Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space

Instability and nonuniqueness for the $2d$ Euler equations in vorticity form, after M. Vishik

In this expository work, we present Vishik's theorem on non-unique weak solutions to the two-dimensional Euler equations on the whole space, \[ \partial_t ω+ u \cdot \nabla ω= f \, , \quad u = \frac{1}{2π} \frac{x^\perp}{|x|^2} \ast ω\, , \] with initial vorticity $ω_0 \in L^1 \cap L^p$ and $f \in L^1_t (L^1 \cap L^p)_x$, $p < \infty$. His theorem demonstrates, in particular, the sharpness of the Yudovich class. An important intermediate step is the rigorous construction of an unstable vortex, which is of independent physical and mathematical interest. We follow the strategy of Vishik but allo…

Vishik非唯一性定理的证明：强迫二维欧拉方程在涡度类L¹∩L^p中的情况

A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation

本文给出强迫二维欧拉方程在涡度类L¹∩L^p中Vishik非唯一性定理的更简单证明，通过构造稳定的涡旋来简化证明过程。

Uniqueness and Nonuniqueness in Inverse Hyperbolic Problems and The Black Hole Phenomenon

This paper consists of two parts. In the first part we describe the recent works on the inverse problems for the wave equation in $(n+1)$-dimensional space equipped with pseudo-Riemannian metric with Lorentz signature. We study the conditions of the existence of black (or white) holes for these wave equations. In the second part we prove energy type estimates on a finite time interval in the presence of black or white holes. We use these estimates to prove the nonuniqueness of the inverse problems.

具有可积涡量的二维欧拉流的柔性

Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity

该研究提出新型凸积分格式，引入Lamb-Chaplygin偶极子构建的基础模块与基于时间平均的误差抵消法，证明二维欧拉方程一类弱解的柔性与非唯一性，突破标准凸积分的临界标度。

$\varepsilon$-regularity criteria in Lorentz spaces to the 3D Navier-Stokes equations

In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $\varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, or deformation tensor in Lorentz spaces. As an application, this allows us to extend the result involving Leray's blow up rate in time, and to show that the number of singular points of weak solutions belonging to $ L^{p,\infty}(-1,0;L^{q,l}(\mathbb{R}^{3})) $ and $ {2}/{p}+{3}/{q}=1$ with $3<q<\infty$ and $q\leq l <\infty$ is finite.

Lorentz Space Estimates and Jacobian Convergence for the Ginzburg-Landau Energy with Applied Magnetic Field

In this paper we continue the study of Lorentz space estimates for the Ginzburg-Landau energy started in our previous paper, \cite{p1}. We focus on getting estimates for the Ginzburg-Landau energy with external magnetic field $h_{ex}$ in certain interesting regimes of $h_{ex}$. This allows us to show that for configurations close to minimizers or local minimizers of the energy, the vorticity mass of the configuration $(u,A)$ is comparable to the $L^{2,\infty}$ Lorentz space norm of $\nabla_A u$. We also establish convergence of the gauge-invariant Jacobians (vorticity measures) in the dual of …