Dispersive Estimates, Strichartz Estimates and Smoothing Effects
نویسنده
چکیده
In this short expository note, we will discuss three subjects: dispersive estimates, Strichartz estimates and smoothing effects which are of great importance in the study of dispersive equations. We will focus on the Euclidean setting and try to derive the interplay among above three objects. We will mainly focus on linear problems in various contexts. Some nonlinear application will also be mentioned. As this note is written as a topic proposal, many detailed proofs will be omitted.
منابع مشابه
Smoothing - Strichartz Estimates for the Schrödinger Equation with Small Magnetic Potential Vladimir Georgiev, Atanas Stefanov and Mirko Tarulli
The work treats smoothing and dispersive properties of solutions to the Schrödinger equation with magnetic potential. Under suitable smallness assumption on the potential involving scale invariant norms we prove smoothing Strichartz estimate for the corresponding Cauchy problem. An application that guarantees absence of pure point spectrum of the corresponding perturbed Laplace operator is disc...
متن کاملSmoothing - Strichartz Estimates for the Schrödinger Equation with Small Magnetic Potential
The work treats smoothing and dispersive properties of solutions to the Schrödinger equation with magnetic potential. Under suitable smallness assumption on the potential involving scale invariant norms we prove smoothing Strichartz estimate for the corresponding Cauchy problem. An application that guarantees absence of pure point spectrum of the corresponding perturbed Laplace operator is disc...
متن کاملJ un 2 00 8 MAGNETIC VIRIAL IDENTITIES , WEAK DISPERSION AND STRICHARTZ INEQUALITIES
We show a family of virial-type identities for the Schrödinger and wave equations with electromagnetic potentials. As a consequence, some weak dispersive inequalities in space dimension n ≥ 3, involving Morawetz and smoothing estimates, are proved; finally, we apply them to prove Strichartz inequalities for the wave equation with a non-trapping electromagnetic potential with almost Coulomb decay.
متن کاملun 2 00 8 MAGNETIC VIRIAL IDENTITIES , WEAK DISPERSION AND STRICHARTZ INEQUALITIES
We show a family of virial-type identities for the Schrödinger and wave equations with electromagnetic potentials. As a consequence, some weak dispersive inequalities in space dimension n ≥ 3, involving Morawetz and smoothing estimates, are proved; finally, we apply them to prove Strichartz inequalities for the wave equation with a non-trapping electromagnetic potential with almost Coulomb decay.
متن کاملA Singular Critical Potential For The Schrödinger operator
We construct a potential V on R, smooth away from one pole, and a sequence of quasimodes for the operator −∆+V , which concentrate on this pole. No smoothing effect, Strichartz estimates nor dispersive inequalities hold for the corresponding Schrödinger equation.
متن کامل