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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 018 085 590
Staffilani, Gigliola; Tataru, Daniel:
Strichartz estimates for a Schroedinger operator with nonsmooth coefficients
Commun. partial Differential Equations 27, No. 7-8, 1337-1372 (2002).
35B45A priori estimates
Primary Classification:
35K15Initial value problems for second-order, parabolic equations
Secondary Classification:
35B35Stability, boundedness
81U30Dispersion theory, dispersion relations
Schroedinger equation; non-smooth coefficients; Cauchy proboem; Strichartz estimates

Consequences of the stability analysis of the Cauchy problem for
partial differential equations of Schroedinger 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 quasi-linear 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. The
subject is alive these days; for me, this became clear during the
Kenji Yajima's seminar ``Dispersive properties of Schr\"odinger
equations with potentials periodic in time" a week ago (cf. its
abstract at http://gemma.ujf.cas.cz/~exner/qc.html).
Remarks to the editors:

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