On Schrodinger equations with modified dispersion

TL;DR

This paper investigates the well-posedness of nonlinear Schrödinger equations with modified dispersion, revealing ill-posedness in certain cases and limitations on Strichartz estimates, especially with bounded or homogeneous Fourier multipliers.

Contribution

It provides new ill-posedness results for Schrödinger equations with modified dispersion, highlighting the impact of Fourier multiplier properties on solution regularity.

Findings

No Strichartz estimates improve Sobolev embedding for bounded multipliers

Ill-posedness occurs at critical regularity with small initial data

Scaling arguments may not determine the critical value for homogeneous degree one symbols

Abstract

We consider the nonlinear Schrodinger equation with a modified spatial dispersion, given either by an homogeneous Fourier multiplier, or by a bounded Fourier multiplier. Arguments based on ordinary differential equations yield ill-posedness results which are sometimes sharp. When the Fourier multiplier is bounded, we infer that no Strichartz-type estimate improving on Sobolev embedding is available. Finally, we show that when the symbol is bounded, the Cauchy problem may be ill-posed in the case of critical regularity, with arbitrarily small initial data. The same is true when the symbol is homogeneous of degree one, where scaling arguments may not even give the right critical value.