Non-Lipshitz flow of the nonlinear Schr\"odinger equation on surfaces

TL;DR

This paper constructs non-Lipschitz flows for the cubic nonlinear Schrödinger equation on certain surfaces, revealing limitations of well-posedness in specific Sobolev spaces depending on the metric smoothness.

Contribution

It demonstrates the existence of non-Lipschitz flows in $H^s$ for the NLS on surfaces with different metrics, identifying critical Sobolev exponents for well-posedness.

Findings

Non-Lipschitz flow exists for $s<2/3$ with Lipschitz metrics.

Non-Lipschitz flow exists for $s<1/2$ with smooth metrics.

Critical Sobolev exponents match those for uniform local well-posedness.

Abstract

We construct non-Lipshitz flow in $H^s$ for the cubic nonlinear Schr\"odinger equation on the 2-torus of revolution with a Lipshitz or smooth metric . The non-Lipshitz property holds for all $s<2/3$ for Lipshitz metric and $s<1/2$ for smooth metric. Both coincide with the Sobolev exponents for uniform local well-posedness.