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We prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation
where is a typical smooth Fourier multiplier and is analytic in both variables. More precisely, we prove that if the initial datum is analytic in a strip of width whose norm on this strip is equal to , then if is small enough, the solution of the nonlinear Schrödinger equation above remains analytic in a strip of width , with norm bounded on this strip by over a very long time interval of order , where is arbitrary and and are positive constants depending on and .
We prove bounds on the restriction of spectral clusters to submanifolds in Riemannian manifolds equipped with metrics of regularity for . Our results allow for Lipschitz regularity when , meaning they give estimates on manifolds with boundary. When , the scalar second fundamental form for a codimension 1 submanifold can be defined, and we show improved estimates when this form is negative definite. This extends results of Burq, Gérard, and Tzvetkov and Hu to manifolds with low regularity metrics.
We consider a twisted magnetic Laplacian with Neumann condition on a smooth and bounded domain of in the semiclassical limit . Under generic assumptions, we prove that the eigenvalues admit a complete asymptotic expansion in powers of .
We study the stability/instability of the subsonic traveling waves of the nonlinear Schrödinger equation in dimension one. Our aim is to propose several methods for showing instability (use of the Grillakis–Shatah–Strauss theory, proof of existence of an unstable eigenvalue via an Evans function) or stability. For the latter, we show how to construct in a systematic way a Liapounov functional for which the traveling wave is a local minimizer. These approaches allow us to give a complete stability/instability analysis in the energy space including the critical case of the kink solution. We also treat the case of a cusp in the energy-momentum diagram.
The purpose of this note is to investigate the high-frequency behavior of solutions to linear Schrödinger equations. More precisely, Bourgain (1997) and Anantharaman and Macià (2011) proved that any weak- limit of the square density of solutions to the time-dependent homogeneous Schrödinger equation is absolutely continuous with respect to the Lebesgue measure on . The contribution of this article is that the same result automatically holds for nonhomogeneous Schrödinger equations, which allows for abstract potential type perturbations of the Laplace operator.
We consider a viscous fluid of finite depth below the air, occupying a three-dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The fluid dynamics are governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is neglected on the free surface. The long-time behavior of solutions near equilibrium has been an intriguing question since the work of Beale (1981).
This is the second in a series of three papers by the authors that answers the question. Here we consider the case in which the free interface is horizontally infinite; we prove that the problem is globally well-posed and that solutions decay to equilibrium at an algebraic rate. In particular, the free interface decays to a flat surface.
Our framework utilizes several techniques, which include
a priori estimates that utilize a “geometric” reformulation of the equations;
a two-tier energy method that couples the boundedness of high-order energy to the decay of low-order energy, the latter of which is necessary to balance out the growth of the highest derivatives of the free interface;
control of both negative and positive Sobolev norms, which enhances interpolation estimates and allows for the decay of infinite surface waves.
Our decay estimates lead to the construction of global-in-time solutions to the surface wave problem.