Linear wave estimates

• Fixed-time estimates for free solutions f :
  • (Energy estimate) If $f(0)$ is in $H^{s}$, then $f(t)$ is also.
  • (Decay estimate) If $f(0)$ has more than $(d+1)/2$ derivatives in $L^{1}$, then $\| f (t)\|_{L^\infty}$ decays like $\langle t\rangle^{-(d-1)/2}$. One can obtain the endpoint of $(d+1)/2$ derivatives if one is willing to localize in frequency or use Hardy spaces and BMO.
  • One can interpolate between these estimates to get $(L^{p},L^{p'})$ estimates with the sharp loss of regularity Br1975. This is useful for Strichartz estimates and for scattering theory.
• Strichartz estimates: A free $\dot{H}^s$ solution is in $L^q_t (\mathbb{R}; L^r_x(\mathbb{R}^d))$ if
  • (Scaling) $d/2-s=1/q+d/r$
  • (Parallel interactions) $(d-1)/4 \geq 1/q + (d-1)/2r$
  • (Increase of integrability) $q, r \geq 2$
  • (No double endpoints) $(d,q,r) \neq (3, 2, \infty)$
    • This estimate 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
    • Actually even when $d>3$, the $(q,r) = (2,\infty)$ estimate is slightly subtle; one has BMO and Besov space estimates but not directly $L^\infty$ estimates; it can be recovered for radial functions FgWgc2006. However, the endpoint $(q,r) = (2, 2(d-1)/(d-3))$ is OK; see KeTa1998.
    • When $d=2$, 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 FgWgc2011 Corollary 1.19 or $L^4_t L^\infty L^2_\omega(\mathbb{R}^n, |x|^{d-1}d |x| d\omega)$ SmhSoWgc-p.
  • In the case $s=1/2,d=3,q=r=4$, a maximizer exists (e.g. with initial position zero and initial velocity given by the Cauchy distribution $1/1+|x|^{2})$, witih best constant $(3pi/4)^{1/4}$ [Fc-p4]
  • These results extend globally outside of a convex obstacle [Bu-p], SmhSo1995, [SmhSo-p], [Met-p]; see [So-p] for a survey of this issue and applications to nonlinear wave equations outside of an obstacle.
  • For data which is radial (or otherwise enjoys additional angular regularity) a much larger range of Strichartz estimates is possible (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 $1/q<(d-1)(1/2-1/r)$, $q,r\geq 2$ and $(q,r)\neq (\infty,\infty)$. 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 $r=\infty$). 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.
    • 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 ($d\geq 3$, $r<\infty$, the case $d=3$ comes from an interpolation from the endpoint estimates in MacNkrNaOz2005) and FgWgc2011) Theorem 1.18 ($d\geq 2$ or r=\infty).
    • If we use instead the norm $L^q_t(\mathbb{R}; L^r L^2_\omega(\mathbb{R}^n, |x|^{d-1}d |x| d\omega)$, the same set of Strichartz estimates is true as the radial case. That is, if $1/q<(d-1)(1/2-1/r)$, $q,r\geq 2$ and $(q,r)\neq (\infty,\infty)$. This set of estimates, with control of the solution in $L^q_t(\mathbb{R}; L^r L^2_\omega(\mathbb{R}^n, |x|^{d-1}d |x| d\omega)$, can be called generalized Strichartz estimates. This results 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.
  • These estimates extend to some extent to the Klein-Gordon equation $\Box u=m^{2}u$.A useful heuristic to keep in mind is that this equation behaves like the Schrodinger equation $u+iu_{t}+1/(2m)\Delta u=0$ when the frequency $\xi$ has magnitude less than $m$, but behaves like the wave equation for higher frequencies.Some basic Strichartz estimates here are in MsSrWa1980; see for instance [Na-p], [MacNaOz-p], [MacNkrNaOz-p] for more recent treatments.
  • For inhomogeneous estimates it is known that a solution with zero initial data and forcing term containing s-1 derivatives in a dual space ${L_{t}^{Q'}}{L_{x}^{R'}}$ will lie in $L_{t}^{q}L_{x}^{r}$ if both $(q,r)$ and $(Q,R)$ are admissible in the above sense, if $s\geq 0$, and if one has the scaling condition $1/Q+d/R+1/q+d/r=d+s+1$.The \u201cs-1\u201d represents a smoothing effect of one derivative, though this full gain is only attainable if one uses the energy exponents $L_{t}^{1}L_{x}^{2}$ and ${L_{t}^{\infty }}{L_{x}^{2}}$. It is possible to obtain inhomogeneous estimates in which only one of the exponents are admissible; this phenomenon was first observed in Har1990, Ob1989 (see also KeTa1998).More recently in [Fc-p2], inhomogeneous estimates are obtained with the above scaling condition assuming the weaker conditions $1\leq q,r\leq \infty$ and $1/q<(n-1)(1/2-1/r)$ or $(q,r)=(\infty ,2)$ and similarly for $Q,R$, and if the following additional conditions hold:
    • In $d=1,2$ no further conditions are required;
    • When $d=3$, r,R are required to be finite;
    • When $d>3$, either $1/q+1/Q<1$, $(n-3)/r\leq (n-1)/R$, and $(n-3)/R\leq (n-1)/r$, or $1/q+1/Q=1$, $(n-3)/r<(n-1)/R,(n-3)/R<(n-1)/r,r\geq q$, and $R\geq Q$.
  • Strichartz estimates extend to situations in which there is a potential or when the metric is variable.For local-in-time estimates and smooth potentials or metrics this is fairly straightforward (the potential can be treated by iterative methods, and the metric by parametrix methods). More interesting issues arise for global-in-time estimates with smooth potentials/metrics or local-in-time estimates with rough potentials/metrics (the two types of results are linked by scaling).
    • For potentials of power-type decay, the global results are as follows: For potentials of the form $V=a/|x|^{2}$ with $d\geq 3$ and $a>-(d-2)^{2}/4$, one has global Strichartz estimates [BuPlStaTv-p]; a simplified proof and more general result dealing with inverse square-like potentials which are not too negative is in [BuPlStaTv-p2]. The condition on a is necessary to avoid bound states.For potentials decaying slower than this, Strichartz estimates can fail.
    • For potentials decaying an epsilon faster than $1/|x|^{2}$ and assumed to be nonnegative, dispersive and Strichartz estimates were obtained when $d=3$ in GeVis2003.
    • (More results to be added in future).