数学中心“偏微分方程与非线性泛函分析”报告会

会议时间：2020年8月22日-23日 08:30-17:30

腾讯会议ID：691 917 767

（一）报告人： 李自来 （河南理工大学）

报告题目：Global strong solutions and non-resistive limit for 1D MHD equations with no vacuum at infinity

摘要：In this talk, we consider the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity. By deriving a priori $\nu $-independent estimates, the non-resistive limit of the global solutions with large initial data and vacuum is justified, where $\nu>0$ is the resistivity coefficient. As a by-product, the global well-posedness of strong solutions for both the compressible resistive and non-resistive MHD equations are established, respectively.

（二）报告人： 赵娜 （北京应用物理与计算数学研究所）

报告题目：Decay and Vanishing of some D-Solutions of the Navier–Stokes Equations

报告摘要：An old problem since Leray asks whether homogeneous D-solutions of the 3 dimensional Navier–Stokes equation in R^3 or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two cases: (1) the full 3 dimensional slab case R^2 × [0, 1] with Dirichlet boundary condition; (2) when the solution is axially symmetric and periodic in the vertical variable. Also, for the slab case, we prove that even if the Dirichlet integral has some growth, axially symmetric solutions with Dirichlet boundary condition must be swirl free, namely u_\theta = 0, thus reducing the problem to essentially a “2 dimensional” problem. In addition, a general D-solution (without the axial symmetry assumption) vanishes in R^3 if, in spherical coordinates, the positive radial component of the velocity decays at order -1 of the distance. This is a joint work with Dr. Bryan Carrillo, Prof. Xinghong Pan and Prof. Qi S. Zhang.

（三）报告人： 杜利怀 （温州大学）

报告题目： Local and global existence of pathwise solution for the stochastic Boussinesq equations with multiplicative noises

报告摘要：In this talk, we consider the stochastic Boussinesq equations in $\mathbb{T}^d$ with the nonlinear multiplicative noises. We establish the local existence of pathwise solutions. Furthermore, we establish the global existence of pathwise solution when the noises are non-degenerate, which show that the non-degenerate multiplicative noises would provide a regularizing effect: the global existence of solution occurs with high probability if the initial data are sufficiently small, or if the noise coefficients are sufficiently large. This is a joint work with Prof.Ting Zhang.

（四）报告人： 廖勇凯 （中国地质大学（武汉））

报告题目：Nonlinear stability of rarefaction waves for a viscous radiative and reactive gas with large initial perturbation

报告摘要：In this talk, we consider the Cauchy problem of a model for the one-dimensional viscous radiative and reactive gas. For such a specific gas motion, a somewhat surprising fact is that, general speaking, the pressure $\widetilde{p} (v,s)$ is not a convex function of the specific volume $v$ and the specific entropy $s$. Even so, we will show that the rarefaction waves are time-asymptotically stable for large initial perturbation provided that the radiation constant $a$ and the strength of the rarefaction waves are sufficiently small.

（五）报告人： 孙家伟 （北京应用物理与计算数学研究所）

报告题目：The Behaviors of the Solutions to the Compressible Navier-Stokes (Euler)-Fokker-Planck Equations

报告摘要：We study the time-asymptotic behaviors of the solutions for the compressible Navier-Stokes/Fokker-Planck (NSFP) equations and the compressible Euler/Fokker-Planck (EFP) equations in spatial three dimension. First, we establish the spectrum structure of the linearized compressible NSFP equations and the linearized compressible EFP equations around an equilibrium state. It is shown that the coupling friction force terms play an key role in the analysis, so that the spectrum structure is genuinely different from the ones of the Fokker-Planck equation and the isotropic Euler equations with damping. Based on the decay properties of the semigroup of both the linearized compressible NSFP equations and the linearized compressible EFP equations, we prove that the global strong solutions of the Cauchy problem for both the compressible NSFP equations and the compressible EFP equations tend in $L^{2}$-norm to the equilibrium state at the optimal time decay rate $(1+t)^{-\frac{3}{4}}$. Then, we establish the pointwise estimates of the global strong solutions to the Cauchy problem of bth the compressible NSFP equations and the compressible EFP equations. Our analysis shows that, due to the interaction of the coupling external force terms, the global solution admit the macroscopic nonlinear diffusion waves and the Huygen's type sound wave propagation (the weak Huygen's principle). These behaviors have an essential difference from the ones for the Fokker-Planck equation and the isotropic Euler equations with damping. To our knowledge, this is the first time to extend the pointwise approach to study the time-asymptotic behaviors of the solutions of the fluid-particles interaction models. Some new strategies are introduced to deal with the difficulties caused by the coupling drag force terms. This is a joint work with Hai-Liang Li, Guojing Zhang and Mingying Zhong

（六）报告人： 张志朋 （南京大学）

报告题目：Energy conservation for the weak solutions to the ideal inhomogeneous magnetohydrodynamic equations

报告摘要：We prove the energy conservation for the weak solution of the three-dimensional ideal inhomogeneous magnetohydrodynamic (MHD) equations in a bounded domain. Two types of sufficient conditions on the regularity of the weak solutions are provided to ensure the energy conservation. Due to the appearance of the boundary, we need to impose the smallness and the Besov-type continuity for the velocity and magnetic fields near the boundary.

（七）报告人： 穆鹏程 （北京应用物理与计算数学研究所）

报告题目：Dispersive estimates for the rotating stratified Boussinesq equations in stratification-dominant limit

报告摘要：We consider the Cauchy problem for the rotating stratified Boussinesq equations. We establish the dispersive estimates for the linear propagator related to the system subjected to stratification-dominant limit. Different from the previously studied cases, the phase $p(\xi)$ in this paper is a perturbation around a function which has singularity and degeneracy. Thus we develop a new and technical frequency cut-off method based on a detailed study on the phase. As an application, we then prove the long time existence and the stratification-dominant limit of the strong solution to the system.