2022非线性偏微分方程理论及应用研讨会

作者: 时间:2022-11-22 点击数:

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,厦门理工学院世界杯正规买球app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: 303-672-748)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

8:50-9:00

翟绍辉

开幕式

9:00-9:45

王焰金

梁之磊

A Kato-Type Criterion for Vanishing Viscosity Near Onsager’s Critical Regularity

9:55-10:40

吴国春

Non-existence of strong solutions with low-energy to the Cauchy problem of two-dimensional radially symmetric isentropic compressible Navier-Stokes equations

10:50-11:35

江杰

The Effect of Signal-dependent Motility in a Keller--Segel System of Chemotaxis

14:00-14:45

金海洋

李莉

Inviscid limit of homogeneous solutions of the Navier-Stokes equations


14:55-15:40


王玉兰

Global solvability in a  chemotaxis-fluid system with prescribed signal on the boundary


15:50-16:35


 

袁迪凡

On the existence and stability of 2D compressible current-vortex sheets

16:35-16:40

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,0592-6291276


四、报告信息(按姓名拼音排序)


The Effect of Signal-dependent Motility in a Keller--Segel System of Chemotaxis

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 KellerSegel system of chemotaxis involving signal-dependent 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 Bio-physics work published in Science and PRL.

From a mathematical point of view, the model features a non-increasing signal-dependent motility function, which may vanish as the concentration becomes unbounded, leading to a possible degenerate problem. We develop systematic new methods to study the well-posedness 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 infinite-time blowup was verified in the two-dimensional 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 Navier-Stokes equations

李莉   宁波大学

摘要:In the three dimensional case, there is a one-to-one correspondence between homogeneous axisymmetric no-swirl solutions, which are smooth on the unit sphere minus two poles, of the stationary incompressible Navier-Stokes equations and a four dimensional hypersurface with boundary. In this talk, I will discuss the inviscid limit of such solution.

 

 

A Kato-Type Criterion for Vanishing Viscosity Near Onsagers 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 NavierStokes equations of incompressible fluids in a bounded domain. In Katos 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 Katos criterion holds for the Hölder continuous solutions with the regularity index arbitrarily close to Onsagers critical exponent.  The proof is based on a new boundary layer foliation and a global mollification technique.

 

 

Global solvability in a chemotaxis-fluid system with prescribed signal on the boundary

王玉兰   西华大学

摘要:In this talk, we shall consider a chemotaxis-fluid model involving Dirichlet boundary condition for the signal. The solution theory is well-developed in the case when the chemotaxis-fluid system is accompanied by homogeneous boundary conditions of no-flux-Neumann-Dirichlet 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 chemotaxis-fluid system involving Dirichlet boundary condition for the signal.

 

 

Non-existence of strong solutions with low-energy to the Cauchy problem of two-dimensional radially symmetric isentropic compressible Navier-Stokes equations

吴国春    华侨大学

摘要:In this talk, we are concerned with the well-posedness of strong solutions to the Cauchy problem of two-dimensional isentropic compressible Navier-Stokes equations. The global existence of strong solutions in homogeneous Sobolev space (without the information of velocity in L2-norm) were established by Li-Xin (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 37:1333-1352, 2014) showed that the two-dimensional radially symmetric isentropic compressible Navier-Stokes system has no non-trivial 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 two-dimensional 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 current-vortex 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 two-dimensional current-vortex sheets in ideal compressible magnetohydrodynamics. Under a suitable stability condition for the background state, we show that the linearized current-vortex sheets problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Then we establish the local-in-time existence and nonlinear stability of current-vortex sheets by a suitable Nash-Moser 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.

 

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