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Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. (English) Zbl 1010.35015
Consequences of the stability analysis of the Cauchy problem for partial differential equations of Schrödinger type are numerous; besides their immediate dispersive interpretation and relevance they may concern, via inverse scattering method, the study (e.g., existence and uniqueness) of solitons, nonlinear KdV waves etc. A few mathematical challenges emerge in this context, one being the loss of an easy form of the localization argument which works easily for the hyperbolic wave equations. This is settled here via the assumption of an asymptotically constant form of the coefficients. Another technical difficulty emerges in connection with the non-smoothness of the coefficients in quasilinear problems.
In this context, the main subject of this paper is the re-derivation of the fundamental (so-called Strichartz) inequalities (which, rougly speaking, inter-relate the $$q$$- and $$r$$-norms with different “admissible” $$q$$ and $$r$$, and provide an estimate for an “improved” norm of the solutions in terms of the “standard” $$r=2$$ norm of the initial state) under the properly weakened assumptions. The proof starts from the “smooth plus non-smooth” split of the solution, with emphasis on the compact spatial support of the (mutually cancelled) inhomogeneous terms. Its main idea lies in a construction (by a suitable integral transformation) of the so-called microlocal parametrix, followed by the estimate obtained by the method of the stationary phase.

MSC:
 35B45 A priori estimates in context of PDEs 81U30 Dispersion theory, dispersion relations arising in quantum theory
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