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LOSS OF REGULARITY FOR SUPERCRITICAL NONLINEAR SCHRODINGER EQUATIONS
 

Summary: LOSS OF REGULARITY FOR SUPERCRITICAL NONLINEAR
SCHR¨ODINGER EQUATIONS
THOMAS ALAZARD AND R´EMI CARLES
Abstract. We consider the nonlinear Schr¨odinger equation with defocusing,
smooth, nonlinearity. Below the critical Sobolev regularity, it is known that
the Cauchy problem is ill-posed. We show that this is even worse, namely
that there is a loss of regularity, in the spirit of the result due to G. Lebeau
in the case of the wave equation. As a consequence, the Cauchy problem for
energy-supercritical equations is not well-posed in the sense of Hadamard. We
reduce the problem to a supercritical WKB analysis. For super-cubic, smooth
nonlinearity, this analysis is new, and relies on the introduction of a modulated
energy functional `a la Brenier.
1. Introduction
We consider the following defocusing nonlinear Schr¨odinger equation on Rn
:
(1.1) it +
1
2
= ||2
; |t=0 = ,

  

Source: Alazard, Thomas - Département de Mathématiques, Université de Paris-Sud 11

 

Collections: Mathematics