Linear wave estimates

• Fixed-time estimates for free solutions f :
  • (Energy 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)} 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 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^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 radial variable [MacNkrNaOz-p].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 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 > 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. 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.
  • 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); see [Stz-p4] for further discussion.

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 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_t L^2_x} 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 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} , either 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 + 1/Q < 1} , 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-3)/r \leq (n-1)/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 (n-3)/R \leq (n-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 1/q + 1/Q = 1} , 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-3)/r < (n-1)/R, (n-3)/R < (n-1)/r, r \geq q} , 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 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 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 V = a/|x|^2} with 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} 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).