【03月10日-03月13日】偏微分方程新进展会议

主办单位：中国科学技术大学中法数学中心

组委会：陈世炳，李嘉禹，刘勇，麻希南，章志飞

会议支持：科技部重点研发项目

会议地点：安徽高速徽风皖韵酒店

会议时间：2022年3月10日-13日，3月10日报到，13日离会

会议日程

3月10日下午14:00-20:00 报到。晚餐：一楼全日餐厅（自助）

3月11日（会议室：三楼百合厅）

3月12日（会议室：三楼百合厅）

3月13日 全天离会

报告信息

张平（中科院数学与系统科学研究院）

Title: On the instantaneous radius of analyticity of \(L^p\) solutions to 3D Navier-Stokes system

Abstarct: In this talk, we first investigate the instantaneous radius of space analyticity for the solutions of 3D Navier-Stokes system with initial data in the Besov spaces \(\dot B^s_{p,q}(\mathbb R^3)\) for \(p\in (1,\infty)\), \(q\in [1,\infty]\) and \(s\in [-1+\f3p,\f3p)\). Then for initial data \(u_0\in L^p(\mathbb R^3)\) (with \(p\in(3,6)\), we  prove that 3D Navier-Stokes system has a unique solution \(u=u_L+v\) with \(u_L:=e^{t\mathrm D}u_0\) and \(v\in {\widetilde{L}^\infty_T(\dot{B}^{1-\frac{3}{p}}_{p,\f{p}2})}\cap {\widetilde{L}^1_T(\dot{B}^{3-\frac{3}{p}}_{p,\f{p}2})}\) for some positive time \(T\).

王超（北京大学）

题目: Inviscid limit of Navier Stokes equations with vortex sheet initial data

摘要: In this talk, I will talk about the inviscid limit of 2-D incompressible Navier-Stokes equations in the whole space with vortex sheet initial data. Although the initial data have different velocities along the tangential direction of one interface, we still can prove that the Navier-Stokes equations admit a smooth solution whose regularity depends on the viscosity. When the viscosity equals to zero, the system becomes the Euler equations whose velocities have a jump across a boundary. Due to the mismatch of the velocity on the interface, same phenomenon as boundary layer will appear. Like the boundary layer theory, to justify the inviscid limit, there appears one derivative loss. In this paper, we improve the energy method to justify the inviscid limit in the analytic setting. This work is joint with Prof. Yuxi Wang and Prof. Zhifei Zhang.

彭双阶（华中师范大学）

题目：Qualitative analysis for Moser-Trudinger nonlinearities

摘要：In this talk, we are concerned with the Moser-Trudinger problem. By  using a variety of local Pohozaev identities, we qualitatively analyze the positive solutions of Moser-Trudinger problem with a low energy, which contains the Morse index, non-degeneracy, asymptotic behavior, uniqueness and symmetry of solutions. This is a joint work with Peng Luo and Kefan Pan.

王伟（浙江大学）

题目：Sharp interface limit of phase transition models with high dimensional energy wells

摘要：The asymptotics of scalar phase transition models in the limit of vanishing interface width have been widely studied over the past decades. We will talk about some analogous results on phase transition models with high dimensional energy wells.

谢春景（上海交通大学）

题目: Analysis of steady flows with stagnation points for the incompressible Euler system in an infinitely long nozzle

摘要: Stagnation point in flows is an interesting phenomenon in fluid mechanics. It induces many

challenging problems in analysis. We first derive a Liouville type theorem for Poiseuille flows in the class of incompressible steady inviscid flows in an infinitely long strip, where the flows can have stagnation points. With the aid of this Liouville type theorem, we show the uniqueness of solutions with positive horizontal velocity for steady Euler system in a general nozzle when the flows tend to the horizontal velocity of Poiseuille flows at the upstream. Finally, this kind of flows are proved to exist in a large class of nozzels and we also prove the optimal regularity of boundary for the set of stagnation points. This talk is based on joint work with Congming Li, Yingshu Lv, and Henrik Shahgholian.

缪爽（武汉大学）

题目：A priori estimates for a class of free boundary problems and application 

摘要：In this talk we shall present some a priori estimates for a class of free boundary problems of inviscid fluids and their application. We will review some previous results and then discuss our recent progress on this direction.

李进开（华南师范大学）

题目：Boundedness and unboundedness of the entropy of the ideal gases with far field vacuum

摘要：In the presence of vacuum, the physical entropy for polytropic gases behave singularly and it is thus hard to study its dynamics. In this talk, we present some recent studies on the propagation of the boundedness of the entropy to the viscous compressible ideal gas in the presence of vacuum. It will be shown in this talk that, in the case that the vacuum presents at the far fields only, the uniform boundedness of the entropy can be propagated locally or globally if the initial density decays slowly, while if the initial density decays sufficiently fast, the entropy becomes unbounded immediately after the initial time.

王勇（中科院数学与系统科学研究院）

题目：Global solutions on compressible Euler and Euler-Poisson equations

摘要：I will talk some results on the global existence of solutions to compressible Euler and Euler-Poisson of large initial data with spherical symmetry.

刘双乾（华中师范大学）

题目:Some progress on the shear flow problem of the Boltzmann equation

摘要:In this talk, I will report our recent study on the shear flow problem on the Boltzmann equation. We will focus on the stability of the shear flow of the Boltzmann equation for both the Maxwell molecular model and the hard potentials, the plane Couette flow as well as the heat transfer of the Boltzmann equation with diffusive reflection boundary condition in a spatially interval will also be discussed.

叶东（华东师范大学）

题目：On sharp discrete Hardy-Rellich inequalities

摘要：Although the history of Hardy inequalities found its origin somehow in the discrete setting, the discrete Hardy-Rellich inequalities are much less understood comparing to the continuous situation. We will show discrete Hardy-Rellich inequalities on \(\mathbb{N}\) with \(\Delta^\frac{\ell}{2}\) and optimal constants, for any \(\ell \geq 1\). Our approach is to establish some sharp first order Hardy inequalities using weighted equality, and then to handle the higher order cases by iteration. We provide also a first order Leray type inequality on \(\N\) with the same constant as the continuous setting. The main idea to get weighted equalities works also for general graphs. This is a joint work with Xia Huang at ECNU.

王志张（复旦大学）

题目：The geometric flow in Minkowski space

摘要：In this talk, we study fully nonlinear curvature flows of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface is strictly convex and satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling the flow converges to a self-expander. We also will study the existence of self-expander with prescribed asymptotic behavior.

李东升（西安交通大学）

题目:A Bernstein Type Theorem for Fully Nonlinear Elliptic Equations in 2D

摘要:For fully nonlinear elliptic equations on exterior domains, Li, Li and Yuan give the asymptotic behavior of solutions at infinity as the space dimensions are greater than two. The method does not work for two dimensional spaces since in this case there is no nontrivial super-solution tending to zero at infinity. In this talk, we will prove the counter result in spaces of dimension two by conformal mapping. The results can be used to study Monge-Ampere equations, special Lagrange equations, etc.

熊金钢（北京师范大学）

题目：Optimal regularity and fine asymptotics for the porous medium equation in bounded domains

摘要：We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with zero Dirichlet boundary conditions after certain waiting time. Previously, only space-time Hölder and spatial Lipschitz regularity were known, dating back to Caffarelli-Friedman 1980 and Aronson-Peletier 1981. Such optimal regularity allows us to refine the asymptotics of solutions for large times, which is new even in 1D. This is joint with Tianling Jin and Xavier Ros-Oton.

黄耿耿（复旦大学）

题目:The Guillemin boundary problem for Monge-Ampere equation in the polygon

摘要:In this talk, we will talk about the existence of solutions for the two dimensional Monge-Amp\`ere equation in the polygons with Guillemin boundary condition.

李维喜（武汉大学）

题目:Analytic regularization effect for the Non-cutoff Boltzmann equation

摘要:We verify that the spatially inhomogeneous Boltzmann equation with strong angular singularity admits the analytic smoothing effect, just like its diffusive models such as the Landau and Fokker-Planck equations. To overcome the degeneracy in the spatial variable, a family of well-chosen vector fields with time-dependent coefficients will play a crucial role, and the analytic regularization effect of weak solutions relies on a quantitative estimate on directional derivatives in these vector fields. 

王克磊（武汉大学）

题目:Nondegeneracy for stable solutions to one phase free boundary problem

摘要:Since the seminal work of Alt-Caffarelli in 1981, the one phase free boundary problem has been studied by many people. To study the regularity and singularity of free boundaries, the blow up analysis is a standard method. It turns out for this free boundary problem, the nondegeneracy condition is crucial for the application of this method. Although the nondegeneracy condition has been known for energy minimizers for a long time, it's not true for general solutions. In this talk, I will discuss a proof of the nondegeneracy for stable solutions. This is based on a joint work with N. Kamburov.

周渊（北京师范大学）

题目：无穷调和，无穷变分及Aronsson方程正则性研究进展

摘要：本报告回顾无穷调和函数，无穷变分绝对极小子及相应的Aronsson方程黏性解的正则性方面已经取得的研究成果。同事汇报我们最近几年来取得的一些进展。

唐春雷（西南大学）

题目：Least energy sign-changing solutions for Schrödinger-Poisson system

摘要：In this talk, we will discuss the following Schrödinger-Poisson system.

Some background knowledge is introduced about Schrödinger-Poisson system firstly. Then, under various assumptions on V(x), K(x) and f(x,u), we will present some recent results about the existence of least energy sign-changing solutions for above system.

邹文明（清华大学）

题目:Normalized solutions to Schrödinger equations with potential and inhomogeneous nonlinearities on large convex domains 

摘要: The talk addresses an open problem raised in Bartsch-Molle-Rizzi-Verzini (CommPDE,2021) on the existence of normalized solutions to Schrodinger equations with potentials and inhomogeneous nonlinearities, defined both on RN as well as on an open bounded convex domain. The nonlinearity is a combination of a mass subcritical and a mass supercritical term. We develop a method to study the existence of normalized solutions. This is a joint work with T. Bartsch and S. Qi.