On the asymptotics of the Boltzmann equation and fluid-dynamic limits

Transcription

1 On the asymptotics of the Boltzmann equation and fluid-dynamic limits Kazuo Aoki Department of Mechanical Engineering and Science, Kyoto University, Kyoto , Japan We consider a rarefied gas in the near continuum regime (i.e., small mean free paths) and discuss the formal asymptotics and fluid-dynamic limits of the Boltzmann equation based on the Hilbert-type expansion. In general, it is believed that the Hilbert Expansion leads to the (compressible) Euler equations at the leading order, and the higher-order equations are the linear and inhomogeneous equations of the Euler type. However, if we consider steady boundary-value problems, different-types of equations are obtained depending on the physical situation under consideration. We will show an example giving the (compressible) Navier-Stokes equations at the leading order (cylindrical Couette flow) and another example giving the equations containing non-navier-stokes stress terms as the leading-order equations (thermal creep flow caused by large temperature variations). In this connection, the inverted velocity profile in the former example and the ghost effect in the latter will be discussed.

2 Partial convexity for partial differential equations Baojun Bian Department of Mathematics, Tongji University, P.R.China In this talk, we consider partial convexity for solution of partial differential equations. We establish a microscopic partial convexity principle for partially convex solution of nonlinear elliptic and parabolic equations. As application, we discuss the partial convexity preserving of solution for parabolic equations. This talk is based on the joint works with Pengfei Guan.

3 Renormalized Resonances and Turbulence in Nonlinear Dispersive Waves David Cai Shanghai Jiao Tong University, China We discuss the non-perturbative nature of wave turbulence in equilibrium in strongly nonlinear regimes and show how the wave spectrum of nonlinear dispersive waves is determined by an intertwining self-consistent renormalization process. We demonstrate that nonlinear wave interactions renormalize the dynamics, leading to (i) a drastic deformation of the resonant manifold even at weak nonlinearities, and (ii) the creation of nonlinear resonance quartets in wave systems for which there would be no resonances as predicted by the linear dispersion relation. Finally, we present an extension of the weak turbulence kinetic theory to systems with strong nonlinearities.

4 Heat Flow Method in the Critical Point Theory Chang Kung Ching School of Math. Science, Peking University, China We study the heat semi-flow for elliptic problems with variational structure, for example, the harmonic map problem, the minimal surfaces problem, the prescribing mean curvature problem as well as some semilinear problems. In some cases when the Palais Smale condition fails, it is used as a replacement of the pseudo gradient flow in deforming the level sets of the energy functionals, so that the Morse theory is established.

5 Vanishing Viscosity Limit for Nonlinear Conservation Laws Gui-Qiang G. Chen University of Oxford and Fudan University The vanishing viscosity limit is one of the most classical, longstanding fundamental issues in the theory of nonlinear conservation laws. In this talk, we will discuss some of old and recent developments in the study of this issue. These especially include the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible flow, among others.

6 Weighted Sobolev Spaces and Semilinear Degenerated Elliptic Equations on Manifolds with Conical Singularities Hua Chen Wuhan University, China In this talk, we would report some recent results on existence of solutions for a class of semilinear degenerated elliptic equations on manifolds with conical (or edge) singularities.

7 Large time behavior of some degenerately dissipative systems with the electromagnetic field Renjun Duan The Chinese University of Hong Kong, China This talk is concerned with the large time behavior of solutions to the Cauchy problem on systems of some fluid or kinetic equations under the self-consistent electromagnetic field satisfying the Maxwell equations. Different from previous work about the study of systems of hyperbolic-parabolic type under the Shizuta- Kawashima condition, we establish a new dissipative structure of the considered degenerately dissipative linearized systems due to appearance of the coupled hyperbolic Maxwell equations. A typical feature for such systems is that solutions over the high frequency domain can decay in time with polynomial rates up to any order depending on regularity of initial data. This is a partially joint work with S. Kawashima and Y. Ueda.

8 Scattering for the focusing energy-subcritical Nonlinear Schrödinger equations Daoyuan Fang Department of Mathematics, Zhejiang University, Hangzhou , China For the 3D focusing cubic nonlinear Schrödinger equation, Scattering of H 1 solutions inside the (scale invariant) potential well was established by Holmer and Roudenko (radial case) and Duyckaerts, Holmer and Roudenko (general case). In this paper, we extend this result to arbitrary space dimensions and focusing, masssupercritical and energy-subcritical power nonlinearities, by adapting the method of the authors before. This is a joint work with Thierry Cazenave and Jian Xie.

9 Regularity for the parabolic obstacle problem with fractional Laplacian Alessio Figalli Department of Mathematics, The University of Texas at Austin, Austin TX 78712, USA. In recent years, there has been an increasing interest in studying constrained variational problems with a fractional diffusion. One of the motivations comes from mathematical finance: jump-diffusion processes where incorporated by Merton into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock s prices, in contrast with the classical lognormal diffusion model of Black and Scholes. These models allow to take into account large price changes, and they have become increasingly popular for modeling market fluctuations, both for risk management and option pricing purposes. In a joint paper with Luis Caffarelli we study the parabolic version of the fractional obstacle problem, i.e. where the elliptic part of the operator is given (at least at the leading order) by a fractional laplacian. We prove optimal spatial regularity and almost optimal time regularity of the solution, recovering in particular the optimal regularity for the stationary case. To obtain this result, we crucially exploit the fact that the solution coincides with the obstacle at the initial time, which corresponds to the fact that (for the backward operator) the stock s price coincides with the payoff at the final time.

10 Exact boundary controllability of nodal profile in a tree-like network Qilong Gu Shanghai Jiao Tong University, China ql gu@yahoo.com.cn We give the definition of a kind of exact boundary controllability, which is called the exact boundary controllability of nodal profile. In a tree-like network, we show the basic principles of getting the controllability for general first order quasilinear hyperbolic systems in dimension 1 with nonlinear boundary conditions and interface conditions. We give the conditions of compatibility of giving nodal profiles, and the method to find the controls corresponding to the giving nodal profiles.

11 Compressible Euler equations with damping Feimin Huang Chinese Academy of Sciences, China In this lecture, I will introduce the recent progress on the large time behavior of entropy solutions of compressible Euler equations with damping. It is shown that any bounded entropy solutions time asymptotically tend to the Barenblatt solution in L 1 norm when the initial mass is finite.

12 Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations Song Jiang, Chunhui Zhou Institute of Applied Physics and Computational Mathematics, Beijing, China We prove the existence of a spatially periodic weak solution to the steady compressible isentropic Navier-Stokes equations in IR 3 for any specific heat ratio γ > 1. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in new higher integrability of the density, and the method of weak convergence.

13 Vanishing viscosity problem for 3-D Navier-Stokes equations with helical symmetry Quansen Jiu School of Mathematical Sciences, Capital Normal University, Beijing ,PRC In this talk, we discuss vanishing viscosity problem for the 3-D Navier-Stokes equations with helical symmetry. We make an appropriate decomposition of the helical vector fields to obtain the uniform estimates of the vorticity and furthermore establish the convergence of the solutions of the Navier-Stokes equations with helical symmetry to the ones of the corresponding Euler equations as the viscosity vanishes. Both the bounded and whole space cases are discussed. This is joint with Milton C. Lopes Filho, Helena J. Nussenzveig Lopes and Dongjuan Niu.

14 L p estimate of the Stokes equations Dongsheng Li Xi an Jiao Tong University, China lidsh@mail.xjtu.edu.cn In this talk, we will give a new proof of the L p estimate of the Stokes equations.

15 Why do you study 2-D Riemann problems for Euler equations? Jiequan Li School of Mathematical Sciences, Beijing Normal University, , China li Two-dimensional (2-D) Riemann problems for compressible fluid flows assume the simplest initial state but provide the most fundamental wave configurations, including the dam collapse, the reflection of oblique shocks and vortex-shock interaction etc. Mathematically they are formulated as mixed-type boundary value problems for most cases. In this talk I discuss the mission of 2-D Riemann problems, main mathematical difficulties and potentially accessible problems for analysts.

16 Stability of Large-amplitude Traveling Waves Arising from Chemotaxis Tong Li University of Iowa USA Traveling wave (band) behavior driven by chemotaxis was observed experimentally by Adler and was modeled by Keller and Segel. We establish the existence and the nonlinear stability of large-amplitude traveling wave solutions to a system of nonlinear conservation laws which is derived from the well-known Keller-Segel model describing cell(bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. This is a joint work with Zhi-an Wang.

17 On the Boltzmann Equation for Bose-Einstein Particles Xuguang Lu Tsinghua University, China Although physical experiments show that the Bose-Einstein condensation occurs in finite time, this has not been clear for the present Boltzmann equation: Does there exist a mass conserved solution of the equation such that a condensation occurs in finite time? To this problem there has been no definite answer even in a formal level. In this talk we will present some recent results, including a theorem of alternative ( concentration or oscillation ) and further singular properties of the EMV solution, and we will also show that this problem of finite time condensation relies heavily on the grazing effect: if the collision kernel is given an angular-cutoff, there will be no condensation in finite time, which is very different from the case of long time behavior where the condensation always happens as time goes to infinity.

18 Dynamical Theory for Oceanic Thermohaline Circulation Tian Ma Sichuan University, China The oceanic thermohaline circulation is a remarkable phenomenon in Oceanography, which is one of key sources to influence the global climate. Recently,we establish a set of dynamic transition theory for the oceanic thermohaline circulation, which can help us to understand well the dynamical behavior of this natural phenomenon. In this report, we shall give a briefly introduction to this theory.

19 The spacetime convexity of the solutions of parabolic equation Xinan Ma Chinese University of Science and Technology, China We study the spacetime convexity of the solution of parabolic equation, we use the technique of constant rank lemma. We shall give some applications.

20 Smoothing effect of weak solutions for spatially homogeneous non-cutoff Boltzmann equation Yoshinori Morimoto Kyoto University, Japan In this talk we consider the Cauchy problem for the spatially homogeneous Boltzmann equation without angular cutoff t f(t, v) = Q(f, f)(t, v), t R +, v R 3, f(0, v) = f 0 (v), where f(t, v) 0 is the distribution of particles at time t with velocity v. right hand side of the equation is given by the Boltzmann bilinear collision operator Q(g, f) = B (v v R S 3 2, σ) {g(v )f(v ) g(v )f(v)} dσdv, where v = v+v 2 + v v 2 σ, v = v+v 2 v v 2 σ for σ S 2. We assume the collision cross section B(v v, cos θ) has the form B = Φ( v v )b(cos θ), cos θ = v v v v σ, 0 θ π 2, where the kinetic factor Φ( v v ) = v v γ and the angular factor containing a singularity, for some constant K > 0. b(cos θ) Kθ 2 2s when θ 0+, 0 < s < 1, The In the case γ + s > 0, we show any weak solution having the finite energy and entropy lies in the Schwartz space if it has the mass conservation and the moments of arbitrary order. The main ingredients of the proof are the suitable choice of the mollifiers composed of pseudo-differential operators, and sharp estimates of the commutators of mollifiers and the Boltzmann operator whose cross section has singularities in both kinetic and angular factors. The content of this talk is based on the theory of non-cutoff Boltzmann equation developed in the joint-works with R. Alexandre, S. Ukai, C.-J. Xu and T. Yang since 2006.

21 Diffusion and Directed Movements in Heterogeneous Environments Wei-Ming Ni East China Normal University and University of Minnesota In this talk I will use the Lotka-Volterra competition system to illustrate first the interaction between diffusion and spatial inhomogeneity, and then incorporate directed movements into consideration.

22 On a Quasilinear System Involving Curl Xingbin Pan East China Normal University, China xbpan@math.ecnu.edu.cn This talk concerns a quasilinear system in a three dimensional domain which involves curl. This system describes the Meissner states of type II superconductors. We shall see that this system has many Meissner solutions, and we shall examine the convergence of the Meissner solutions to a solution of the limiting system as the Ginzburg-Landau parameter increases to infinity.

23 Large-time Behavior of Solutions to the Inflow Problem of Full Compressible Navier-Stokes Equations Xiaohong Qin Nanjing University of Science and Technology, China Large-time behavior of the solutions to the inflow problem of full compressible Navier-Stokes equations is considered on the half line R + = (0, ). First, we give the existence (or non-existence) of the boundary layer solution to the inflow problem when the right end state (ρ +, u +, θ + ) belongs to the subsonic, transonic and supersonic regions, respectively. Then some wave structures involving the the boundary layer solution (subsonic and transonic cases), the viscous contact wave and the rarefaction waves to the inflow problem are described and the asymptotic stability of these wave patterns are proved under some smallness conditions. The proofs are given by the elementary energy method based on underlying wave structures.

24 Uniform attractors for a 3D non-autonomous Navier-Stokes-Voight Equations Yuming Qin Dong University, China, yuming qin@hotmail.com This work is jointly with Xin-guang Yang and Xin Liu. In this paper, under suitable assumptions on the external force f and initial data u τ, we prove the global existence of solutions and the existence of the uniform attractor for a 3D non-autonomous Navier-Stokes-Voight equations u t ν u α 2 u t + (u )u + p = f(t, x), x Ω, t R, u = 0, x Ω, t R, u(t, x) Ω = 0, u(τ, x) = u τ (x), τ R, by establishing the asymptotical compactness or uniform condition-(c).

25 Two-dimensional Riemann problems for compressible Euler equations and Zheng s patch Wancheng Sheng Department of Mathematics, Shanghai University Shanghai, P.R.China mathwcsheng@shu.edu.cn In this talk, I will show you the numerical simulation on Two-dimensional Riemann problems for compressible Euler equations by use of numerical generalized characteristic analysis method. Some building blocks, such as Zhengs Patch (semihyperbolic patch), Shock Reflection, Richtmyer-Meshkov Instability, etc., are shown. (Jointed with G.D.Wang and T.Zhang)

26 Interplays between self-diffusion, cross-diffusion and logistic source in chemotaxis systems Youshan Tao Department of Applied Mathematics, Dong Hua University, Shanghai , P. R. China This talk mainly addresses parabolic-parabolic chemotaxis system. Some previous results on this system are firstly reviewed. Then, two recent works are reported: One studies the interaction between nonlinear diffusion, chemotaxis and logistic dampening; the other discusses the interplay between self-diffusion and crossdiffusion. Finally, this talk is closed with two potentially very interesting models: a combined chemotaxis-haptotaxis model describing cancer invasion with tissue remodeling, and a coupled chemotaxis-fluid model reflecting the motion of oxygendriven swimming bacteria in an incompressible fluid; the former was initially proposed by Chaplain and Lolas (M3AS 2005), and the latter was originally developed by Tuval et al. (PNAS 2005). Two recent results reported in this talk are joint works with Michael Winkler (University of Duisburg-Essen, Germany).

27 On a 3D Model for the Incompressible Euler and Navier-Stokes Equations Shu Wang College of Applied Sciences, Beijing University of Technology, P.R.China wangshu@bjut.edu.cn In this talk, we will discuss some properties of the incompressible Euler and Navier-Stokes equations by studying a 3D model for axisymmetric 3D incompressible Euler and Navier-Stokes equations with swirl. The 3D model is derived by reformulating the axisymmetric 3D incompressible Euler and Navier-Stokes equations and then neglecting the convection term of the resulting equations. Some properties of this 3D model are reviewed. Some potential features of the incompressible Euler and Navier-Stokes equations discussed here includes the stabilizing effect of the convection, the effects of the sign of the vorticity and the boundary of the fluids on globally dynamic stability, the role of the dimension of the space to capture the finite time blowup of fluids.

28 Some analysis of contact angle hysteresis Xiao-Ping Wang Hong Kong University of Science and Technology, China We analyze the wetting hysteresis on rough and chemically patterned surfaces from a phase-field model for immiscible two phase fluid. By matched asymptotic analysis of the model, we derive some equations that describe the structure of the interface on the chemically patterned surface. These equations describe directly the the contact angle hysteresis and force hysteresis. The limit can also be justified from the theory of Gamma convergence.

29 Some results on control problems of 1-D hyperbolic systems Zhiqiang Wang Fudan University, China We first present a result on boundary controllability of 1-D hyperbolic system with a vanishing characteristic speed (joint work with Jean-Michel Coron and Olivier Glass). Then we deal with a conservation law with nonlocal velocity which models a highly re-entrant manufacturing system (joint work with Jean-Michel Coron and Matthias Kawski). For this nonlocal model, we show some results on controllability and optimal control problems.

30 On Compressible Navier Stokes Systems Zhouping Xin The Chinese University of Hong Kong, Hong Kong, China

31 Microlocal analysis of Kinetic equations Chao-Jiang XU Wuhan University, China In this talk, we establish the hypoellipticity of the non homogeneous Boltzmann equation without angular cutoff. By using the nonlinear microlocal analysis, we can study this equation as a generalized Kolmogrove equation which is non linear and anisotropic. The key step to obtain the regularizing effect is a generalized version of the uncertainty principle, it is a very strong results of microlocal analysis, from which we proved the hypoellipticity of a transport equation.

32 Semi-hyperbolic patches in transonic flows Hongmei Xu Hohai University, China xxu We study the time asymptotic behavior of solutions to the nonlinear wave equation with viscosity in even multi spatial dimension. Our study is based on the detailed analysis of the Green function of the linearized system. This is used to study the coupling of nonlinear diffusion waves. Time asymptotic shape of the solutions are obtained and shown to exhibit the generalized Huygen principle.

33 Fluid Dynamic Limits of Boltzmann Equation to Riemann Solutions of Euler Equations Tong Yang Department of mathematics, City university of Hong Kong, Hong Kong, P.R. China Fluid dynamic limit to the compressible Euler equations from the Boltzmann equation is a problem with long history. Under the assumption of slab symmetry, even though intensive studies have been made when the solution of the Euler equations has non-interacting single waves, the problem on the genuine Riemann problem is still unsolved let alone the general weak solutions. In this talk, we present some recent results on this problem when the Riemann solution is superposition of either shock-rarefaction wave or rarefaction wave-contact discontinuity. Convergence rates in terms of Knudsen number is also given. The follows from some recent joint works with Feimin Huang and Yi Wang.

34 Variational Methods for Real Ultrasound Image Despeckling and Segmentation Xiaoping YANG School of Science, Nanjing University of Sci. and Tech., Nanjing, , China Speckle appears in all conventional ultrasound images, and it generally tends to reduce the image resolution and contrast, thereby reducing the diagnostic value of the imaging modality. In this talk, we focus on real ultrasound image despeckling and segmentation. We discuss some corresponding variational models. The existence and uniqueness of minimizers of the variational problems and the associated evolution problems are studied. We also show the capability of our models on some numerical experiments.

35 Waiting Time for a non-newtonian Polytropic Filtration Equation with Convection Yin Jingxue South China Normal University, P.R.China yjx@scnu.edu.cn This is a joint work with Professor Yang Tong and Doctor Jin Chunhua. We discuss the waiting time phenomena for a class of non-newtonian polytropic filtration equation with convection. According to the influence of the convection on the waiting time property, we divide the convection into three cases, that is the strong convection, the mild convection and the weak convection. For different cases, the sufficient and necessary conditions on the initial data are given for the existence of waiting time respectively.

36 Nonlinear stability of rarefaction waves to the Landau equation Hongjun Yu School of Mathematical Sciences, South China Normal University, Guangzhou, China The Landau equation, which was proposed by Landau in 1936, is a fundamental equation to describe collisions among charged particles interacting with their Coulombic force. Rarefaction waves in gas dynamics can be described by the Euler, Navier-Stokes, Boltzmann or Landau equations. In this paper we show that rarefaction waves for the Landau equation are time asymptotic stable and tend to the rarefaction waves of the Euler and Navier-Stokes equations for the first time. The method combines the analytic techniques for viscous conservation laws, properties of Burnett functions and energy method through the micro-macro decomposition of the Landau equation. Our result also covers a class of generalized Landau equations, include Coulomb potential case, which describes grazing collisions in a dilute gas.

37 Construction of Green s functions for the Boltzmann equations Shih-Hsien Yu National University of Singapore, Singapore matysh@nus.edu.sg In this talk, we will review the development on constructing Green s functions of linearized Boltzmann equation around a global maxwellian and Boltzmann shock profile and its applications to various problems.

38 Well-posedness of Hydrodynamics on the Moving Elastic Surfaces Pingwen Zhang School of Mathematical Sciences, Peking University, Beijing , China The dynamics of a membrane is a coupled system of a moving elastic surface and an incompressible membrane fluid. The difficulties in analyzing such a system include the nonlinearity of the moving curved space (geometric nonlinearity), the nonlinearity of the elastic force (elastic nonlinearity) and the nonlinearity of the fluid dynamics (fluid nonlinearity). The coupled system includes parabolic equations, wave equations and elliptic equations, but energy dissipation is satisfied. Here we prove the local existence and uniqueness of the solution to the coupled system by getting regularity through reformulating the system into a new system in the isothermal coordinates and constructing a suitable iterative scheme. This work is joined with Wei Wang and Zhifei Zhang.

39 One-dimensional Compressible Navier-Stokes Equation with Large Density Variation Huijiang Zhao School of Mathematics and Statistics, Wuhan University, China This talk is concerned with one-dimensional compressible Navier-Stokes equation with large density variation. It is based on recent works joint with Lili Fan, Hongxia Liu and Tao Wang.

40 Semi-hyperbolic patches in transonic flows Yuxi Zheng Yeshiva University and Penn State University, U.S.A. In construction of solutions to the self-similar Euler systems for compressible gases in two space dimensions, one finds domains of solutions whose characteristics are trapped away from given boundary or initial conditions. These domains are what we call semi-hyperbolic patches. These patches are quite common in solutions to the Riemann problems. They are related to solutions to the classical Tricomi problem or Keldysh problem. We show a couple of ways to construct these solutions through nonlinear methods. This talk is based on joint work with Tong Zhang, Jiequan Li, Xiaomei Ji, Mingjie Li, Zhicheng Yang, Xiao Chen and Zhen Lei.

41 On Non-degeneracy of Solutions to SU(3) Toda System Feng Zhou East China Normal University, China In this talk, we discuss the solution to the following SU(3) Toda system u + 2e u e v = 0, v e u + 2e v = 0 in R 2, e u <, e v <, R 2 R 2 We prove that it is nondegenerate, i.e., the kernel of the associated linearized operator is exactly eightdimensional. This is a joint work with J.C.Wei and C.Y.Zhao.

42 Entropy and renormalized solutions for quasilinear elliptic (parabolic) equations with L 1 data Shulin Zhou Peking University, CHANA szhou@math.pku.edu.cn In this talk we study the existence and uniqueness of both entropy solutions and renormalized solutions for quasilinear elliptic (parabolic) equations with L 1 data. And moreover, we will show the equivalence of entropy solutions and renormalized solutions for such equations.

43 Life Span of Nonlinear Wave equations with Small Initial Data Yi Zhou Fudan University, China In this talk, we will review results concerning global existence or long-time existence for nonlinear wave equation with small initial data. we will also present our new result for the initial boundary value problem exterior to non-trapping obstacles in three or four space dimensions.