2017年春季偏微分方程讨论班

讨论班简介


时间:每周三下午16:00到17:00。
教室:1208。

讨论班预告

Time: 2月14日 16:30-17:30

Speaker: 陈世炳(澳大利亚国立大学)

Title: Partial $W^{2,p}$ regularity in optimal transportation.

Abstract: I will present some recent work on the partial $W^{2,p}$ regularity result on the optimal transport problem. In particular, we prove that under mild conditions (ensuring the existence of optimal maps), the potential functions of the optimal transport map is locally $W^{2,p}$ outside a closed singular set of measure 0. This is a joint work with Alessio Figalli.

Time: 2月15日 16:00-17:00

Speaker: 方道元(浙江大学)

Title: Almost global solutions for semilinear Klein-Gordon equations on the circle

Abstract: We study the Cauchy problem of the semi-linear Klein-Gordon equation on the circle with a nonlinearity of polynomial type. We show that the solution exists almost globally in many cases, for almost every positive mass. In particular, we obtain the almost global existence except only one case if the nonlinearity is monomials. The results are based on the method of normal forms. The difficulty is to find a structure of the nonlinearity so that the process of normal forms can be performed up to any order. These are the joint works with Zheng Han and Qidi Zhang.

Time:2月22日

Speaker:王志张(复旦大学)

Title: The curvature estimates for sum type of Hessian equations

Abstract: In this talk, we will introduce a new type of Hessian equation, sum type equation. We will generalize the similar concavity appearing in sigma Hessian equations. We also use these concavity to obtain curvature estimates for right hand side depending on the gradient term. These estimates also can be used to discuss the existence results of prescribe curvature problems.

Time:3月1日

Speaker:江宁(武汉大学)

Title: Well-posedness of Ericksen-Leslie's hyperbolic liquid crystal model

Abstract: We present in this talk a joint work with Yilong Luo on well-posedness of Ericksen-Leslie's hyperbolic liquid crystal model. For heneral model, local existence with initial data is proved, while for a special case which is Navier-Stokes equations coupled with wave map to S^2, we show local existence for large bounded data. With additional structure assumption, we finally establish a global existence with small data. A relation between the Lagrangian multiplier and the geometric constraint plays a key role in the proof.

Time:3月8日

Speaker:Tak Kwong Wong(香港大学)

Title: Axisymmetric Flow of Ideal Fluid Moving in a Narrow Domain

Abstract: In applications in blood flow and pipeline transport, the radial length scale of the underlying flow is usually small compared to the horizontal length scale. In this talk, we will introduce a new model called the axisymmetric hydrostatic Euler equations, which describe the leading order behavior of an ideal and axisymmetric fluid moving in such narrow channel. After providing the formal derivation, we will discuss the mathematical analysis of this model under a new sign condition. This is a joint work with Robert M. Strain.

Time:3月15日

Speaker:丁琪(复旦大学)

Title: Dirichlet problem of minimal graphs.

Abstract: Dirichlet problem of minimal graphs has been investigated for over a century. After successive efforts by several mathematicians, Jenkins-Serrin gave the classical sharp criterion for solvability of Dirichlet problem of the minimal surface equation (codimension 1 case). In higher codimensions, however, the situation is much more complicated shown in a seminal paper by Lawson-Osserman. In this paper, we obtain the existence of minimal graphs over arbitrary mean convex bounded $C^2$ domains for a large class of prescribed boundary data. In contrast, we also construct a class of prescribed boundary data on just mean convex domains, for which Dirichlet problem in codimension 2 is not solvable.

Time:3月22日

Speaker:雷震(复旦大学)

Title: Strong Null Condition and Incompressible Fluids

Abstract: We introduce the notion of "strong null condition" and discover that many incompressible fluid system inherently satisfies such a condition, instead of the well-known standard null condition. Using this structure, we solved the long-standing global existence problem of incompressible elastodynamic system in two dimensions. The method can also be applied to study the inviscid limit of incompressible viscoelastic system.

Time:3月29日

Speaker:来米加(上海交通大学)

Title: A note on convex hypersurfaces with singularities.

Abstract: We study singular closed convex hypersurfaces with constant Weingarten curvature. The main result is that if singularities are point singularities and H_k=1 for k\in [1,n), then singularities are removable. For k=n, there exist convex hypersurfaces with point singularities with constant curvature. If there are only two singular points, then it must be a rotational symmetric football. This is a joint work with H. Fang and W. Wo.

Time:3月29日

Speaker:李玉祥(东南大学)

Title: Blow-up and global existence of attraction-repulsion chemotaxis systems

Abstract: In this talk, I will review blow-up problems for the famous Keller-Segel system which describes the chemotactic movement of the bacteria. Then I give two recent results about blow-up and global existence fro attraction-repulsion chemotaxis systems.

Time:4月5日

Speaker:唐岚(华中师范大学)

Title: On the Fractional Navier-Stokes Equations

Abstract: We study the suitable weak solutions to the fractional Navier-Stokes equations and we give some partial regularity results.

Time:4月12日 16:00-17:00

Speaker:王明新(哈尔滨工业大学)

Title: A free boundary problem for the prey-predator model with different free boundaries

Abstract: This talk concerns with a free boundary problem for the prey-predator model with different free boundaries. Main contents are: global existence, uniqueness, regularity and some estimates of (u,v, g, h); long time behaviors of (u,v); spreading and vanishing; estimates of speeds of g and h and asymptotic spreading speeds of u and v.

Time:4月12日 17:00-18:00

Speaker: 王梦(浙江大学)

Title: Weak convergence of the Landau-de Gennes flow to motion by mean curvature

Abstract: We study the asymptotic bahaviour of the Landau-de Gennes flow as the elastic constant tends to zero. We show that vorticity evolves according to motion by mean curvature in Brakke’s weak formulation.

Time:4月19日

Speaker:马跃(西安交通大学)

Title: Nonlinear stability of Minkowski space-time with massive scalar field

Abstract: In this talk we will present some recent work about the system of Einstein equation coupled with a massive scalar field (partially published in [2]). More precisely, on the nonlinear global stability of the Minkowski space-time. In a PDE point of view, this is equivalent to theglobal existence of a special class of quasi-linear wave-Klein-Gordon system with small initial data. The wave-Klein-Gordon system, due to its lack of scaling invariance and its lack of Klein-Gordon structure, can not be treated by a simple combination of the classical techniques applied on wave equations and Klein-Gordon equations. To solve this difficulty, we introduced the “hyperboloidal foliation method” introduced by the author in [1] combined with some newly developed tools such as L∞ estimates on Klein-Gordon equations in curved space-time and L∞ estimates on wave equations based on the expression of spherical means. We also adapt some tools developed in classical framework for the analysis of Einstein equation into our hyperboloidal foliation framework, such as the estimates based on wave gauge conditions and the L∞ estimates on wave equations based on integration along characteristics. References

[1] P. LeFloch and Y. MA The hyperboloidal foliation method, World Scientific, 2015

[2] P. LeFloch and Y. MA The nonlinear stablity of Minkowski space for self-gratitating massive field. The wave-Klein-Gordon model, Communications in Mathematical Physics September 2016, Volume 346, Issue 2, pp 603-665

Time:4月26日 16:00-17:00

Speaker:尹会成(南京师范大学)

Title: A class of degenerate hyperbolic equations and their related applications

Abstract: In this talk, I focus on the studies on a class of degenerate hyperbolic equations and their related applications in compressible fluid dynamics, semilinear wave equations with time-depending damping, compressible Euler equations with time-depending damping and so on.

Time:4月26日 17:00-18:00

Speaker:盛为民(浙江大学)

Title: FLOW BY GAUSS CURVATURE TO THE ALEKSANDROV AND DUAL MINKOWSKI PROBLEMS

Abstract: In this talk, I will introduce our recent work on Gauss curvature flow with Xu-Jia Wang and Qi-Rui Li. In this work we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\R^{n+1}$ with the speed $f r^{\alpha} K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. We prove that if $\alpha\ge n+1$, the flow exists for all time and converges smoothly after normalization to a hypersurface, which is a sphere if $f\equiv 1$. Our argument provides a new proof for the classical Aleksandrov problem ($\alpha = n+1$) and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang recently, for the case $q<0$ ($\alpha>n+1$). If $\alpha< n+1$, corresponding to the case $q > 0$, we also establish the same results for even function $f$ and origin-symmetric initial condition, but for non-symmetric $f$, counterexample is given for the above smooth convergence.

Time:4月27日 11:00-12:00; 1208室

Speaker:周春琴(上海交通大学)

Title: Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian

Abstract: TBA

Time:5月10日

Speaker:袁伟(中山大学)

Title: Brown-York mass and vacuum static space-time

Abstract: In 1950s, Arnowit, Deser and Misner gave the Hamiltonian formalism of Einstein equations and proposed the well known positive mass conjecture. It was solved by Schoen-Yau and Witten using different technics almost 30 years later. After their works, the study in this area turned to localization of ADM mass gradually. In this talk, I will give a brief introduction of the study of mass problem and vacuum static space-time. Also, I will talk about the idea of the proof of positive mass theorem for Brown-York mass with respect to generic three dimensional vacuum static metric under certain physical assumptions.

Time:5月17日 16:00-17:00

Speaker:吴国强(浙江理工大学)

Title: Heat kernel estimate along Ricci-harmonic flow

Abstract: In this talk we discuss Ricci-harmonic flow on which the scalar curvature is bounded. At first, we establish a time derivative bound for solution to the heat equation, based on this, we derive the distance distortion estimate and the existence of a cutoff function. At last we use these to get the heat kernel upper bound and lower bound along Ricci-harmonic flow. This is joint with Prof Yi Li.

Time:5月24日 16:30-17:30

Speaker:蒲学科(重庆大学)

Title: Long wavelength limit for the quantum Euler-Poisson equation

Abstract: In this talk, we will introduce our recent work on the long wavelength limit for the quantum Euler-Poisson equation. Under the Gardner{Morikawa transform, we derive the quantum Korteweg{de Vries (KdV) equation by a reductive perturbation method. We show that the KdV dynamics can be seen at time interval of order $O(\epsilon^{-3/2})$. When the nondimensional quantum parameter H = 2, it reduces to the inviscid Burgers equation. The 2D case will also be considered.

Time:6月6日 9:00-10:00 地点:1518

Speaker:王邵东(加拿大McGill大学 )

Title: Infinitely many solutions for cubic schrodinger equation in dimension 4

Abstract: In this talk, I will present some recent results in the existence of blowing-up solutions to a cubic schrodinger equation on the standard sphere in dimension 4. This is the joint work with Jerome Vetois.

Time:6月13日 15:00-16:00

Speaker:桂长峰(University of Texas at San Antonio and Hunan University)

Title: The Sphere Covering Inequality and its application to a Moser-Trudinger type inequality and mean field equations

Abstract: In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two {\it distinct} surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least $4 \pi$. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. Other applications of this inequality include the classification of certain Onsager vortices on the sphere, the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. The resolution of several open problems in these areas will be presented. The talk is based on joint work with Amir Moradifam from UC Riverside.

Time:6月13日 16:00-17:00

Speaker:金天灵(香港科技大学)

Title: Holder gradient estimates for parabolic homogeneous p-Laplacian equations.

Abstract: We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation $$ u_t=|\nabla u|^{2-p} div (|\nabla u|^{p-2}\nabla u), $$ where 1

Time:6月30日

Speaker:尹洪明(Washington State University)

Title: Recent progress for some nonlinear problems arising from microwave and inductive heating processes

Abstract: In this talk I will discuss some nonlinear problems in electromagnetic fields. The focus will be on the problems related to the microwave heating and inductive heating in industries. The mathematical models consist of Maxwell’s equations coupled with a nonlinear heat equation. Some recent results on existence and regularity of weak solution will be discussed in my talk.