An entropy satisfying relaxation/hllc solver for gas dynamics that computes vacuum without correction

Transcription

1 An entropy satisfying relaxation/hllc solver for gas dynamics that computes vacuum without correction François Bouchut CNRS & Ecole Normale Supérieure, Paris, France 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 conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics series, Birkhäuser, 2004,

2 Zero-electron-mass limit in the hydrodynamic model for plasmas Li Chen Tsinghua University, P. R. China, Abstract: The limit of vanishing ratio of the electron mass to the ion mass in the isentropic transient Euler-Poisson equations with periodic boundary conditions is proved. The equations consist of the conservation laws for the electron density and current density for given ion density, coupled to the Poisson equation for the electrostatic potential. The limit is related to the low-mach-number limit of Klainerman and Majda. In particular, the limit velocity satisfies the incompressible Euler equation with damping. The difference to the zero-mach-number limit comes from the electrostatic potential which needs to be controlled. This is done by a reformulation of the equations in terms of the enthalpy, higher-order energy estimates and a careful use of the Poisson equation.

3 Some Reviews on Nonlinear System of Conservation Laws Shuxing Chen Fudan University, P.R.China)

4 Existence of global solutions to regular shock reflection for potential flow Gui-Qiang Chen Northwestern University, USA Mikhail Feldman University of Wisconsin-Madison, USA We prove existence of global solutions to regular shock reflection for potential flow in the case when the wedge half-angle is close to ninety degrees. Presented by Author 2

5 Effects of surface tension on the stability of dynamical liquid-vapor interfaces Heinrich Freistuehler Max-Planck Institute University of Leipzig, Germany Abstract: The persistence of sharp and flat interfaces under multidimensional perturbations is investigated from a viewpoint analogous to Majda s treatment of classical shocks. The main novelty here is that jump conditions for the free interfaces take surface tension into account. This means that, unlike classical jump conditions, they are non-homogeneous, containing first-order and second-order differentials of the front. A normal modes analysis shows that neutral modes may propagate along the front. In the standard setting, this would imply a weak stability result, involving energy estimates with loss of derivatives. In our case the lack of homogeneity of the underlying boundary value problem implies that neutral modes can only be of large enough wave length. Suitable frequency cut-offs then yield energy estimates without loss of derivatives - for the constant-coefficients linearized problem, as in the case of uniformly stable classical shocks. (Joint work with S. Benzoni-Gavage.)

6 Collision potentials and L -stability of Some kinetic equations Seung-Yeal Ha Department of Mathematical Sciences, Korea syha@math.snu.ac.kr In this talk, I will present collision potentials and L -stability for the Boltzmann equation and Vlasov-Poisson system. Collision potentials measures the possible future collisions between particles and are employed to the L -scattering and stability analysis of aforementioned equations.

7 Stability of contact discontinuity for gas motion Feimin Huang Chinese Academy of Sciences, P.R.China Abstract: The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations is a long standing open problem. One of the reasons is that contact discontinuities are linearly degenerate waves in the nonlinear settings, like the Navier-Stokes equations and the Boltzmann equation. The nonlinear diffusion waves generated by the perturbations in soundwave families couple and interact with the contact discontinuity and then cause analytic difficulties. Another reason is that in contrast to the basic nonlinear waves, shock waves and rarefaction waves, for which the corresponding characteristic speeds are strictly monotone, the characteristic speed is constant across a contact discontinuity, and the derivative of contact wave decays slower than the one for rarefaction wave. In this talk, I will present some recent works on the time asymptotic stability of a damped contact wave pattern with an convergence rate for the Navier-Stokes equations and the Boltzmann equations. One of the key observations is that even though the energy estimate involving the lo wer order may grow at the rate (+t) 2, it can be compensated by the decay in the energy estimate for derivatives which is of the order of ( + t) 2. Thus, these reciprocal order of decay rates for the time evolution of the perturbation are essential to close the priori estimate containing the uniform bounds of the L norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern.

8 Weak Shock Mach Reflection John Hunter University of California at Davis, USA Abstract: We will survey weak shock Mach reflection and the von Neumann triple point paradox. We will show numerical solutions (obtained jointly 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 complex flow immediately behind it, thus resolving the triple point paradox. We will also show some very recent experimental results of Skews and Ashworth that support these theoretical results.

9 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 for one-dimensional compressible heat-conducting fluids with centered rarefaction data of arbitrary strength exists globally in time, and moreover, as the viscosity and heat-conductivity coefficients tend to zero, the global solution converges to the center rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.

10 On Strong Convergence in Vortex Sheets Problem for 3-D Axisymmetric Euler Equations Quansen JIU Department of Mathematics, Capital Normal University, Beijing 00037, PRC Zhouping XIN IMS and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong Abstract: In this report, we summarize some of the main results on the structures and convergence properties of the sequences of approximate solutions to the vortex-sheets problem for axisymmetric flows without swirls, in both unsteady and steady case. The main results include two aspects: First, we will show that if the approximate solutions of the 3-D unsteady axisymmetric Euler equations converge strongly (in L 2 -space) over the region outside the symmetry axis, then they must converge strongly in the whole space. This implies if there would appear energy concentration in the process of limit for the approximate solutions, the set of energyconcentration must contain points in the region outside the symmetry axis. There is no restriction on the signs of the initial vorticity here. Second, we will give a strong convergence criterion for the approximate solutions of the 3-D steady axisymmetric Euler equations.

11 Transonic problems for two dimensional self-similar potential flow Eun Heui Kim California State University Long Beach,USA Abstract: In this talk we discuss analytical tools to understand transonic potential flows.

12 Propagation of Viscous Shock Waves Away From Boundary Chiu-Ya Lan National Sun Yat-sen University, Taiwan Abstract: In this talk, I will focus on studying the propagation of shock waves away from the boundary for scalar viscous conservation law. We will introduce an iteration scheme to decouple two effects on solution behavior: the compressibility of the shock and the presence of the boundary. Through this scheme we can obtain pointwise description of the perturbation of the shock profile.

13 Large-Time Behavior of Solutions for the Boltzmann Equation with Hard Potantials Ming-Yi Lee Institute of Mathematics, Academia Sinica, Taiwan Abstract: We study the solutions of the one-dimensional Boltzmann equation for hard potential models with Grad s angular cutoff. The collision frequency ν(ξ) ( + ξ ) γ makes the spectrum of the operator iξ η + L non-analytic in η. We resolve this complication using the real analytic method in the estimates of the fluidlike long waves. We devise a new energy method to account for the sub-exponential behavior of waves outside fluid region.

14 INTRODUCTION TO NONCLASSICAL SHOCKS AND KINETIC RELATIONS PHILIPPE G. LEFLOCH UNIVERSITY OF PARIS 6 & CNRS Abstract. I will review results on undercompressive, nonclassical shock waves and discuss various tools that have been introduced to study the effect of small scales and the selection of physically meaningful, discontinuous solutions. I will explain how to characterize zero diffusion-dispersion limits for hyperbolic systems that are strictly hyperbolic but not globally genuinely nonlinear, and for systems of mixed (hyperbolic-elliptic) type. Solutions typically contain undercompressive shocks or subsonic phase transitions. These waves are fundamental in, for instance, phase transition dynamics (van der Waals fluids, martensitic materials) when both viscosity and capillarity effects come into play. While classical shocks are compressive, independent of small-scale regularization mechanisms, and can be characterized by a single entropy inequality, by contrast nonclassical waves are undercompressive, are very sensitive to diffusive and dispersive mechanisms. The selection of the latter is more delicate and requires an additional jump condition, referred to as a kinetic relation. Among the issues of interest we can mention : the role of the kinetic relation for solving the Riemann problem, the derivation of kinetic relations from a traveling wave analysis, the existence of nonclassical entropy solutions via Glimm scheme and wave front tracking, the uniqueness of admissible solutions in the class of solutions with tame variation, the nucleation criterion for thin films, the design of difference schemes based on entropy conservative flux. References [] P.G. LeFloch, Propagating phase boundaries: formulation of the problem and existence via the Glimm scheme, Arch. Rational Mech. Anal. 23 (993), [2] P.G. LeFloch, Hyperbolic systems of conservation laws: The theory of classical and nonclassical shock waves, Lectures in Mathematics, ETH Zürich, Birkäuser, [3] P.G. LeFloch and M. Shearer, Nonclassical Riemann solvers with nucleation, Proc. Royal Soc. Edinburgh 34A (2004), lefloch@ann.jussieu.fr. Homepage :

15 Quantum hydrodynamics: well-posedness and asymptotics Hailiang Li Department of Mathematics, Capital Normal University, P.R.China hailiang l i@mail.cnu.edu.cn Abstract: We review the recent progress on the quantum hydrodynamical model for charge transport.

16 RECENT DEVELOPMENT OF THE GENERALIZED RIEMANN PROBLEM (GRP) SCHEME JIEQUAN LI We will talk about the recent development of the generalized Riemann problem scheme for hyperbolic balance laws. The concept of Riemann invariants are extensively used in the analytic resolution of the generalized Riemann problem associated with high resolution Godunovtype schemes. The resulting scheme is independent of the Eulerian or Lagrangian formulation of physical systems. The Delicate sonic case are simply treated and the multidimensional extension are straightforward.

17 On Traffic Instabilities Tong Li Department of Mathematics, University of Iowa, USA We will review recent results on traffic instabilities. An innovative approach to traffic dynamics is proposed. The self-organized oscillatory and chaotic behavior of traffic flow are identified and formulated. The results agree with the empirical findings for freeway traffic and with the previous numerical simulations. Thus the work gives a justification for observed and simulated traffic instabilities and some insight into their meanings.

18 The Green s function for one-dimensional Broadwell equations Huey-Er Lin National Taiwan Normal University, Taiwan helin@math.ntnu.edu.tw Abstract: The Broadwell model system is one type of the discrete Boltzmann equation which appears in the kinetic theory for rarified gases. In this talk, we plan to devise a Picard- type iteration and use Fourier method to obtain pointwise estimate for the Green function for the linearized Broadwell model system in one-dimensional space around a local Maxwellian.

19 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 the study of Boltzmann equation in two regards: First, the shock wave theory naturally gives rise to corresponding problems for Boltzmann equation. Some of the techniques for shock wave theory can be applied to Boltzmann equation. Secondly, the understanding in conservation laws helps to highlight the striking differences between fluid mechanics and kinetic theory. This is particularly so for the behavior near solid boundary. We plan to give some examples to exhibit these two regards.

20 ANALYSIS OF QUANTUM HYDRODYNAMIC MODELS Pierangelo Marcati Universita di L Aquila, Italy marcati@univaq.it Abstract: We present the analysis of QHD models arising in the physics of semiconductor devices and regarding models of various condensation phenomena. The analysis regards dissipative and dispersive properties of nonlinear waves. References: []Hailiang Li, Marcati, Pierangelo, Existence and asymptotic behavior of multi-dimensional quantum hydrodynamical model for semiconductors Comm. Math. Phys. 245 (2004), no. 2, [2]Donatelli, Donatella; Marcati, Pierangelo, in preparation

21 On a Shockley-Read-Hall Model for Semiconductors: Convergence to Equilibrium Vera Miljanović, Christian Schmeiser Abstract We are considering a drift-diffusion and a kinetic model for the flow of electrons in a semiconductor crystal, incorporating the effects of recombination-generation via traps distributed in the forbidden band. In mathematical terms, model consists of a reactiondiffusion-convection equation for the electric field and an integro-differential equation for the distribution of occupied traps. We derive formal and rigorous asymptotics, and show convergence.

22 Conservation laws and completely conservative difference schemes for the nonlinear kinetic Landau-Fokker-Planck equation I. F. Potapenko Keldysh Institute for Applied Mathematics, RAS, Department of Kinetic Equations, Miusskaya Pl., 4, Moscow, Russian Federation. The kinetic Landau - Fokker - Planck (LFP) equation is widely used for the description 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 difference 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 distribution function of particles with the power law interaction potentials U = α/r s, where 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. 759.I.F.Potapenko, C.A. de Azevedo. J.of Comp. and Appl. Math.,03(999) 5 2.I.F.Potapenko, A.V.Bobylev, C.A. de Azevedo,A.S. de Assis. Phys.Rev. E 56 (997) 3. A.V.Bobylev, K.Nanbu Phys.Rev. E 6 (2000) 4576

23 Global Attractor for A Nonlinear Thermoviscoelasticity with A Non-convex Free Energy Density Yuming Qin Department of Applied Mathematics, College of Science, Donghua University, Shanghai 20005,P.R. China. yuming@dhu.edu.cn Abstract This paper is concerned with the existence of a global attractor for a semiflow governed by the weak solutions to a nonlinear one-dimensional thermoviscoelasticity with a non-convex free energy density. The constitutive assumptions for the Helmholtz free energy include the model for the study of martensitic phase transitions in shape memory alloys. To describe physically phase transitions between different configurations of crystal lattices, we work in a framework in which the strain u belongs to L. New approaches are introduced and more delicate estimates are derived to establish the crucial L -estimate of strain u in deriving the compactness of the orbit of the semiflow and existence of an absorbing set.

24 Linear and nonlinear viscoelastic models with memory term Bruno Rubino University of L Aquila, Italy rubino@univaq.it In the talk I will recall the nonlinear 3-D viscoelastic model with memory term and consider the Cauchy problem. The assumptions on the memory kernel includes integrable singularities at zero and polynomial decay at infinity. I will give the results of existence and the time decay of the solutions both for the linear and nonlinear problems. Furthermore, I will discuss a preliminary idea to obtain the stability of planar travelling waves.

25 Acceleration Waves and Weaker Kawashima Condition Jie Lou Department of Mathematics, Shanghai University, China Tommaso Ruggeri Research Center of Applied Mathematics (CIRAM), University of Bologna, Italy We consider dissipative hyperbolic systems of balance laws in which a block of equations are conservation laws which arise typically in the Extended Thermodynamics []. In this case, a coupling condition firstly introduced by Kawashima (Kcondition) [2], play a fundamental role for the global existence of smooth solution for small initial data and for the stability of constant state [3]-[5]. Nevertheless the counterexample 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 required only for the genuine non linear characteristic velocities and not for the linear degenerate 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 at least for smooth solution in the class of discontinuity wave (C -piecewise) [7]. References: [] I. Müller, T. Ruggeri, Rational Extended Thermodynamics. 2nd ed. Springer Tracts in Natural Philosophy 37, Springer-Verlag, New York, 998. [2] S. Kawashima, Proc. Roy. Soc. Edimburgh, 06A, 69 (987). [3] B. Hanouzet, R. Natalini, Arch. Rat. Mech. Anal. 69, 89 (2003). [4] W.-A. Yong, Arch. Rat. Mech. Anal. 72 no. 2, 247 (2004). [5] T. Ruggeri, D. Serre, Quarterly of Applied Math., 62(), 63 (2003). [6] Y. Zeng, Arch. Rat. Mech. Anal. 50 no. 3, 225 (999). [7] J. Lou, T. Ruggeri, Acceleration Waves and Weaker Kawashima Condition. Submitted. Presented by T. Ruggeri

26 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 transonic shocks in the steady flows through a two-dimensional nozzle with varing cross-sections. The flow is governed by the steady full Euler equations. We show that the solutions behind the shock front remain subsonic in a downstream region and the shock front is smooth. The problem is approached by a one-phase free boundary problem where the shock front is a free boundary. The steady full Euler equations are decomposed into elliptic equations and a system of transport equations for the free boundary problem(jointwork with G.-Q. Chen and Jun Chen).

27 Hamilton-Jacobi Equations in Infinite Dimension for Approximation of Optimal Control and Hydrodynamical Limits Anders Szepessy Royal Institute of Technology, Sweden Abstract: Optimal control problems for low, d, dimensional differential equations, can be solved computationally by their corresponding Hamilton-Jacobi-Bellman partial differential equation in R d+. I will show how to use Hamilton-Jacobi equations in infinite dimension to find stochastic hydrodynamical limits of microscopic Master equations and to solve inverse optimal design problems for partial differential equations: Master equations. Even small noise can have substantial influence on the dynamics of differential equations, e.g. for nucleation and coarsening in phase transformations. I will present a simple derivation of an accurate model for the noise in macroscopic differential equations, related to phase transformations/reactions, derived from more fundamental microscopic Master equations. Optimal Design Problems. Many inverse problems for differential equations can be formulated as optimal control problems. It is well known that inverse problems often need to be regularized to obtain good approximations. I will presents a method to regularize and establish error estimates for some control problems in high dimension, based on symplectic approximation of the Hamiltonian system for the control problem: Controls can be discontinuous due to a lack of regularity in the Hamiltonian or due to colliding backward paths, i.e. shocks; the error analysis, for both these cases, is based on consistency with the Hamilton-Jacobi-Bellman equation, in the viscosity solution sense, and a discrete Pontryagin principle where the characteristic Hamiltonian system is solved approximately with a R 2 approximate Hamiltonian.

28 Approximation by diffusion and homogenization for semiconductor Boltzmann-Poisson system Mohamed Lazhar TAYEB Faculty of Sciences of Tunis, University of Tunis El-Manar, 060, Tunisia Laboratory of Engineering Mathematics, Polytechnic School of Tunisia. Abstract The subject of the communication is to study the approximation by diffusion and homogenization of semiconductor Boltzmann-Poisson system. Let f ε (t, x, v) a solution of the rescaled Boltzmann equation: t f ε + ε (v. xf ε x Φ ε T. v f ε ) = Q(f ε ) ε 2, (t, x, v) R + ω R d where the collision operator is the linear BGK approximation (Boltzmann Statistics) of electron-phonon collision: Q(f ε )(v) = σ(v, v )(M(v)f(v ) M(v )f(v))dv, R d σ is the cross section and M is the normalized Maxwellian. We are interested in the generalization of results obtained in [2, 3, 5] by coupling diffusion and homogenization and considering the potential: Φ ε T (t, x) := Φ H (t, x, x ε ) + Φε p(t, x) where Φ H is bounded, smooth and Y = [0, ] d )-periodic in y = x/ε and Φ ε P is self-consistent: x Φ ε P = ρ ε = f ε dv, Φ ε P ω = 0. R d In one dimension and when the incoming boundary data are well prepared: f ε b (t, x, v) = ρ b (t, x) exp( v 2 /2 Φ H (t, x, x/ε)), v.n(x) < 0 we prove that the distribution function f ε converges in two scale strong [, 4]: f ε 2 s f ε := ρ(t, x) exp( v 2 /2 + Φ 0 (t, x) Φ H (t, x, y)) where (ρ, Φ P ) is a solution of an homogenized Drift-Diffusion-Poisson system corresponding to an effective potential Φ 0 and a non symmetric diffusion matrix: t ρ + x.j(ρ) = 0, J(ρ) = D(t, x) [ x ρ + ρ( x Φ 0 + x Φ P )], x 2 Φ 2 P = ρ, ρ ω = ρ b D(t, x) = v L (v exp( v 2 /2 + Φ 0 Φ H )dvdy R Y L = v y Φ H y Φ H v Q. Acknowledgments. I would like to thank Naoufel Ben Abdallah and Nader Masmoudi for helpful discussions and encouragements. References [] G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (992), no. 6, [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 [3] N. Ben Abdallah and M. L. Tayeb, Diffusion approximation and homogenization of the semiconductor Boltzmann equation, to appear in SIAM MMS. [4] T. Goudon, F. Poupaud: Approximation by diffusion and homogenization of kinetic equations, Comm. PDE, 26 (200), pp [5] N. Masmoudi and M. L. Tayeb, Diffusion limit for semiconductor Boltzmann-Poisson system, preprint.

29 On a multidimensional model for the dynamic combustion of compressible reacting fluids Konstantina Trivisa University of Maryland USA trivisa@math.umd.edu Abstract: In this work we present a multidimensional model for the dynamic combustion of compressible reacting fluids formulated by the Navier Stokes equations in Euler coordinates. For the chemical model we consider a one way irreversible chemical reaction governed by the Arrhenius kinetics. The existence of globally defined weak solutions of the Navier-Stokes equations for compressible reacting fluids is established by using weak convergence methods, compactness and interpolation arguments in the spirit of Feireisl and P.L. Lions. This is joint work with D. Donatelli

30 L Stability of Shock Waves in the Keyfitz-Kranzer System Huiying Wang Zhejiang University, P.R.China Abstract: In this paper, we study the long time L stability of the viscous shock wave solution of a 2 2 system the so-called Keyfitz- Kranzer system. We do not use the pointwise semigroup method, but only use the elementary analysis method combined with the lap number theory, and establish the purely L stability result without restraint on the shock strength and the size of the initial perturbation, much like in the scalar case.

31 On Multi-Dimensional Transonic Shock Waves in A Nozzle by Zhouping Xin The IMS and Department of Mathematics The 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 shock waves 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 transonic shock wave patterns in a nozzle with incoming supersonic flow and a given subsonic pressure at the exit. The existence and stability of such transonic wave patterns will be discussed. This is joint work with Huicheng Yin. W orkinthistalkwassupportedinpartbygrantsfromtheresearchgrantscouncilofhongkongspeciladministrativeregioncuhk4028/04p,cuhk404

32 CAUCHY PROBLEM FOR VLASOV-POISSON-BOLTZMANN SYSTEM TONG YANG The dynamics of the dilute electrons can be modelled by the fundamental Vlasov-Poisson-Boltzmann system when the electrons interact with themselves through collisions in the self-consistent electric field. In this talk, we will present the result on the Cauchy problem for this system which shows that any smooth perturbation of a global Maxwellian leads to a unique global-in-time classical solution. A convergence rate in time will also be given. This is a joint work of Huijiang Zhao which generalizes our previous result on this problem together with Hongjun Yu. Finally, we will also mention our current work on the perturbation around a stationary solution. Department of Mathematics, City University of Hong Kong address: matyang@cityu.edu.hk

33 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 constant states which are separated by a smooth curve or surface. Nonselfsimilar M-D elementary waves, their new structures and properties are disclosed, for example the contour surface of M-D rarefaction wave is a family of cylindrical surface etc., which are essentially different from that in M-D selfsimilar case. Furthermore, global solutions of a class of 2-D systems of conservation laws will be also presented and are formulated by implicit function, their structure combining 2-D non-selfsimilar elementary waves and non-constant intermediate states will be shown.

34 AGlobal singularity structures of weak solutions to the semilinear dispersive wave equations Huicheng Yin Dept. of Math. & IMS, Nanjing University, Nanjing 20093, China In this talk, we are concerned with the global singularity structures of weak solutions to the semilinear dispersive wave equations whose initial data are chosen to be discontinuous on the unit sphere. Combining Strichartz s inequality with the commutator argument techniques, we show that the weak solutions are globally C 2 regular away from the focusing cone surface x = t and the outgoing cone surface x = t +.

35 Fundamental Solutions of Hyperbolic-Parabolic Systems and Shock Wave Stability Tai-Ping Liu Stanford University, USA Yanni Zeng University of Alabama at Birmingham, USA We construct the fundamental solution for a general hyperbolic-parabolic system of conservation laws along a weak shock profile. Our formulation has explicit dependence on the shock strength. This allows us to perform nonlinear stability analysis for the shock wave, and obtain detailed asymptotic behavior of the solution. The result applies to compressible Navier-Stokes equations and the magnetohydrodynamics, even in the case of having multiple eigenvalues in the transversal fields. Presented by Author 2

36 Steady Supersonic Flow Past a Curved Cone Yongqian Zhang School of Mathematical Sciences, Fudan University, Shanghai, , 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 flow past an infinite cone. It has been studied by Lien and Liu, Chen, Xin and Yin when the open angle of the cone is sufficiently small. We relax the restriction on the sharp angle. By obtaining more delicate estimates for the interaction of the elementary waves, we get the existence result for the cone with the open angle less than a critical value.

37 A Case Study of Global Stability of Strong Rarefaction Waves for 2 2 Hyperbolic Conservation Laws with Artificial Viscosity Huijiang Zhao Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences P.O. Box 700, Wuhan 43007, China Abstract This paper is concerned with global stability of strong rarefaction waves for a class of 2 2 hyperbolic conservation laws with artificial viscosity. It is based on a recent work joint with Dr. Ran Duan.

38 Two-dimensional regular shock reflection on a wedge Yuxi Zheng Penn. State University, USA yzheng@math.psu.edu I will present recent work on the regular shock reflection on a wedge for the adiabatic Euler system and the pressure gradient system.

39 GLOBAL CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS WITH WEAK LINEAR DEGENERACY YI ZHOU Consider the following Cauchy problem for the first order quasilinear strictly hyperbolic system u t + A(u) u x = 0 t = 0 : u = f(x). We let M = sup x R f (x) < +. The main result of this paper is that under the assumption that the system is weakly linearly degenerated, there exists a positive constant ε independent of M, such that the above Cauchy problem admits a unique global C 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 equations when the inhomogenous term satisfying a matching condition by Wu.