一、研讨会主旨
为增进非线性偏微分方程及其相关领域的前沿问题和最新研究成果的交流，促进相关学科的发展，厦门理工学院威尼斯棋牌2299正版和福建省高校数学学科联盟于2022年11月26日以腾讯会议在线的方式（会议ID: 303672748）举办“2022 非线性偏微分方程理论及应用研讨会”。
二、日程安排
时间 
主持人 
报告人 
报告题目 
8:509:00 
翟绍辉 
开幕式 
9:009:45 
王焰金 
梁之磊 
A KatoType Criterion for Vanishing Viscosity Near Onsager’s Critical Regularity 
9:5510:40 
吴国春 
Nonexistence of strong solutions with lowenergy to the Cauchy problem of twodimensional radially symmetric isentropic compressible NavierStokes equations 
10:5011:35 
江杰 
The Effect of Signaldependent Motility in a KellerSegel System of Chemotaxis 
午 休 
14:0014:45 
金海洋 
李莉 
Inviscid limit of homogeneous solutions of the NavierStokes equations 
14:5515:40

王玉兰 
Global solvability in a chemotaxisfluid system with prescribed signal on the boundary 
15:5016:35

袁迪凡 
On the existence and stability of 2D compressible currentvortex sheets 
16:3516:40 
闭幕式 
三、组委会(按姓名拼音排序)
陈卿 司新 王剑苹 王明海
会务联系人：王明海，wangminghai@xmut.edu.cn，05926291276
四、报告信息(按姓名拼音排序)
The Effect of Signaldependent Motility in a KellerSegel System of Chemotaxis
江 杰 中国科学院精密测量科学与技术创新研究院
摘要：In this talk, we would like to report our recent work on a Keller—Segel system of chemotaxis involving signaldependent motility. This model was originally proposed by Keller and Segel in their seminal work in 1971, and has been used to provide a new mechanism for pattern formation in some recent Biophysics work published in Science and PRL.
From a mathematical point of view, the model features a nonincreasing signaldependent motility function, which may vanish as the concentration becomes unbounded, leading to a possible degenerate problem. We develop systematic new methods to study the wellposedness problem. The key idea lies in an introduction of an elliptic auxiliary problem which enables us to apply delicate comparison arguments to derive the upper bound of concentration. Moreover, new iteration as well as monotonicity techniques are developed to study the global existence of classical solutions and their boundedness in any dimension. It is shown that the dynamic of solutions is closely related to the decay rate of the motility function at infinity. In particular, a critical mass phenomenon as well as an infinitetime blowup was verified in the twodimensional case if the motility is a negative exponential function.
The talk is based on my recent joint works with Kentarou Fujie (Tohoku University), Philippe Laurençot (University of Toulouse and CNRS), Yanyan Zhang (ECNU), and Yamin Xiao (IAPCM).
Inviscid limit of homogeneous solutions of the NavierStokes equations
李莉 宁波大学
摘要：In the three dimensional case, there is a onetoone correspondence between homogeneous axisymmetric noswirl solutions, which are smooth on the unit sphere minus two poles, of the stationary incompressible NavierStokes equations and a four dimensional hypersurface with boundary. In this talk, I will discuss the inviscid limit of such solution.
A KatoType Criterion for Vanishing Viscosity Near Onsager’s Critical Regularity
梁之磊 西南财经大学
摘要：We consider a vanishing viscosity sequence of weak solutions for the three dimensional Navier–Stokes equations of incompressible fluids in a bounded domain. In Kato’s seminal paper (Seminar on nonlinear partial differential equations, Springer, New York, 1983), he showed that for sufficiently regular solutions, the vanishing viscosity limit is equivalent to having vanishing viscous dissipation in a boundary layer of width proportional to the viscosity. We prove that Kato’s criterion holds for the Hölder continuous solutions with the regularity index arbitrarily close to Onsager’s critical exponent. The proof is based on a new boundary layer foliation and a global mollification technique.
Global solvability in a chemotaxisfluid system with prescribed signal on the boundary
王玉兰 西华大学
摘要：In this talk, we shall consider a chemotaxisfluid model involving Dirichlet boundary condition for the signal. The solution theory is welldeveloped in the case when the chemotaxisfluid system is accompanied by homogeneous boundary conditions of nofluxNeumannDirichlet type. However, if in line with what is suggested by the modeling literature, the boundary condition for the signal is changed to a nonhomogeneous Dirichlet one, the corresponding solution theory is much less understood. We shall discuss the global solvability in a 3D chemotaxisfluid system involving Dirichlet boundary condition for the signal.
Nonexistence of strong solutions with lowenergy to the Cauchy problem of twodimensional radially symmetric isentropic compressible NavierStokes equations
吴国春 华侨大学
摘要：In this talk, we are concerned with the wellposedness of strong solutions to the Cauchy problem of twodimensional isentropic compressible NavierStokes equations. The global existence of strong solutions in homogeneous Sobolev space (without the information of velocity in L^{2}norm) were established by LiXin (Ann PDE 5: 7, 2019), provided the smooth initial data are of small total energy. In particular, the initial density can even have compact support. However, Luo (Math Methods Appl Sci 37:13331352, 2014) showed that the twodimensional radially symmetric isentropic compressible NavierStokes system has no nontrivial global smooth solution in the inhomogeneous Sobolev space if the initial density is compactly supported. The main purpose of this presentation is to prove that the twodimensional radially symmetric strong solution does not exist in the inhomogeneous Sobolev space for any short time if the smooth initial data are of small total energy and the initial density has compact support.
On the existence and stability of 2D compressible currentvortex sheets
袁迪凡 北京师范大学&牛津大学
摘要：In this talk, I will present the existence and the stability of twodimensional currentvortex sheets in ideal compressible magnetohydrodynamics. Under a suitable stability condition for the background state, we show that the linearized currentvortex sheets problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Then we establish the localintime existence and nonlinear stability of currentvortex sheets by a suitable NashMoser iteration, provided the stability condition is satisfied at each point of the initial discontinuity. This is a joint work with Alessandro Morando, Paolo Secchi, Paola Trebeschi.