Stability of Mach Configuration

Transcription

1 Stability of Mach Configuration Suxing CHEN Fudan University We prove the stability of Mach configuration, which occurs in moving shock reflection by obstacle or shock interaction for compressible flow. The compressible flow is described by full steady Euler system of gas dynamics. The unperturbed Mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity. Among four regions divided by these four lines in the neighborhood of the intersection, two are supersonic region, and other two are subsonic regions. We prove that if the constant states in the supersonic region are slightly perturbed, then the structure of the whole configuration holds, while the other two shock fronts and the slip line, as well as the flow field in the subsonic region, are also slightly perturbed. Such a conclusion asserts the stability of the Mach configuration in shock theory. In order to prove the result we reduce the problem to a free boundary value problem, where two unknown shock fronts are free boundary, while the slip line is transformed to a fixed line by a Lagrange transformation. In the region where the solution is to be determined we have to deal with an elliptic-hyperbolic composed system. By decoupling this system and combining the technique for both hyperbolic equation and elliptic equation we establish the required estimates, which are crucial in the proof of the existence of solution to the free boundary value problem.

2 The study on kinetic equations Ling HSIAO Institute of Mathematics and System Sciences, Chinese Academy of Science Several kinetic equations will be discussed,such as Boltzmann equation,relativistic Boltzmann equation,landau equation,relativistic Landau equation,...the emphasis is on Boltzmann and Relativistic Boltzmann equations in this talk.

3 L p L q decay estimates for linear type-2 models of thermoelasticity with lower order terms Kay JACHMANN Institute of Applied Analysis, TU Bergakademie Freiberg, Germany jachmann@math.tu-freiberg.de We are interested in studying so-called type-2 models of thermoelasticity, introduced by Green & Naghdi in 1993, both in the one- and three-dimensional linear case. They describe a thermoelasticity without energy dissipation. After studying some properties for solutions to the Cauchy problems for the type-2 models without any lower order terms, some results concerning L p L q decay estimates for such type-2 models with an additional dissipation and an additional mass term will be discussed. While we can apply a diagonalization technique to prove parabolic decay rates both in the one- and three-dimensional case for solutions to the Cauchy problem of the type-2 model with an additional dissipation term, we will have to apply the stationary phase method to deal with the case of an additional mass term. The latter is the more interesting problem and much harder to deal with, and thus the emphasis of the talk will be put on this.

4 Asymptotics of a kinetic model for granular flow Hailiang LI Capital Normal University, Beijing hailiang li@mail.cnu.edu.cn In this talk, we consider the uniqueness and long time asymptotics of weak solutions of the model.

5 Some understanding about pseudo-steady fluid flows Jiequan LI Capital Normal Universiey,Beijing We will show our recent understanding about the pseudo-steady compressible fluid flows, which have self-similar solutions.

6 Linear Lagrangian systems of conservation laws Yue-Jun PENG Laboratoire de Mathématiques, CNRS UMR 6620 Université Blaise Pascal (Clermont-Ferrand 2), Aubière cedex, France We introduce a system of conservation laws via Euler-Lagrangian change of variables in one space dimension. For such a system supposed to be hyperbolic, we give a precise description of its mathematical structure on the eigenvalues, linear degeneracy of the characteristic fields, classical Riemann invariants and entropy-flux pairs. Existence and uniqueness of entropy solutions of Cauchy problem are proved and their explicit expressions are given. Furthermore, we show that the entropy solution is weakly stable in bounded space of functions and satisfies entropy equalities for all entropy-flux pair of the system. Applications concern the Chaplygin gas dynamics and Born-Infeld equations.

7 A cartoon for climbing ramp problem of a shock and von Neumann paradox Wancheng SHENG Shanghai University wcsheng@mail.shu.edu.cn Climbing ramp problem of a shock for two dimensional scalar conservation laws are considered. By use of the generalized characteristic analysis method, we prove constructively that there are only three configurations of solutions: regular reflectionlike, Mach reflection-like and von Neumann reflection-like for a kind of flux functions of the conservation laws and the states at wave front and back of the shock. The criteria among these three configurations are obtained. Furthermore, a necessary condition of appearance of regular reflection in gas dynamics is delivered.

8 The Boltzmann Equation in the Space L 2 L β Seiji UKAI 1 and Tong YANG Department of Mathematics and Liu Bie Ju Centre for Mathematical Sciences City University of Hong Kong mcukai@cityu.edu.hk We present a function space in which the Cauchy problem for the Boltzmann equation is well-posed globally in time near an absolute Maxwellian in a mild sense without any regularity conditions. The asymptotic stability of the absolute Maxwellian is also established in this space and, moreover, it is shown that the higher order spatial derivatives of the solutions vanish in time faster than the lower order derivatives. No smallness assumptions are imposed on the derivatives of the initial data, and the optimal decay rates are derived. Furthermore, the Boltzmann equation with a time-periodic source term is solved in the same space on the unique existence and stability of a time-periodic solution which has the same period as the source term. The proof is based on the spectral analysis of the linearized Boltzmann operator. 1 Presenting author

9 The ring blow up solutions of the nonlinear Schrödinger equation Xiao-Ping WANG Hong Kong University of Science and Technology We present some results on a new type of singular solutions of the critical and supercritical nonlinear Schrödinger equation, that collapse with a quasi self-similar ring profile at a square root blowup rate. We find and analyze the equation of the ring profile. We observe that the self-similar ring profile is an attractor for a large class of radially-symmetric initial conditions, but is unstable under symmetry-breaking perturbations. The equation for the ring profile admits also multi-ring solutions that give rise to collapsing self-similar multi-ring solutions, but these solutions are unstable even in the radially-symmetric case, and eventually collapse with a single ring profile.

10 Fluid Dynamic Equations and Multi-scale Analysis Shu WANG 1 College of Applied Sciences, Beijing University of Technology wangshu@bjut.edu.cn In this talk asymptotic limits and multiscale analysis problems of some macroscopic Fluiddynamic PDEs are studied. These asymptotic limits contain quasineutral limit and nonrelativistic limit etc while the nonlinear Fluiddynamic PDEs concern some models like Euler-Maxwell system, e-mhd system, Euler-Poisson system, Navier-Stokes-Poisson system and Drift-Diffusion system etc, widely used in the area of applied sciences such as semiconductors, plasmas, fluid dynamics and so on, and they are full of multiscale phenomena. Some formal and rigorous convergence results are given and some new methods or ideas are reviewed. 1 This work is partially supported by the Program for New Century Excellent Talents in University, by the NSFC(Grant no ), by Beijing Natural Science Foundation (Grant no ).

11 Optimal convergence rates for the compressible Navier-Stokes equations with potential forces Tong YANG City University of Hong Kong By combining the L p L q estimates on the solutions to the linearized compressible Navier-Stokes equations and the energy method, in this talk, we will present some recent results on the optimal convergence rates of the solutions to the Navier-Stokes equations with potential force to the stationary solutions in the whole space. The analysis will also lead to the ongoing study on the Boltzmann equation. This is a joint work with Renjun Duan, Seiji Ukai and Huijiang Zhao.

12 Existence of travelling wave solutions for hyperbolic systems of balance laws Wen-An YONG Tsinghua University This work is concerned with traveling wave solutions for hyperbolic systems of balance laws satisfying a stability condition and a Kawashima-like condition. We are interested in the case where the traveling wave equations have a singularity, which is absent for 2 2-systems satisfying the two conditions. To deal with the singularity, we reduce the problem to a parametrized one without singularity by using the center manifold theorem. For the parametrized problem, we prove the existence of solutions by modifying an existing argument in the literature. In this way, we show the existence of traveling wave solutions.

13 Global Existence of Classical Solutions to the Vlasov-Poisson-Boltzmann System Huijiang ZHAO Wuhan University The time evolution of the distribution function for the charged particles in a dilute gas is governed by the Vlasov-Poisson-Boltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this talk, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible Navier-Stokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwallian introduced in Liu-Yang-Yu (Physica D., 2004) and elaborated in Yang-Zhao (J. Math. Phys.,2006).

14 On the global existence and uniqueness of solutions to Prandtl s system Xinying XU and Junning ZHAO 1 Xiamen University jnzhao@jingxian.xmu.edu.cn In this paper, we consider the prandtl system for the non-stationary boundary layer in the vicinity of a point where the outer flow has zero velocity. It is assumed that U(t, x, y) = x m U 1 (t, x), where 0 x L and m 1. We establish the global existence of the weak solution to this problem. Moreover the uniqueness of the weak solution is proved. 1 Presenting author

15 Local exact boundary controllability for nonlinear wave equation Yi ZHOU Fudan University This paper deals with the local exact boundary controllability for the dynamics governed by nonlinear wave equation, subject to Dirichlet or Neumann or any other kind of boundary controls which result in wellposedness of the initial-boundary value problem. A constructive method is developed. The local exact boundary controllability for semi-linear wave equation is constructed in the case of both three (odd) and two (even) space dimensions, and the boundary control is time optimal when the space dimension is three (odd). Especially, the local exact boundary controllability is established for quasi-linear wave equation in several space dimensions by using the constructive method.