Strichartz estimate

In applied mathematics, Strichartz estimates are a family of inequalities for linear dispersive partial differential equations. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces. They were first noted by R. Strichartz and arose out of contentions to the Fourier restriction problem.[1]

Examples

Consider the linear Schrödinger equation with h = m = 1. Then the solution for initial data \( u_0 \) is given by \( e^{it\Delta/2}u_0 \).

In this case Strichartz estimates take the form:

Suppose Strichartz equation is posed in \( \mathbb{R}^d \) and let q and r be real numbers satisfying \( 2\leq q, r \leq \infty; \frac{2}{q}+\frac{d}{r}=\frac{d}{2} \); and \( (q,r,d)\neq(2,\infty,2) \). Then the homogeneous Strichartz estimate takes the form:[2]

\( \|e^{it\Delta/2} u_0\|_{L^q_tL^r_x}\leq C_{d,q,r} \|u_0\|_2. \)

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:[2]

\( \left\| \int_\mathbb{R} e^{-is\Delta/2}F(s)\,ds\right\|_{L^2_x}\leq C_{d,\tilde q, \tilde r}\|F\|_{L^{\tilde q'}_tL^{\tilde r'}_x}.\)

The inhomogeneous Strichartz estimates are:[2]

\( \left\| \int_{s<t} e^{-i(t-s)\Delta/2}F(s)\,ds\right\|_{L^q_tL^r_x}\leq C_{d, q, r, \tilde q, \tilde r}\|F\|_{L^{\tilde q'}_tL^{\tilde r'}_x}.\)

References

R.S. Strichartz (1977), "Restriction of Fourier Transform to Quadratic Surfaces and Decay of Solutions of Wave Equations", Duke Math. J 44: 705–713
Tao, Terence (2006), Nonlinear dispersive equations: Local and global analysis, CBMS Regional Conference Series in Mathematics 106, ISBN 0-8218-4143-2