Strichartz estimates

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

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

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

$\left\|u\right\|_{L^{4}({{\mathbb {R}}^{3+1}}_{+})}\leq C\left(\left\|f\right\|_{{{\dot {H}}^{1/2}}({\mathbb {R}}^{3})}+\left\|g\right\|_{{{\dot {H}}^{-1/2}}({\mathbb {R}}^{3})}+\int \limits _{0}^{\infty }\left\|F\right\|_{L^{2}({\mathbb {R}}^{3})}\right)$