Related papers: Global regularity for the 3D finite depth capillar…
We prove a unique continuation from infinity theorem for regular waves of the form $[ \Box + \mathcal{V} (t, x) ]\phi=0$. Under the assumption of no incoming and no outgoing radiation on specific halves of past and future null infinities,…
Two-dimensional periodic surface waves propagating under the combined influence of gravity and surface tension on water of finite depth are considered. Within the framework of small-amplitude waves, we find the exact solutions of the…
In this paper, we examine the formation of small capillary waves (parasitic ripples) on the surface of steep steadily-travelling gravity waves. Previously, authors have developed ad-hoc analytical procedures for describing the formation of…
In this paper, we study solitary waves propagating along the surface of an infinitely deep body of water in two or three dimensions. The waves are acted upon by gravity and capillary effects are allowed --- but not required --- on the…
We study the Cauchy problem for 3-D nonlinear elastic waves satisfying the null condition with low regularity initial data. In the radially symmetric case, we prove the global existence of a low regularity solution for every small data in…
We consider the two-dimensional capillary water waves with nonzero constant vorticity in infinite depth. We first derive the Babenko equation that describes the profile of the solitary wave. When the velocity $c$ is close to a critical…
In this paper we consider the steady water wave problem for waves that possess a merely $L_r-$integrable vorticity, with $r\in(1,\infty)$ being arbitrary. We first establish the equivalence of the three formulations--the velocity…
A key feature of $(1+1)$-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave…
We study two-dimensional periodic capillary-gravity water waves propagating at the free surface of water in a flow with arbitrary, prescribed vorticity over a flat bed. Using conformal mappings and a new Babenko-type reformulation of…
We consider the propagation of acoustic waves in a medium with a periodic array of small inclusions of arbitrary shape. The inclusion size $a$ is much smaller than the array period. We show that global gaps do not exist if $a$ is small…
In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a…
The nonlinear wave equation $u_{tt}-\Delta u +|u_t|^{p-1}u_t=0$ is shown to be globally well-posed in the Sobolev spaces of radially symmetric functions $H^k_{\rm rad}({\bf R}^3)\times H^{k-1}_{\rm rad}({\bf R}^3)$ for all $p\geq 3$ and…
We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the relative velocity field is a Beltrami field,…
We prove the global stability of small perturbation near the constant equilibrium for the two dimensional simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model, where the direction function of liquid crystal…
In this Letter we consider long capillary-gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially…
Stability criterion for the surface gravity capillary waves in a flowing two-layered fluid system with viscous dissipation is investigated. It is seen that the dissipative instability of negative energy waves is absent,- contrary to what…
In this article, we consider the Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model without kinematic transport in three spatial dimensions, which is a nonlinear coupling of incompressible Navier-Stokes equations…
We investigate the existence and properties of traveling waves for the Euler-Korteweg system with general capillarity and pressure. Our main result is the existence in dimension two of waves with arbitrarily small energy. They are obtained…
In this paper, we study the nonexistence of global weak solutions for a wave equation with nonlinear memory and damping terms. We give an answer to an open problem posed in [M. D'Abbicco, A wave equation with structural damping and…
We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i \pt_t u = \sqrt{-\Delta} \, u - |u|^{p-1} u, \quad \mbox{with $(t,x) \in \R \times \R^d$} $$ with exponents $1 < p < \infty$ for $d=1$ and $1 < p <…