Strichartz estimates are a family of inequalities for linear dispersive partial differential equations, that arose out of contentions to the Fourier restriction problem. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces.
Strichartz estimates are a family of inequalities for linear dispersive partial differential equations, that arose out of contentions to the Fourier restriction problem. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces.
Consider the linear Schrödinger equation in $ \mathbb R ^ d $ with $ h = m = 1 $. Then the solution for initial data $ u _ 0 $ is given by $ e ^ { \frac { i t \Delta} 2 } u _ 0 $. Let $ q $ and $ r $ be real numbers satisfying $ 2 \le q , r \le \infty $, $ \frac 2 q + \frac d r = \frac d 2 $ and $ ( q , r , d ) \ne ( 2 ,\infty , 2 ) $. In this case the homogeneous Strichartz estimates take the form:
$$ \left\lVert e ^ { \frac { i t \Delta} 2 } u _ 0 \right\rVert _ { L _ t ^ q L _ x ^ r } \le C _ { d , q , r } \lVert u _ 0 \rVert _ 2 \text . $$
Further suppose that $ \tilde q , \tilde r $ satisfy the same restrictions as $ q , r $ and $ { \tilde q }' , {\tilde r }' $ are their dual exponents, then the dual homogeneous Strichartz estimates take the form:
$$ \left\lVert \int _ { \mathbb R } e ^ { - \frac { i s \Delta } 2 } F ( s ) \operatorname d s \right\rVert _ { L _ x ^ 2 } \le C _ { d , \tilde q , \tilde r } \lVert F \rVert _ { L _ t ^ { { \tilde q } ' } L _ x ^ { { \tilde r } ' } } \text . $$
The inhomogeneous Strichartz estimates are:
$$ \left\lVert \int _ { s < t } e ^ { \frac { i ( t - s ) \Delta } 2 } F ( s ) \operatorname d s \right\rVert _ { L _ t ^ q L _ x ^ r } \le C _ { d , q , r , \tilde q , \tilde r } \lVert F \rVert _ { L _ t ^ { { \tilde q } ' } L _ x ^ { { \tilde r } ' } } \text . $$
Source: Wikipedia
