女王调教

4月23日下午行健楼665

4月24日上午行健楼665

摘 要

Time-asymptotic stability of composite wave of viscous shock and rarefaction for barotropic Navier-Stokes equation

王益（中国科*女王调教-女王调教视频-女王 调教小说数学与系统科学研究院）

We prove the time-asymptotic stability of composite waves consisting of the superposition of a viscous shock and a rarefaction for the one-dimensional compressible isentropic Navier-Stokes equation. Our result solves a long-standing problem first mentioned in 1986 by Matsumura and Nishihara in [Japan J. Appl. Math., 3, 1-13, 1986]. The same authors introduced it officially as an open problem in 1992 in [Comm. Math. Phys., 144, 325-335, 1992] and it was again described as a very challenging open problem in 2018 in the survey paper [A. Matsumura, Handbook of mathematical analysis in mechanics of viscous fluids, 2495-2548, Springer, 2018]. The main difficulty is due to the incompatibility of the standard anti-derivative method, used to study the stability of viscous shocks, and the energy method used for the stability of rarefactions. Instead of the anti-derivative method, our proof uses the $a$-contraction with the time-dependent shifts to control the compressibility of viscous shocks in the original perturbation framework for the stability of rarefactions. This method is energy based, and can seamlessly handle the superposition of waves of different kinds.

Microscopic conservation laws for the derivative nonlinear Schrodinger equation

徐桂香（北京师范大学）

Using the perturbation determinant introduced by B. Simon, we show the microscopic conservation laws for the Schwartz solutions of the derivative nonlinear Schrodinger equation (DNLS) with small mass, which can be used to show local smoothing estimate and global well-posedness of DNLS in the lower regularity space.

Commutator estimates on bounded domains and applications to IBVP to compressible Navier-Stokes equations

李进开（华南师范大学）

In this talk, we will present estimates on two kinds of commutators defined on bounded domains: one is a natural extension in the case of general bounded domains of the classic Riesz commutator and the other is that of the spatial derivatives with the solution mapping of the co-normal derivative problem of the Laplacian. These two kinds of commutators arise in studying the IBVP to the compressible Navier-Stokes equations with Navier slip boundary conditions. As an application of these commutator estimates as well as a BMO estimate for the gradient of solutions to the Lame system subject to the Navier slip boundary conditions, we establish the global well-posedness of strong solutions to the isentropic compressible Navier-Stokes equations with general Navier slip boundary conditions on general boundary domains, under the condition that the initial energy is small. The initial vacuum is allowed.

Global well-posedness of the viscous surface wave problem

王焰金（厦门大学）

Consider a viscous incompressible fluid below the air and above a fixed bottom. The fluid dynamics is governed by the gravity-driven incompressible Navier-Stokes equations, and the effect of surface tension is neglected on the free surface. The global well-posedness and long-time behavior of solutions near equilibrium have been intriguing questions since Beale (1981). It was proved by Guo and Tice (2013) that with certain additional low horizontal frequency assumption of the initial data in 3D an integrable decay rate of the velocity is obtained so that the global unique solution can be constructed, while the global well-posedness in 2D was left open. We prove the global well-posedness in both 2D and 3D, without any low frequency assumption of the initial data. The key ingredients are a nonlinear cancellation by using Alinhac good unknowns and the improved anisotropic decay rates of the velocity, which are even not integrable.

Layer potentials and homogenization of Stokes system in perforated domains

荆文甲（清华大学）

In this talk I will present a unified homogenization method for Stokes systems in periodically perforated domains with Dirichlet boundary conditions at the boundaries of the holes. The method is based on the layer potentials for Stokes system and a quantitative analysis of the rescaled cell problem. It treats various asymptotic regimes of the hole-cell ratio in a unified manner, and it provides natural correctors and quantitative estimates. The talk is based on a joint work with Yong Lu and Christophe Prange.