Workshop on PDEs Arising from Kinetic Theory and Fluid Dynamics
Date: March 18, 2017
Venue: LargeLecture Room, School of Mathematical Sciences, SJTU
Organizers: PDE group
Time / Speaker / Affiliation / Title14:00-14:40 / Prof. Tong Yang / City University of Hong Kong / Boundary layers for MHD
14:40-15:20 / Prof. Hui-Jiang Zhao / Wu Han University / Convergence to the Self-similar Solutions to theHomogeneous Boltzmann Equation
15:20-15:40 / Coffee Break
15:40-16:20 / Prof. Tao Luo / City University of Hong Kong / Existenceof Magnetic Compressible FluidStars
16:20-17:00 / Prof. Chun-Lai Mu / Chong Qing University / Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude
17:00-18:00 / Free Discussion
Abstract
Professor Tong YANG
Title: Boundary layers for MHD
Abstract: The classical theory of Oleinik for the Prandtl boundary layer equations states that the monotonicity of the tangential velocity field in the normal direction to the boundary leads to local in time well-posedness. In this talk, we will present our recent work on the MHD boundary layer that shows the tangential magnetic field stabilizes the boundary layer so that the monotonicity condition is no longer needed.Furthermore, this physical condition leads to justification of the high Reynolds numbers limit. The talk includes some recent joint works with Chengjie Liu and FengXie.
The research is partially supported by General Research Fund of Hong Kong, CityU 11320016.
Professor Huijiang ZHAO
Title: Convergence to the Self-similar Solutions to theHomogeneous Boltzmann Equation
Abstract:The Boltzmann H-theorem implies that the solution to the Boltzmann equation tends to an equilibrium, that is, a Maxwellian when time tends to infinity. This has been proved in varies settings when the initial energy is fi nite. However, when the initial energy is infinite, the time asymptotic state is no longer described by a Maxwellian, but a self-similar solution obtained by Bobylev-Cercignani. The purpose of this paper is to rigorouslyjustify this for the spatially homogeneous problem with Maxwellian molecule type cross section without angular cutoff .
Professor Tao LUO
Title: Existence of Magnetic Compressible FluidStars
Abstract: In this talk, I will present a result of the existence of magnetic star solutions which are axi-symmetric stationarysolutions for the Euler–Poisson system of compressible fluids coupled to amagneticfield , proved by a variational method. The method of proof consistsin deriving an elliptic equation for the magnetic potential in cylindrical coordinatesin R^3, and obtaining the estimates of the Green’s function for this elliptic equationby transforming it to 5-Laplacian.
This is a joint work with Paul Federbushand and JoelSmoller.
Professor Chunlai MU
Title: Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude
Abstract:In this talk, we study continuity and persistence for a nonlinear evolution equationdescribing the free surface of shallow water wave with a moderate amplitude.By the approach for approximate solutions and well-posedness estimates, we obtain two sequences of solution for Constantin-Lannes equation, whichare bounded in the Sobolev space $H^s(\mathbb{R})$ with $s>3/2$, and the distance between the two sequences is lower-bounded by a positive constant for any time $t$, but converges to zero at the initial time.This implies thatthe solution map is not uniformly continuous.Furthermore, the solution mapfor Constantin-Lannes equationis shown H\"{o}lder-continuous in $H^r$-topology for all$0\leq r< s$ with exponent $\alpha$ depending on $s$ and $r$.In addition, we also investigatethe asymptotic behaviors of the strong solutions to Constantin-Lannesequation at infinity within its lifespan provided the initial data in weighted $L_{\phi}^p:=L^p (\mathbb{R},\phi^pdx)$ spaces.