoo Song
Department of Mathematics,
Kyung Hee University, Korea
Abstract: We establish the existence and uniqueness of transonicshocks in the steady flows through a two-dimensional nozzle with varingcross-sections. The flow is governed by the steady full Euler equations.We show that the solutions behind the shock front remain subsonic in adownstream region and the shock front is smooth. The problem is ap-proached by a one-phase free boundary problem where the shock frontis a free boundary. The steady full Euler equations are decomposedinto elliptic equations and a system of transport equations for the freeboundary problem(jointwork with G.-Q. Chen and Jun Chen).
1
Hamilton-Jacobi Equations in Infinite Dimension
for Approximation of Optimal Control and Hydrodynamical
Abstract: Optimal control problems for low, d, dimensional differ-ential equations, can be solved computationally by their correspondingHamilton-Jacobi-Bellman partial differential equation in Rd+1. I willshow how to use Hamilton-Jacobi equations in infinite dimension to findstochastic hydrodynamical limits of microscopic Master equations and tosolve inverse optimal design problems for partial differential equations:Master equations. Even small noise can have substantial influence onthe dynamics of differential equations, e.g. for nucleation and coarseningin phase transformations. I will present a simple derivation of an accu-rate model for the noise in macroscopic differential equations, relatedto phase transformations/reactions, derived from more fundamental mi-croscopic Master equations.Optimal Design Problems. Many inverse problems for differential equa-tions can be formulated as optimal control problems. It is well knownthat inverse problems often need to be regularized to obtain good ap-proximations. I will presents a method to regularize and establish er-ror estimates for some control problems in high dimension, based onsymplectic approximation of the Hamiltonian system for the controlproblem: Controls can be discontinuous due to a lack of regularity inthe Hamiltonian or due to colliding backward paths, i.e. shocks; theerror analysis, for both these cases, is based on consistency with theHamilton-Jacobi-Bellman equation, in the viscosity solution sense, anda discrete Pontryagin principle where the characteristic Hamiltonian sys-tem is solved approximately with a R2 approximate Hamiltonian.
1
Approximation by diffusion and homogenization for semiconductor Boltzmann-Poisson system
Mohamed Lazhar TAYEB
Faculty of Sciences of Tunis, University of Tunis El-Manar, 1060, TunisiaLaboratory of Engineering Mathematics, Polytechnic School of Tunisia.
The subject of the communication is to study the approximation by diffusion and homogenization of semi-conductor Boltzmann-Poisson system. Let fε(t, x, v) a solution of the rescaled Boltzmann equation:
∂tfε +
1ε(v.∇xfε −∇xΦε
T .∇vfε) =Q(fε)
ε2, (t, x, v) ∈ R+ × ω ×Rd
where the collision operator is the linear BGK approximation (Boltzmann Statistics) of electron-phonon collision:
Q(fε)(v) =∫Rd
σ(v, v′)(M(v)f(v′)−M(v′)f(v))dv′,
σ is the cross section and M is the normalized Maxwellian. We are interested in the generalization of resultsobtained in [2, 3, 5] by coupling diffusion and homogenization and considering the potential:
ΦεT (t, x) := ΦH(t, x,
x
ε) + Φε
p(t, x)
where ΦH is bounded, smooth and Y = [0, 1]d)-periodic in y = x/ε and ΦεP is self-consistent:
−∆xΦεP = ρε =
∫Rd
fε dv, ΦεP |∂ω = 0.
In one dimension and when the incoming boundary data are well prepared:
where (ρ,ΦP ) is a solution of an homogenized Drift-Diffusion-Poisson system corresponding to an effectivepotential Φ0 and a non symmetric diffusion matrix:
Acknowledgments. I would like to thank Naoufel Ben Abdallah and Nader Masmoudi for helpful discussionsand encouragements.
References[1] G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), no. 6, 1482–1518.[2] N. Ben Abdallah and M. L. Tayeb, Diffusion Approximation for the one dimensional Boltzmann-Poisson
system, Discrete & Continuous Dynamical Systems-Ser. B, Volume 4, No. 4, November 2004, pp. 1129-1142.[3] N. Ben Abdallah and M. L. Tayeb, Diffusion approximation and homogenization of the semiconductor Boltz-
mann equation, to appear in SIAM MMS.[4] T. Goudon, F. Poupaud: Approximation by diffusion and homogenization of kinetic equations, Comm. PDE,
26 (2001), pp. 537-569.[5] N. Masmoudi and M. L. Tayeb, Diffusion limit for semiconductor Boltzmann-Poisson system, preprint.
On a multidimensional model for the dynamic combustion of
Abstract: In this work we present a multidimensional model for thedynamic combustion of compressible reacting fluids formulated by theNavier Stokes equations in Euler coordinates. For the chemical modelwe consider a one way irreversible chemical reaction governed by the Ar-rhenius kinetics. The existence of globally defined weak solutions of theNavier-Stokes equations for compressible reacting fluids is establishedby using weak convergence methods, compactness and interpolation ar-guments in the spirit of Feireisl and P.L. Lions.This is joint work with D. Donatelli
1
L1 Stability of Shock Waves in the Keyfitz-Kranzer System
Abstract: In this paper, we study the long time L1 stability of theviscous shock wave solution of a 2 × 2 system–the so-called Keyfitz-Kranzer system. We do not use the pointwise semigroup method, butonly use the elementary analysis method combined with the lap numbertheory, and establish the purely L1 stability result without restraint onthe shock strength and the size of the initial perturbation, much like inthe scalar case.
1
On Multi-Dimensional Transonic Shock Waves in A Nozzle
by Zhouping Xin The IMS and Department of MathematicsThe Chinese University of Hong Kong
(International Conference on Conservation Laws and Kinetic Theory)July/2005, Shanghai
In this talk, I will discuss some recent progress in the studies of transonic shockwaves in a nozzle with general sections in both 2-dimension and three dimensions.We will focus on the on the conjecture due to Courant-Friedriches on the transonicshock wave patterns in a nozzle with incoming supersonic flow and a given subsonicpressure at the exit. The existence and stability of such transonic wave patternswill be discussed. This is joint work with Huicheng Yin.
W orkinthistalkwassupportedinpartbygrantsfromtheResearchGrantsCouncilofHongKongSpecilAdministrativeRegionCUHK4028/04P,CUHK4040/02P,andCUHK4279/00P.
1
CAUCHY PROBLEM FORVLASOV-POISSON-BOLTZMANN SYSTEM
TONG YANG
The dynamics of the dilute electrons can be modelled by the funda-mental Vlasov-Poisson-Boltzmann system when the electrons interactwith themselves through collisions in the self-consistent electric field. Inthis talk, we will present the result on the Cauchy problem for this sys-tem which shows that any smooth perturbation of a global Maxwellianleads to a unique global-in-time classical solution. A convergence ratein time will also be given. This is a joint work of Huijiang Zhao whichgeneralizes our previous result on this problem together with HongjunYu. Finally, we will also mention our current work on the perturbationaround a stationary solution.
Department of Mathematics, City University of Hong KongE-mail address: [email protected]
1
Non-selfsimilar multi-dimensional elementary waves and
global solutions of conservation laws
Xiaozhou Yang
Department of Mathematics, Shantou University,P.R.China
Abstract: In this talk, we will discuss the multi-dimensional (M-D)conservation laws whose Riemann data just contain two different con-stant states which are separated by a smooth curve or surface. Non-selfsimilar M-D elementary waves, their new structures and propertiesare disclosed, for example the contour surface of M-D rarefaction waveis a family of cylindrical surface etc., which are essentially different fromthat in M-D selfsimilar case. Furthermore, global solutions of a class of2-D systems of conservation laws will be also presented and are formu-lated by implicit function, their structure combining 2-D non-selfsimilarelementary waves and non-constant intermediate states will be shown.
1
AGlobal singularity structures of weak solutionsto the semilinear dispersive wave equations
Huicheng YinDept. of Math. & IMS, Nanjing University, Nanjing 210093, China
In this talk, we are concerned with the global singularity structures of weaksolutions to the semilinear dispersive wave equations whose initial data are chosento be discontinuous on the unit sphere. Combining Strichartz’s inequality with thecommutator argument techniques, we show that the weak solutions are globallyC2−regular away from the focusing cone surface |x| = |t− 1| and the outgoing conesurface |x| = t + 1.
Fundamental Solutions of Hyperbolic-Parabolic Systems and Shock Wave Stability
We construct the fundamental solution for a general hyperbolic-parabolic systemof conservation laws along a weak shock profile. Our formulation has explicit depen-dence on the shock strength. This allows us to perform nonlinear stability analysisfor the shock wave, and obtain detailed asymptotic behavior of the solution. Theresult applies to compressible Navier-Stokes equations and the magnetohydrody-namics, even in the case of having multiple eigenvalues in the transversal fields.
1 Presented by Author 2
Steady Supersonic Flow Past a Curved Cone
Yongqian Zhang
School of Mathematical Sciences, Fudan University,
Shanghai, 200433, P.R.China
Abstract: This is a joint work with Professor Shuxing Chen and Dr.Zejun Wang. We are concerned with the problem of 3-D supersonic flowpast an infinite cone. It has been studied by Lien and Liu, Chen, Xinand Yin when the open angle of the cone is sufficiently small. We relaxthe restriction on the sharp angle. By obtaining more delicate estimatesfor the interaction of the elementary waves, we get the existence resultfor the cone with the open angle less than a critical value.
1
A Case Study of Global Stability of Strong
Rarefaction Waves for 2× 2 Hyperbolic Conservation
Laws with Artificial Viscosity
Huijiang ZhaoWuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences
P.O. Box 71010, Wuhan 430071, China
Abstract
This paper is concerned with global stability of strong rarefaction waves for a class of2× 2 hyperbolic conservation laws with artificial viscosity. It is based on a recent work jointwith Dr. Ran Duan.
1
Two-dimensional regular shock reflection on a wedge
I will present recent work on the regular shock reflection on a wedge for theadiabatic Euler system and the pressure gradient system.
GLOBAL CLASSICAL SOLUTIONS TO QUASILINEARHYPERBOLIC SYSTEMS WITH WEAK LINEAR
DEGENERACY
YI ZHOU
Consider the following Cauchy problem for the first order quasilinearstrictly hyperbolic system
∂u
∂t+ A(u)
∂u
∂x= 0
t = 0 : u = f(x).
We letM = supx∈R|f ′(x)| < +∞.
The main result of this paper is that under the assumption that thesystem is weakly linearly degenerated, there exists a positive constantε independent of M, such that the above Cauchy problem admits aunique global C1 solution u = u(t, x) for all t ∈ R, provided that∫ +∞
−∞|f ′(x)|dx ≤ ε,∫ +∞
−∞|f(x)|dx ≤ ε
M.
This result is then generalized to the inhomogenous systems of equa-tions when the inhomogenous term satisfying a matching condition byWu.
