 
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 illposed. 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
energysupercritical equations is not wellposed in the sense of Hadamard. We
reduce the problem to a supercritical WKB analysis. For supercubic, 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 = ,
