【70周年院庆学术论坛】非线性偏微分方程研讨会日程安排

会议报告摘要

南京大学：吕勇

题目：Space-time resonances and high-frequency Raman instabilities in the two-fluid Euler-Maxwell system

摘要：We apply the symbolic flow method to the two-fluid Euler-Maxwell system, and show that space-time resonances induce high-frequency Raman and Langmuir instabilities. A consequence is that the Zakharov WKB approximation to Euler-Maxwell is unstable for non-zero group velocities. A key step in the proof is the reformulation of the set of resonant frequencies as the locus of weak hyperbolicity for linearized equations around the WKB solution. Due to large transverse variations in the WKB profile, the equation satisfied by the symbolic flow around resonant frequencies is a linear  partial differential equation. At space-time resonances corresponding to the Raman or Langmuir instability, we observe a fast growth of the symbolic flow, which translates into an instability result for the original system.

中国科学院大学：邱国寰

题目：Very weak solutions of the Dirichlet problem for 2-Hessian equation
摘要：Weyl's lemma states that every weak solution of Laplace's equation is also a smooth solution.  We find that there is no Weyl's lemma for 2-Hessian equations. For any α small, we construct infinitely many C 1,α very weak solutions to the 2- Hessian equation with prescribed boundary value. This is joint work with Tongtong Li.

北京大学：王超

题目: The interaction between rough vortex patch  and boundary layer

摘要: In this talk, we talk about the asymptotic behavior of the solution of  the Navier-Stokes equations in the half plane at high Reynolds number regime, when the initial vorticity belongs to the Yudovich class and is supported away from the boundary.  We prove the $L^p$ ($2\leq p< \infty$) convergence from the Naiver-Stokes equations to the Euler equations. The key point is to introduce a good functional framework to control the interaction between rough vortex patch and boundary layer.

华东师范大学： 张德凯

题目：Homogeneous k-Hessian equations and some applications

摘要：In this talk, we consider the homogeneous k-Hessian equation in an exterior domain and in a punctured domain respectively. We prove the existence of  solutions by showing that the approximating solutions are uniformly controlled pointwisely by the fundamental solution up to second order. As an application, we prove  monotonicity formulae along the the level set of the approximating solution and obtain some integral inequalities on the boundary of the domain. This talk is based on joint work with Prof. Xi-Nan Ma and Prof. Zhenghuan Gao.