Related papers: Splash singularity for water waves
We show that given an initial vorticity which is bounded and $m$-fold rotationally symmetric for $m \ge 3$, there is a unique global solution to the 2D Euler equation on the whole plane. That is, in the well-known $L^1 \cap L^\infty$ theory…
We investigate the formation of singularities in the incompressible Navier-Stokes equations in $d\geq 2$ dimensions with a fractional Laplacian $|\nabla |^\alpha$. We derive analytically a sufficient but not necessary condition for…
We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging and no assumptions regarding the absence of stagnation…
We study the problem of evaporating drops contracting to a point. Going back to Maxwell and Langmuir, the existence of a spherical solution for which evaporating drops collapse to a point in a self-similar manner is well established in the…
For the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x~=~0$, it is well known that solutions can develop singularities in finite time. For an open dense set of initial data, the present paper provides a detailed asymptotic…
We present the recent result [8] concerning the existence of quasi-periodic in time traveling waves for the 2d pure gravity water waves system in infinite depth. We provide the first existence result of quasi-periodic water waves solutions…
We consider the Shallow Water equations in the supercritical and subcritical cases in one space variable,posed in a finite spatial interval with characteristic boundary conditions at the endpoints, which, as is well known, are…
In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to…
We investigate a two-dimensional transmission model consisting of a wave equation and a Kirchhoff plate equation with dynamical boundary controls under geometric conditions. The two equations are coupled through transmission conditions…
This paper aims to give a refined wave breaking description of the Cauchy problem to the one-dimensional nonlinear shallow water equations providing a sharp estimate of the lifespan of the solutions depending on the amplitude and topography…
We consider the two-dimensional deep gravity-capillary water waves with point vortices. We first formulate the question in the holomorphic coordinates. Then, we derive an a priori energy estimate for water waves, and show that the water…
This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\p_t^2u-\ds\sum_{i=1}^2\p_i(c_i^2(u)\p_iu)$ $=0$, where $c_i(u)\in C^{\infty}(\Bbb R^n)$, $c_i(0)\neq 0$, and…
We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional…
In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that…
We provide numerical evidence for the existence of a cascade of filament instabilities in the surface quasigeostrophic system for atmospheric and oceanic motions near a horizontal boundary. The cascade involves geometrically shrinking…
In the paper, the shock formation for the two-dimensional rotating shallow water system is established. We construct a large class of initial data which leads to the finite-time blow-up for the solutions. Moreover, the solutions are allowed…
The stability of naked singularities in self-similar collapse is probed using scalar waves. It is shown that the multipoles of a minimally coupled massless scalar field propagating on a spherically symmetric self-similar background…
We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space $H^s$ with $s>3/2$. The main idea is to consider two suitable sequences of…
We consider the Cauchy problem for the equations of pressureless gases in two space dimensions. For a generic set of smooth initial data (density and velocity), it is known that the solution loses regularity at a finite time $t_0$, where…
This paper considers two-dimensional steady continuous stratified periodic water waves. Firstly, we prove that each streamline must be symmetric about the crest line when it is strictly monotonous between troughs and crests by exploiting…