Strichartz estimates

From DispersiveWiki

(Redirected from Strichartz)

This article is a stub. You can help DispersiveWiki by expanding it.

Strichartz estimates are spacetime estimates on homogeneous and inhomogeneous linear dispersive and wave equations. They are particularly useful for solving semilinear perturbations of such equations, in which no derivatives are present in the nonlinearity.

Strichartz estimates can be derived abstractly as a consequence of a dispersive inequality and an energy inequality.

Linear Strichartz estimate

Let $\dot H^{\alpha}(\Bbb R^n)$ denote the homogeneous Sobolev space with norm

$\left\| u \right\|_{\dot H^{\alpha}(\Bbb R^n)} = \left\|(-\Delta^{\alpha/2}) u \right\|_{L^2(\Bbb R^n)}$

If $u$ solves the linear wave equation

$\Box u = F(t,x)$

with data

$u(0,\cdot)=f \qquad \partial_t u (0,\cdot )=g$

then the Strichartz estimates states that

$\left\| u \right\|_{L^{4}({{\Bbb R}^{3+1}}_+)} \leq C \left( \left\| f \right\|_{{\dot H^{1/2}}(\Bbb R^3)} + \left\| g \right\|_{{\dot H^{-1/2}}(\Bbb R^3)} + \int\limits_0^{\infty} \left\| F \right\|_{L^2(\Bbb R^{3})} \right)$

Strichartz estimates

From DispersiveWiki

Linear Strichartz estimate

Views

Personal tools

orientation

Categories

development

Search

Toolbox