Loss of regularity for supercritical nonlinear Schrodinger equations

Abstract

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.

More details in the computations. Additional remarks in Section 6

References in corpus (3)

• Geometric optics and instability for semi-classical Schrodinger equations
• WKB analysis for nonlinear Schrödinger equations with potential
• Supercritical geometric optics for nonlinear Schrodinger equations

Cited by in corpus (10)

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• An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit
• Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schr{ö}dinger equation
• Non-regularity in Hölder and Sobolev spaces of solutions to the semilinear heat and Schrödinger equations
• Finite time blowup for a supercritical defocusing nonlinear Schrödinger system
• Semi-classical limit of the massive Klein-Gordon-Maxwell system toward the relativistic Euler-Maxwell system via an adapted modulated energy method
• Nonlinear Schrodinger equation and frequency saturation