Related papers: Strong ill-posedness of logarithmically regularize…
We consider the hydrodynamics of an incompressible fluid on a 2D periodic domain. There exists a family of stationary solutions with vorticity given by $\Omega^*=\alpha\cos (\mathbf{p} \cdot \mathbf{x} )+\beta \sin (\mathbf{p} \cdot…
Let $\Omega \subset \mathbb{R}^n$ be a smooth bounded domain having zero in its interior $0 \in \Omega.$ We fix $0 < \alpha \le 2$ and $0 \le s <\alpha.$ We investigate a sufficient condition for the existence of a positive solution for the…
We consider concentrated vorticities for the Euler equation on a smooth domain $\Omega \subset \mathbf{R}^2$ in the form of \[ \omega = \sum_{j=1}^N \omega_j \chi_{\Omega_j}, \quad |\Omega_j| = \pi r_j^2, \quad \int_{\Omega_j} \omega_j d\mu…
The global well-posedness and stability of solutions to the three-dimensional compressible Euler equations with damping is a longstanding open problem. This problem was addressed in \cite{WY, STW} in the isentropic regime (i.e. $\gamma>1$)…
In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschtiz function and the free surface belongs to…
We construct global-in-time, unique solutions of the two-dimensional Euler equations in a Yudovich type space and a $\rm bmo$-type space. First, we show the regularity of solutions for the two-dimensional Euler equations in the Spanne space…
In this work, we develop weighted Lorentz-Sobolev estimates for viscosity solutions of fully nonlinear elliptic equations with oblique boundary condition under weakened convexity conditions in the following configuration $F(D^{2}u, Du, u,…
We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + \alpha \partial \mathcal{R}(u) \ni f^\delta \end{equation*} for linear and nonlinear ill-posed problems with monotone operators…
This paper investigates the local regularity of solutions to stationary Fokker-Planck equations on an open set $U \subset \mathbb{R}^d$ with $d \geq 2$. A central objective is to relax the classical assumptions on the coefficients by…
We investiage the (slightly) super-critical 2-D Euler equations. The paper consists of two parts. In the first part we prove well-posedness in $C^s$ spaces for all $s>0.$ We also give growth estimates for the $C^s$ norms of the vorticity…
This paper is concerned with the global well-posedness of the two-dimensional incompressible vorticity equation in the half plane. Under the assumption that the initial vorticity $\omega_0\in W^{k,p}(\R^{2}_+)$ with $k\geq3$ and $1<p<2$, it…
We study a class of weakly hyperbolic Cauchy problems on $\mathbb{R}^d$, involving linear operators with characteristics of variable multiplicities, whose coefficients are unbounded in the space variable. The behaviour in the time variable…
We establish that the quadratic non-linear Schr\"odinger equation $$ iu_t + u_{xx} = u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\R)$ when $s \geq -1$ and ill-posed when $s < -1$. Previous work of Kenig, Ponce and…
We construct solutions of the 2D incompressible Euler equations in $\mathds{R}^2\times [0,\infty)$ such that initially the velocity is in the super-critical Sobolev space $H^\beta$ for $1<\beta<2$, but are not in $H^{\beta'}$ for…
We study regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F $ in bounded domains in $\R^n$. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like…
Many questions related to well-posedness/ill-posedness in critical spaces for hydrodynamic equations have been open for many years. In this article we give a new approach to studying norm inflation (in some critical spaces) for a wide class…
In this paper we are concerned with a initial boundary-value problem for a coupled system of two KdV equations, posed on the positive half line, under the effect of a localized damping term. The model arises when modeling the propagation of…
It was shown recently by Ars\'enio and the author that the two-dimensional incompressible Euler--Maxwell system is globally well-posed in the Yudovich class, provided that the electromagnetic field enjoys appropriate conditions, including…
In this note, using the ideas from our recent article \cite{EM}, we prove strong ill-posedness for the 2D Euler equations in $C^k$ spaces. This note provides a significantly shorter proof of many of the main results in \cite{BLi2}. In the…
We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the "(sound) wave-part" of the data in Sobolev space and "transport-part" in…