Remarks on stationary and uniformly-rotating vortex sheets: Rigidity results

关于由测地线或零测地线长度确定洛伦兹度量的注记

Remarks on the determination of the Lorentzian metric by the lengths of geodesics or null-geodesics

本文研究洛伦兹度量由测地线或零测地线长度确定的问题，证明了度量在给定初始数据下的可恢复性及刚性，并推广到零测地线情形。

Remarks on a mean field equation on $\mathbb{S}^{2}$

In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -Δ_{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.

Flexibility and rigidity of free boundary MHD equilibria

We study stationary free boundary configurations of an ideal incompressible magnetohydrodynamic fluid possessing nested flux surfaces. In 2D simply connected domains, we prove that if the magnetic field and velocity field are never commensurate, the only possible domain for any such equilibria is a disk, and the velocity and magnetic field are circular. We give examples of non-symmetric equilibria occupying a domain of any shape by imposing an external magnetic field generated by a singular current sheet charge distribution (external coils). Some results carry over to 3D axisymmetric solutions…

A Kato type criterion for the zero viscosity limit of the incompressible Navier-Stokes flows with vortex sheets data

There are a few examples of solutions to the incompressible Euler equations which are piecewise smooth with a discontinuity of the tangential velocity across a hypersurface evolving in time: the so-called vortex sheets. An important open problem is to determine whether or not these solutions can be obtained as zero viscosity limits of the incompressible Navier-Stokes solutions in the energy space. In this paper we establish a couple of sufficient conditions similar to the one obtained by Kato in [T.~Kato. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. Semi…

关于半线性Neumann问题中模式的一些注记

Some remarks on patterns for semilinear Neumann problems

本文研究半线性椭圆方程的解的结构，证明在非线性满足单调性条件时，所有解均为常数，不依赖于领域凸性，且通过构造例子验证假设的尖锐性。

Remarks of Global Wellposedness of Liquid Crystal Flows and Heat Flows of Harmonic Maps in Two Dimensions

We consider the Cauchy problem to the two-dimensional incompressible liquid crystal equation and the heat flows of harmonic maps equation. Under a natural geometric angle condition, we give a new proof of the global well-posedness of smooth solutions for a class of large initial data in energy space. This result was originally obtained by Ding-Lin in \cite{DingLin} and Lin-Lin-Wang in \cite{LinLinWang}. Our main technical tool is a rigidity theorem which gives the coercivity of the harmonic energy under certain angle condition. Our proof is based on a frequency localization argument combined w…

Boundary properties of fractional objects: flexibility of linear equations and rigidity of minimal graphs

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in~$\R^3$, i.e. nonlocal minimal surfaces with a graphical structure. We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum. This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can …

A remark on constant mean curvature hypersurfaces in warped product manifolds

Alexandrov's theorem asserts that spheres are the only closed embedded constant mean curvature hypersurfaces in space forms. In this paper, we consider Alexandrov's theorem in warped product manifolds and prove a rigidity result in the spirit of Alexandrov's theorem. Our approach generalizes the proofs of Reilly and Ros and, under more restrictive assumptions, it provides an alternative proof of a recent theorem of Brendle.

关于二维欧拉方程定常与匀速旋转解径向对称性的注记

Remarks on radial symmetry of stationary and uniformly-rotating solutions for the 2D Euler equation

本文证明了二维不可压缩欧拉方程中具有紧支集涡量的匀速旋转解在角速度满足特定区间条件时必为径向对称，推广了既有刚性定理并涵盖不规则涡块。

接近圆形涡旋片的 Kelvin-Helmholtz 方程的长时间动力学

Long-time dynamics for the Kelvin-Helmholtz equations close to circular vortex sheets

本文研究圆形稳态解附近的 Kelvin-Helmholtz 系统，利用 Hamiltonian Birkhoff 正规型和仿线性化技术，证明了小振幅解的几乎整体存在性，揭示了速度跳跃与毛细效应共同防止非线性失稳的长时间稳定化现象。