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Science China Mathematics

, Volume 58, Issue 5, pp 891–914 | Cite as

Wellposedness of some quasi-linear Schrödinger equations

  • Jean-Yves CheminEmail author
  • Delphine Salort
Articles Invited Articles
  • 224 Downloads

Abstract

This article is devoted to the study of a quasilinear Schrödinger equation coupled with an elliptic equation on the metric g. We first prove that, in this context, the propagation of regularity holds which ensures local wellposedness for initial data small enough in \(\dot H^{\tfrac{1} {2}} \) and belonging to the Besov space \(\dot B_{2,1}^{\tfrac{3} {2}} \). In a second step, we establish Strichartz estimates for time dependent rough metrics to obtain a lower bound of the time existence which only involves the \(\dot B_{2,\infty }^{1 + \varepsilon } \) norm on the initial data.

Keywords

quasilinear Schrödinger equation Strichartz estimates paradiffential calculus stationary phase method 

MSC(2010)

35Q41 35S50 35Q55 

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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Laboratoire J.-L. Lions, UMR 7598Université Pierre et Marie Curie, JussieuParis Cedex 05France
  2. 2.Laboratoire de Biologie Computationnelle et Quantitative, UMR 7238Sorbonne UniversitésParisFrance
  3. 3.L’Institut Jacques Monod, UMR 7592Université Paris DiderotParisFrance

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