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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 018 085 590  
Author(s):  
Staffilani, Gigliola; Tataru, Daniel:  
Shorttitle:  
Strichartz estimates for a Schroedinger operator with nonsmooth coefficients  
Source:  
Commun. partial Differential Equations 27, No. 78, 13371372 (2002).  
Classification:  
 
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
Schroedinger equation; nonsmooth coefficients; Cauchy proboem; Strichartz estimates  
Review:  
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 nonsmoothness of the coefficients in quasilinear problems. In this context, the main subject of this paper is the rederivation of the fundamental (so called Strichartz) inequalities (which, rougly speaking, interrelate the q and rnorms 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 nonsmooth" 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:  