Related papers: Persistence property in weighted Sobolev spaces fo…
The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr\"odinger type equation. Local…
In this paper, we reconsider the well-known result of Pego-Weinstein \cite{MR1289328} that soliton solutions to the Korteweg-deVries equation are asymptotically stable in exponentially weighted spaces. In this work, we recreate this result…
We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative…
We consider a Nonlinear Schr\"odinger Equation with a very general non linear term and with a trapping $\delta $ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by…
In this work we present a newly developed study of the interpolation of weighted Sobolev spaces by the complex method. We show that in some cases, one can obtain an analogue of the famous Stein-Weiss theorem for weighted $L^{p}$ spaces. We…
We study the initial value problem associated to the dispersion generalized Benjamin-Ono equation. Our aim is to establish well posedness results in weighted Sobolev spaces and to deduce from them some sharp unique continuation properties…
This paper addresses the problem of global well-posedness of a coupled system of Korteweg-de Vries equations, derived by Majda and Biello in the context of nonlinear resonant interaction of Rossby waves, in a periodic setting in homogeneous…
Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann--Hilbert problems and the Dbar approach, the long-time asymptotic behavior of solutions to the fifth-order modified Korteweg-de Vries equation on the…
We consider a number of linear and non-linear boundary value problems involving generalized Schr\"odinger equations. The model case is $-\Delta u=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev…
Strong Feller property and irreducibility are study for a class of non-linear monotone stochastic partial differential equations with multiplicative noise. H\"older continuity of the associated Markov semigroups are discussed in some…
We consider a general nonlinear dispersive equation with monomial nonlinearity of order $k$ over $\mathbb{R}^d$. We construct a rigorous theory which states that higher-order nonlinearities and higher dimensions induce sharper local…
We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to…
We consider the Schroedinger equation with a general interaction term, which is localized in space. The interaction may be x, t dependent and non-linear. Purely non-linear parts of the interaction are localized via the radial Sobolev…
The aim of this work is to consider the internal stabilization of a nonlinear coupled system of two Korteweg--de Vries equations in a finite interval under the effect of a very weak localized damping. The system was introduced by Gear and…
In this article, we investigate observability-related properties of the Korteweg-de Vries equation with a discontinuous main coefficient, coupled by suitable interface conditions. The main result is a novel two-parameter Carleman estimate…
We study the zero-dispersion limit for a class of Korteweg--de Vries (KdV)-type initial-boundary value problems on the half-line, with Dirichlet boundary conditions assigned at \(x=0\). We focus on the outflow regime, where the solution of…
Variable order space-fractional diffusion equation derived as an important model to describe complex anomalous diffusion phenomenon. In this article, well-posedness theory has been constructed for equations with the "Dirichlet" or the…
A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and…
We present extensions of rigidity estimates and of Korn's inequality to the setting of (mixed) variable exponents growth. The proof techniques, based on a classical covering argument, rely on the log-H\"older continuity of the exponent to…
In this paper we prove the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed $L^2$-norm, and the stability of associated solitary waves for two classes of coupled nonlinear…