Related papers: The $\alpha$-SQG patch problem is illposed in $C^{…

We prove well-posedness results for the Dirichlet problem in $\mathbb{R}^{n}_{+}$ for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized H\"older spaces…

We study the Cauchy problem for fractional Schr\"odinger equation with cubic convolution nonlinearity ($i\partial_t u - (-\Delta)^{\frac{\alpha}{2}}u\pm (K\ast |u|^2) u =0$) with Cauchy data in the modulation spaces $M^{p,q}(\mathbb…

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation \[\partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\ u(x,0)=u_0(x),\] is locally well-posed in the Sobolev spaces $H^s$ for $s>1-\alpha$ if $0\leq…

This article is concerned with the well-posedness of the incompressible Euler equations describing a stably stratified ocean, reformulated in isopycnal coordinates. Our motivation for using this reformulation is twofold: first, its quasi-2D…

We prove $C^{2,\alpha}$ regularity of sufficiently flat free boundaries, for the thin one-phase problem in which the free boundary occurs on a lower dimensional subspace. This problem appears also as a model of a one-phase free boundary…

We study the wellposedness of Cauchy problem for the fourth order nonlinear Schr\"odinger equations i\partial_t u=-\eps\Delta u+\Delta^2 u+P((\partial_x^\alpha u)_{\abs{\alpha}\ls 2}, (\partial_x^\alpha \bar{u})_{\abs{\alpha}\ls 2}),\quad…

We study the rigidity problem for $(-\alpha)$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $\Omega_{a, b, \theta_0}:= \{(r,\theta): a<r<b, \ 0<\theta<\theta_0\}$, where…

In this paper we study the well-posedness in Sobolev spaces of the incompressible Euler equations in an infinite strip delimited from below by a non-flat bottom and from above by a free-surface. We allow the presence of vorticity and…

In an open bounded interval $\Omega$, the problem \[ u_{tt} = u_{xx} - \big(f(\Theta)\big)_x, \Theta_t = \Theta_{xx} - f(\Theta) u_{xt}, \] is considered under the boundary conditions $u|_{\partial\Omega}=\Theta_x|_{\partial\Omega}=0$, and…

The general surface quasi-geostrophic equation is the scalar transport equation defined by \begin{equation*} \frac{\partial \theta}{\partial t}+v^\gamma_1 \frac{\partial \theta}{\partial x_1}+v^\gamma_2 \frac{\partial \theta}{\partial x_2}…

In this paper we establish well posedness of the Neumann problem with boundary data in $L^2$ or the Sobolev space $\dot W^2_{-1}$, in the half space, for linear elliptic differential operators with coefficients that are constant in the…

In this paper we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=\lambda|x|^{-\alpha}|u|^{\beta}u$ in $H^1$. The well-posedness theory in $H^1$ has been intensively…

We provide a simple proof that the Cauchy problem for the incompressible Euler equations in $\mathbb{R}^{d}$ with any $d\ge3$ is ill-posed in critical Sobolev spaces, extending an earlier work of Bourgain and Li in the case $d = 3$. The…

We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$, $u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is of class…

We study the global well-posedness of the two-dimensional defocusing fourth-order Schr\"odinger initial value problem with power type nonlinearities $\vert u\vert^{2k}u$ where $k$ is a positive integer. By using the $I$-method, we prove…

Consider the $3$-d primitive equations in a layer domain $\Omega=G \times (-h,0)$, $G=(0,1)^2$, subject to mixed Dirichlet and Neumann boundary conditions at $z=-h$ and $z=0$, respectively, and the periodic lateral boundary condition. It is…

We prove the global existence of solutions with small and smooth initial data of a nonlinear dispersive equation for the motion of generalized surface quasi-geostrophic (GSQG) fronts in a parameter regime $1<\alpha<2$, where $\alpha=1$…

In this article we consider the Euler-$\alpha$ system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary…

We consider second-order elliptic equations in a half space with leading coefficients measurable in a tangential direction. We prove the $W^2_p$-estimate and solvability for the Dirichlet problem when $p\in (1,2]$, and for the Neumann…

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…