Ill-posedness of quintic fourth order Schr\"odinger equation

TL;DR

This paper demonstrates that the solution map for the quintic fourth order nonlinear Schrödinger equation experiences norm inflation at all points in super-critical Sobolev spaces, indicating ill-posedness in these regimes.

Contribution

It establishes the ill-posedness of the equation in super-critical Sobolev spaces through norm inflation phenomena, covering both negative and positive regularity cases across various dimensions.

Findings

Norm inflation occurs in super-critical Sobolev spaces for the equation.

The result applies to both defocusing and focusing cases in negative regularity.

In positive regularity, norm inflation is shown for the defocusing equation in dimensions 3, 4, and 5.

Abstract

We prove that the solution map, associated to the quintic fourth order nonlinear Schr\"odinger equation, exhibits the norm inflation phenomenon at every point in the Sobolev spaces of super-critical regularity. Indeed, we prove this result separately in the cases of negative and of positive regularity. In the negative regularity case, we prove the result for both the defocusing and focusing equations: in the one dimensional case, the associated solution map exhibits norm inflation in Sobolev spaces of super-critical regularity (including the critical index); in the higher dimensional case, the solution map exhibits the same phenomenon in spaces of negative regularity. Meanwhile, in the case of positive regularity, we prove the result for the defocusing equation in dimensions $d=3,4,5$. Our proofs are based on the "high-to-low" and "low-to-high" frequency cascades respectively.