偏微分方程的数学建模与理论研究研讨会

会议时间：5月27日-28日（2017年）；

会议地点：海韵园实验楼105报告厅

学术报告题目与摘要

1. 报告人：辛周平，香港中文大学

题目：高维流体研究中的一些进展和问题

2. 报告人：邓引斌，华中师范大学

题目：Nontrivial solutions for a p-biharmonic equation without (AR) condition

3. 报告人：丁时进，华南师范大学

题目：Stability analysis for imcompressible Navier-Stokes equations with Navier boundary conditions

摘要：This talk is concerned with the instability and stability of the trivial steady states of the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a slab domain in dimension two. The main results show that the stability (or instability) of this constant equilibrium depends crucially on whether the boundaries dissipate energy and the strengthen of the viscosity and slip length. It is shown that in the case that when all the boundaries are dissipative, then nonlinear asymptotic stability holds true. Otherwise, there is a sharp critical viscosity, which distinguishes the linear and nonlinear stability from instability.

4. 报告人：赵会江，武汉大学

题目：On the Vlasov-Poisson-Boltzmann limit of the Vlasov-Maxwell-Boltzamann system in the perturbative framework

摘要：We give a rigorous global in time mathematical justi cation of the limit from the Vlasov-Maxwell-Boltzmann system to the Vlasov-Poisson-Boltzmann system in the perturbative framework as the light velocitytends to infinity.

5. 报告人：温焕尧，华南理工大学

题目：Global weak solution to a two-phase model

摘要：I will talk about the global existence of weak solution to the Dirichlet problem for a two-phase model. By relying on weak compactness tools we obtain an existence result within the class of large data.

6. 报告人：王焰金，厦门大学

题目：On partially dissipative hyperbolic systems violating the Kawashima condition

摘要：Consider the Cauchy problem for the quasilinear hyperbolic system of balance laws in multidimensions. The system is partially dissipative in the sense that there is one and only one eigen-family violating the Kawashima condition. By imposing certain supplementary degeneracy conditions with respect to the non-dissipative eigen-family, global unique smooth solutions near constant equilibria are constructed. The proof is based on the introduction of the partially normalized coordinates, a delicate structural analysis, a family of scaled energy estimates with minimum fractional derivative counts and a refined decay estimates of the dissipative components of the solution.

7. 报告人：王术，北京工业大学应用数理学院

题目：Boundary Layer Problem and Zero Viscosity and/or Diffusion Limit of the Incompressible MHD System

摘要：In this paper, we study the boundary layer problem, zero viscosity-diffusion limit and zero magnetic diffusion vanishing limit of the initial boundary value problem for the incompressible viscous and diffusive MHD system with Dirichlet boundary conditions. The main difficulties overcome here are to deal with the effects of the boundary layers resulted by the Dirichlet boundary condition for the velocity and magnetic field.Firstly, we identify a non-trivial class of initial data for which we can establish the uniform stability of the Prandtl's type boundary layers and prove rigorously that the solutions to the viscous and diffusive incompressible MHD systems converges strongly to the superposition of the solution to the ideal MHD systems with a Prandtl's type boundary layer corrector. One key derivation here is that for the class of initial data we identify here, there exist cancellations between the boundary layers of the velocity field and that of the magnetic fields so that one can use an elaborate energy method to take advantage this special structure. Then, in the case of fixed positive viscosity, we establish the stability of diffusive boundary layer for the magnetic field and convergence of solutions in the limit of zero magnetic diffusion for general initial data. Finally, for general initial data, we also establish the stability of the boundary layers of the incompressible viscousand diffusive MHD system with the different horizontal and vertical viscosities and magnetic diffusions, when they go to zero with thedifferent speeds.

8. 报告人：酒全森，首都师范大学

题目：Remarks on the Global Well-posedness of the 2D Generealized SQG

摘要：In this talk, we consider the 2D generalized SQG:

\begin{eqnarray*}

&&\omega_t+u\cdot\nabla\omega=0, x\in R^2, t>0,\\[3mm]

&&u=K\ast\omega,

\end{eqnarray*}

With$K(x)=\frac{x^\perp}{|x|^{2+2\alpha}},0\le\alpha\le \frac12.$ When $\alpha=0$, it is the two-dimensional Euler equations.  When $\alpha=\frac 12$, it corresponds to SQG. We will prove that if the existence interval of the smooth solution to SQG is $[0,T]$, then when $\alpha<\frac 12$ the existence interval of the generalized SQG will keep on $[0,T]$.

9. 报告人：王春朋，吉林大学

题目：Subsonic jet flows and subsonic-sonic nozzle flows

摘要：In this talk, I will first introduce the subsonic jet flows for a given surrounding pressure from finitely long convergent nozzles. Then, I will present our recent results on subsonic-sonic flows in finitely long and infinitely long nozzles. This talk is based on joint works with Professor Zhouping Xin.

10. 报告人：张挺，浙江大学

题目：Dispersive effects of the incompressible and compressible fluids

摘要：In this talk, we consider the  Cauchy problem of  the $N$-dimensional incompressible viscoelastic fluids with $N\ge2$. It is shown that, in the low frequency part,  thissystem possesses some dispersive properties derived from the one parameter group $e^{\pm it\Lambda}$. Based on this dispersive effect,  we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations.  The proof relies heavily on the dispersive estimates for  the system of acoustics, and a careful study of the nonlinear terms. And we also obtain the similar result for the  isentropic compressible Navier-Stokes equations. Here, the initial velocity with arbitrary $\dot{B}^{\fr{N}{2}-1}_{2,1}$ norm of potential part $\Pe^\bot u_0$  and large highly oscillating are allowed in our results.  (Joint works with Daoyuan Fang and Ruizhao Zi)

11. 报告人：曹广福，华南农业大学

题目：Wandering subspaces of the Hardy-Sobolev spaces over Dn