Related papers: Spatial decay properties for a model in shear flow…
We consider the Cauchy problem for the two-dimensional Novikov-Veselov equation integrable via the inverse scattering problem for the Schr\"odinger operator with fixed negative energy. The associated linear equation is characterized by a…
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…
This paper is devoted to the large time behavior of weak solutions to the three-dimensional Vlasov-Navier-Stokes system set on the half-space, with an external gravity force. This fluid-kinetic coupling arises in the modeling of…
We revisit the local well-posedness theory of nonlinear Schr\"odinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data…
A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version of the…
We consider a model case for a strictly convex domain of dimension $d\geq 2$ with smooth boundary and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay…
We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate…
We consider unique continuation properties of solutions to a family of evolution equations. Our interest is mainly on nonlinear non-local models. This class contains the Benjamin-Ono, the Intermediate Long Wave, the Camassa-Holm, the…
We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized…
We consider the defocusing nonlinear Schr{\"o}dinger equation with a gauge invariant power-like nonlinearity. We prove global dispersive estimates in a semi-classical scaling, after rescaling the solution thanks to a suitable distorsion of…
In recent work the authors proposed a broad global well-posedness conjecture for cubic defocusing dispersive equations in one space dimension, and then proved this conjecture in two cases, namely for one dimensional semilinear and…
Stability of inviscid shear shallow water flows with free surface is studied in the framework of the Benney equations. This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations. It is…
We prove global weighted Strichartz estimates for radial solutions of linear Schr\"odinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields…
This paper is concerned with the energy decay of a viscoelastic variable coefficient wave equation with nonlocality in time as well as nonlinear damping and polynomial nonlinear terms. Using the Lyapunov method, we establish a polynomial…
In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of…
We present several new spatially localized equilibrium and traveling-wave solutions of plane Couette and channel flows. The solutions exhibit strikingly concentrated regions of vorticity that are flanked on either side by high-speed…
We prove that the initial value problem associated to a nonlocal perturbation of the Benjamin-Ono equation is locally and globally well-posed in Sobolev spaces $H^s(\mathbb{R})$ for any $s>-3/2$ and we establish that our result is sharp in…
In this paper we study the regularity properties of solutions to the Davey-Stewartson system. It is shown that for initial data in a Sobolev space, the nonlinear part of the solution flow resides in a smoother space than the initial data…
In this paper, we address the question of the hyperbolicity and the local well- posedness of the multi-layer shallow water model, with free surface, in two dimensions. We first provide a general criterion that proves the symmetrizability of…
In the first part of this work we study the local well-posedness of dispersive equations in the weighted spaces $H^s(\mathbb{R})\cap L^2(|x|^{2b}dx)$. We then apply our results for several dispersive models such as the Hirota-Satsuma…