Ill-posedness for nonlinear Schrodinger and wave equations

TL;DR

This paper proves that nonlinear Schrödinger and wave equations are ill-posed in certain Sobolev spaces when the regularity is too low, extending previous results to higher dimensions and general nonlinearities.

Contribution

It extends previous one-dimensional results to higher dimensions and general nonlinearities, using zero-dispersion limit analysis for ill-posedness.

Findings

Ill-posedness occurs below critical regularity thresholds

Construction of solutions relies on zero-dispersion limit analysis

Results apply to both focusing and defocusing nonlinearities

Abstract

The nonlinear wave and Schrodinger equations on Euclidean space of any dimension, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space of index s whenever the exponent s is lower than that predicted by scaling or Galilean invariances, or when the regularity is too low to support distributional solutions. This extends previous work of the authors, which treated the one-dimensional cubic nonlinear Schrodinger equation. In the defocusing case soliton or blowup examples are unavailable, and a proof of ill-posedness requires the construction of other solutions. In earlier work this was achieved using certain long-time asymptotic behavior which occurs only for low power nonlinearities. Here we analyze instead a class of solutions for which the zero-dispersion limit provides a good approximation.