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Loss of regularity for supercritical nonlinear Schrödinger equations

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Abstract

We consider the nonlinear Schrödinger 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 à la Brenier.

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Authors and Affiliations

  1. CNRS and Université Paris-Sud, Mathématiques, Bât. 425, 91405, Orsay cedex, France

    Thomas Alazard

  2. CNRS and Université Montpellier 2, Mathématiques CC051, 34095, Montpellier cedex 5, France

    Rémi Carles

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  1. Thomas Alazard
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  2. Rémi Carles
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Correspondence to Rémi Carles.

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Alazard, T., Carles, R. Loss of regularity for supercritical nonlinear Schrödinger equations. Math. Ann. 343, 397–420 (2009). https://doi.org/10.1007/s00208-008-0276-6

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  • DOI: https://doi.org/10.1007/s00208-008-0276-6

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