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

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

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $Ω$, such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations

We study convex integration solutions in the context of the modelling of shape-memory alloys. The purpose of the article is two-fold, treating both rigidity and flexibility properties: Firstly, we relate the maximal regularity of convex integration solutions to the presence of lower bounds in variational models with surface energy. Hence, variational models with surface energy could be viewed as a selection mechanism allowing for or excluding convex integration solutions. Secondly, we present the first numerical implementations of convex integration schemes for the model problem of the geometr…

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…

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…

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。

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 …