Related papers: Precise smoothing effect in the exterior of balls
In this paper we prove the smoothing effect for solutions of Schr{\"o}dinger equations with variable coefficients and in a non trapping exterior domain. We allow quadratic potentials at infinity.
We consider refinements of the local smoothing estimates for the Schr\"odinger equation in domains which are exterior to a strictly convex obstacle in $\RR^n$. By restricting the solution to small, frequency dependent collars of the…
We prove, under the exterior geometric control condition, the Kato smoothing effect for solutions of an inhomogenous and damped Schr\"odinger equation on exterior domains.
The smoothing effect states that solutions to the Schr{\"o}dinger equation in the Euclidean space have, for almost-every time, a local-in-space improved regularity (gain of half a derivative in Sobolev spaces). In this note, we show that,…
We show the necessity of the non trapping condition for the plain smoothing effect ($H^{1/2}$) for Schr\"odinger equation with Dirichlet boundary conditions in exterior problems. We also give a class of trapped obstacles (Ikawa's example)…
In this paper a global smoothing property of Schrodinger equations is established in the critical case in dimensions two and higher. It is shown that the critical smoothing estimate is attained if the smoothing operator has some structure.…
In \cite{poiret}, we explain how we can construct global solutions for the cubic Schr\"odinger equation in three dimensional with initial data in $ L^2(\mathds{R}^3) $. The main ingredient of this proof is the existence of the bilinear…
We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary. We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside…
In this article, we consider the focusing cubic nonlinear Schr\"odinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground…
We study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear Schr\"odinger equation with constant coefficients which appears as a model for propagation of pulse in optical…
We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schroedinger and Gross-Pitaevskii equations on the exterior of a non-trapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained…
\rm We obtain the global smooth effects for the solutions of the linear Schr\"odinger equation in anisotropic Lebesgue spaces. Applying these estimates, we study the Cauchy problem for the generalized elliptical and non-elliptical…
We consider the Cauchy problem of a class of higher order Schr\"odinger type equations with constant coefficients. By employing the energy inequality, we show the $L^2$ well-posedness, the parabolic smoothing and a breakdown of the…
The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions.
In this paper, we prove global well-posedness and scattering for the defocusing, cubic nonlinear Schr{\"o}dinger equation in three dimensions when $n = 3$ when $u_{0} \in H^{s}(\mathbf{R}^{3})$, $s > 3/4$. To this end, we utilize a…
We describe regularizing effects in the linearization of a kinetic equation that arises in study of a system of nonlinear waves satisfying the Schr\"odinger equation in terms of weak turbulence and condensate. The problem is first…
In this article, we study the decay of the solutions of Schr\"odinger equations in the exterior of an obstacle. The main situations we are interested in are the general case (no non-trapping assumptions) or some weakly trapping situations
We outline a general method of obtaining exact solutions of Schroedinger equations with a position dependent effective mass. Exact solutions of several potentials including shape invariant potentials have also been obtained.
In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schr\"odinger equation. Recently, Dodson [arXiv:2004.09618] studied the global well-posedness in a critical Sobolev space $\dot{W}^{11/7,7/6}$. In this…
We prove some smoothing effects for the 3-D Navier-Stokes equations for initial data belonging to the critical Sobolev space $H^{1/2}(\R^3)$. Asymptotic behavior of the global solution when the time goes to infinity is studied. We also…