Illposedness for dispersive equations: Degenerate dispersion and Takeuchi--Mizohata condition

TL;DR

This paper investigates two mechanisms causing ill-posedness in dispersive equations, namely degenerate dispersion and the failure of the Takeuchi--Mizohata condition, providing a unified framework and new quantitative results.

Contribution

It introduces a unified energy- and duality-based approach to analyze ill-posedness in dispersive equations, covering both degenerate dispersion and Takeuchi--Mizohata failure.

Findings

Proves strong ill-posedness in high-regularity Sobolev spaces for various quasilinear dispersive equations.

Demonstrates how degenerate dispersion and the Takeuchi--Mizohata condition lead to non-existence and unboundedness of solutions.

Provides a quantitative version of Mizohata's classical $L^{2}$-illposedness result for linear Schrödinger equations with failed Takeuchi--Mizohata condition.

Abstract

We provide a unified viewpoint on two illposedness mechanisms for dispersive equations in one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi--Mizohata condition. Our approach is based on a robust energy- and duality-based method introduced in an earlier work of the authors in the setting of Hall-magnetohydynamics. Concretely, the main results in this paper concern strong illposedness of the Cauchy problem (e.g., non-existence and unboundedness of the solution map) in high-regularity Sobolev spaces for various quasilinear degenerate Schr\"odinger- and KdV-type equations, including the Hunter--Smothers equation, $K(m, n)$ models of Rosenau--Hyman, and the inviscid surface growth model. The mechanism behind these results may be understood in terms of combination of two effects: degenerate dispersion -- which is a property of the principal term in the presence of degenerating coefficients -- and the evolution of the amplitude governed by the Takeuchi--Mizohata condition -- which concerns the subprincipal term. We also demonstrate how the same techniques yield a more quantitative version of the classical $L^{2}$-illposedness result by Mizohata for linear variable-coefficient Schr\"odinger equations with failed Takeuchi--Mizohata condition.