Strichartz estimates for the wave and the Schr_dinger equation
Ramona Anton
ENS, France.
12-09-03 (Institute Colloquium)

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Abstract

This talk follows the article of M.Keel and T.Tao, Endpoint Strichartz estimates.

The solution of the Schr_dinger equation and that of the wave equation can be written as evolution operators. Those operators are bounded on $L^2(\mathbb{R}^d)$, as those equations conserve the energy and they also verify a decay estimates: $||U(t)U*(s)f||_{L^\infty_x} \leq C|t-s|^{d/2} ||f||_{L^1_x}$ for $f\in C^\infty_0$.

Using those estimates and functional analisys technique we prove the Strichartz estimates $$||U(t)f||_{L^p_t L^q_x} \leq C ||f||_{L^2},$$ where $p$, $q$ and $d$ are related by a relation that can be deduce by a scale argument.

Strichartz estimates enable better imbeddings that Sobolev imbeddings. We prove regularity and global existence for the cubic Schr_dinger equation.