Strichartz estimates

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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.

The Strichartz estimates originate from Sz1977. The non-endpoint estimates for the wave equations were proved in LbSo1995 and GiVl1995. The full estimates including the endpoint were proved in KeTa1998. See KeTa1998 for more details on the history.

Strichartz estimates for the wave equations

Let $\dot H^{s}(\Bbb R^n)$ denote the homogeneous Sobolev space with norm $$ \left\| u \right\|_{\dot H^{s}(\Bbb R^n)} = \left\|(-\Delta^{s/2}) u \right\|_{L^2(\Bbb R^n)} $$ (In general, such spaces are defined modulo polynomials. The practical choice of the space $\dot H^{s}(\Bbb R^n)$ is the completion of $C_0^\infty$ with the norm. It can be characterized to be the space modulo polynomials of order less than $s-n/2$. In particular, for $s<n/2$, it is the usual function space.)

Consider 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 can be stated as follows.

Strichartz estimates (with $F=0$)

We have the estimates $$\|u\|_{L^q_t (\mathbb{R}; L^r_x(\mathbb{R}^n))}\le C (\|f\|_{\dot H^s}+\|g\|_{\dot H^{s-1}})$$ if the following conditions are satisfied

  • (Scaling) $n/2 - s = 1/q + n/r$
  • (Parallel interactions, [Knapp example]) $\frac{1}{q}\leq \frac{n-1}{2}(\frac{1}{2}-\frac{1}{r})$
  • (Increase of integrability) $q, r \geq 2$
  • (No double endpoints) $(q,r) \neq (\max( 2, \frac{4}{n-1}), \infty)$
    • The endpoint $(q,r,n)=(2,\infty,3)$ can be recovered for radial functions KlMa1993, or when a small amount of smoothing (either in the Sobolev sense, or in relaxing the integrability) in the angular variable MacNkrNaOz2005.However in the general case one cannot recover the estimate even if one uses the BMO norm or attempts Littlewood-Paley frequency localization Mo1998
    • A similar situation occurs when $n=2$, the endpoint is $(q,r) = (4,\infty)$. It is not true in general, but hold for radial functions FgWgc2006. It is also true with a small amount of angular regularity on the data FgWgc2011 Corollary 1.19 or use the norm $L^4_t L^\infty L^2_\omega(\mathbb{R}^n, |x|^{d-1}d |x| d\omega)$ SmhSoWgc-p.
    • For the case $n > 3$, it is still open if the $L^2_t L^\infty_x$ estimate is true or not. One has BMO and Besov space estimates; it can be recovered for radial functions FgWgc2006. However, another endpoint $(q,r) = (2, 2(n-1)/(n-3))$ is OK; see KeTa1998.
  • $(q,r) \neq (\infty, \infty)$
    • It is because we do not have the Sobolev embedding $\dot H^{n/2}\subset L^\infty_x$. On the other hand, the Besov space estimates is true.

Radial Strichartz estimates

For data which is radial (spherical symmetric), a much larger range of Strichartz estimates is available (basically because the parallel interaction obstruction is substantially weakened).

When the data is radial, the Strichartz estimates is known to be true if and only if $$1/q<(n-1)(1/2-1/r) or (q,r)=(\infty,2), q,r\geq 2, (q,r)\neq (\infty,\infty) .$$ The positive results are due to KlMa1993 ($(q,r,d)=(2,\infty,3)$, [So1995] ($d=3$), [Stz2005] Theorem 1.3 ($d\geq 3, r<\infty$) and FgWgc2006) Theorem 4 ($d=2$ or $r=\infty$) (see also JgWgcYx-p). The counterexample shows that $1/q<(d-1)(1/2-1/r)$ is necessary for radial Strichartz estimates is due to HiKur2008 Corollary 4.3, which basically follows from the fact that the decay estimates for the wave equation are sharp.

Generalized Strichartz estimates

For data with additional angular regularity or the $L^q_t L^r_x$ is replaced with a norm with weaker angular regularity, a similar generalization of Strichartz estimates is possible.

  • If we add some angular regularity to the initial data, the same set of Strichartz estimates is true as the radial case. These results are due to Stz2005 Theorem 1.5 ($n\geq 4$, $r<\infty$), and FgWgc2011) Theorem 1.18 ($n\geq 2$ or r=\infty). The case $n=3$ comes from an interpolation of the estimates in Stz2005 from the endpoint estimates in MacNkrNaOz2005). To be more precise, we have the following estimate

$$\|u\|_{L^q_t (\mathbb{R}; L^r_x(\mathbb{R}^n))}\le C (\|f\|_{\dot H^{s,b}_\omega}+\|g\|_{\dot H^{s-1,b}_\omega})$$ for $(q,r)$ satisfying $(n-1)(1/2-1/r)/2<1/q<(n-1)(1/2-1/r)$ and $q\geq 2$, and $b>b_{kn}=2/q-(n-1)(1/2-1/r)$. Here the Sobolev space with angular regularity $b\geq 0$ is defined as follows, $f\in \dot H^{s,b}_\omega$ if and only if $f\in \dot H^{s}$ and $(1-\Delta_\omega)^{b/2}f\in\dot H^{s}$, with $\Delta_\omega$ be the spherical Laplacian.

  • Instead, if we use the norm $L^q_t(\mathbb{R}; L^r L^2_\omega(\mathbb{R}^n, |x|^{n-1}d |x| d\omega)$, the same set of Strichartz estimates is true as the radial case. That is, if $1/q<(n-1)(1/2-1/r)$ or $(q,r)=(\infty,2)$, $q,r\geq 2$ and $(q,r)\neq (\infty,\infty)$, we have

$$\|u\|_{L^q_t (\mathbb{R}; L^r L^2_\omega(\mathbb{R}^n))}\le C ( \|f\|_{\dot H^{s}}+\|g\|_{\dot H^{s-1}})$$ This set of estimates are due to MacNkrNaOz2005 ($(q,r,d)=(2,\infty,3)$), SmhSoWgc-p ($d=2$) Proposition 1.2, and then generalized to $d\geq 3$ JgWgcYx-p Theorem 1.4.

Inhomogeneous Strichartz estimates

By duality and Christ-Kiselev lemma, it is easy to see that a set of inhomogeneous estimates with $F$ in $L^{q_1'}_t L^{r_1'}_x$ with $(q_1,r_1)$ satisfying the admissible conditions (with condition $q>q_1'$). The technical condition $q>q_1'$ can be relaxed, see KeTa1998.

It is possible to obtain inhomogeneous estimates in which the exponents are even not admissible; this phenomenon was first observed in Har1990, Ob1989 (see also KeTa1998). See Fc2005, Vi2007, Tag2010 for recent breakthrough in this direction.