Linear wave estimates

• Fixed-time estimates for free solutions f :
  • (Energy estimate) If $f(0)$ is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f (t)} is also.
  • (Decay estimate) If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f (0)} has more than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (d+1)/2} derivatives in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^1} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \| f (t)\|_{L^\infty}} decays like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle <t>^{-(d-1)/2}} . One can obtain the endpoint of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot{H}^s} solution is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^q_t L^r_x} if
  • (Scaling) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d/2 - s = 1/q + d/r}
  • (Parallel interactions) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (d-1)/4 \geq 1/q + (d-1)/2r}
  • (Increase of integrability) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q, r \geq 2}
  • (No double endpoints) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n,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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (q,r) = (2,\infty)} estimate is slightly subtle; one has BMO and Besov space estimates but not directly Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^\infty} estimates; it can be recovered for radial functions FgWgc2006. However, the endpoint Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (q,r) = (2, 2(d-1)/(d-3))} is OK; see KeTa1998.
    • When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=2} , the endpoint is (q,r) = (4,\infty). It is not true in general, but hold for radial functions FgWgc2006.
  • In the case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1 / 1 + |x|^2)} , witih best constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (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 know to be true if and only if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/q<(d-1)(1/2-1/r) } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q,r\ge 2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (q,r)\neq (\infty,\infty)} . The positive results are due to [Stz-p4] (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\geq 3} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r<\infty} ) and FgWgc2006) (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=2} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r=\infty} ). The counterexample shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/q<(d-1)(1/2-1/r)} is necessary for radial Strichartz estimates is due to HiKur2008 Corollary 4.3.

These estimates extend to some extent to the Klein-Gordon equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box u = m^2 u} .A useful heuristic to keep in mind is that this equation behaves like the Schrodinger equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u+ i u_t + 1/(2m) \Delta u = 0} when the frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \xi} has magnitude less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {L^{Q'}_t} {L^{R'}_x}} will lie in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^q_t L^r_x} if both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (q,r)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (Q,R)} are admissible in the above sense, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 0} , and if one has the scaling condition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {L^{\infty}_t} {L^2_x}} . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1 \leq q,r \leq \infty} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/q < (n-1)(1/2-1/r)} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (q,r) = (\infty,2)} and similarly for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q,R} , and if the following additional conditions hold:

In Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=1,2} no further conditions are required;

When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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).