Loss of regularity for supercritical nonlinear Schrodinger equations

Details Loss of regularity for supercritical nonlinear Schrodinger equations Loss of regularity for supercritical nonlinear Schrodinger equations

2007-01-29

arxiv.org/abs/math/0701857v3

Math. Ann. 343, 2 (2009) 397-420

Mathematics

Analysis of PDEs

More details in the computations. Additional remarks in Section 6

Scientific paper

10.1007/s00208-008-0276-6

We consider the nonlinear Schrodinger 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: there is a loss of regularity, in the same spirit as the result due to G.Lebeau in the case of the wave equation. We use an isotropic change of variable, which reduces the problem to a super-critical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functional a la Brenier.

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