Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III

Abstract

In an earlier work of the author it was proved that the Strichartz estimates for second order hyperbolic operators hold in full if the coefficients are of class C 2 C^2 . Here we strengthen this and show that the same holds if the coefficients have two derivatives in L 1 ( L ∞ ) L^1(L^\infty ) . Then we use this result to improve the local theory for second order nonlinear hyperbolic equations.

Cubic quasilinear wave equation and bilinear estimates

“…Some results in that direction have been proved by the authors (see [1] and [2]) and also by D. Tataru (see [22]) for quasilinear wave equations of the following type (E)…”

“…H s−1 . This theorem has been proved with 1/4 instead than 1/6 in [1] and then improved a little bit in [2] and proved with 1/6 by D. Tataru in [22]. Strichartz estimates for quasilinear equations are the key point of the proofs.…”

“…By microlocalization of the estimates, we mean that we shall prove estimates that are valid on time intervals whose length depend on the frequency parameter. These techniques have been introduced in [1] and used in [2] and improved by D. Tataru in [22].…”

Propagation of singularities for rough metrics

“…Following [Smith 2006], we work with a scaled wave-packet transform, similar to the FBI transform used in [Tataru 2002], but based on a Schwartz function with Fourier transform of compact support instead of a Gaussian.…”

Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces

“…Such a condition is in fact guaranteed in case of "small perturbations" of the flat metric. On the subject of dispersion for Klein-Gordon and wave equations, we mention, in a non exhaustive way, [1,2,3,16,20,21,22].…”