On the ill-posedness of the Prandtl equation

TL;DR

This paper demonstrates that the Prandtl equation is linearly ill-posed in Sobolev spaces, revealing instability mechanisms at high frequencies and contrasting with well-posedness in analytic or monotonic data.

Contribution

It proves the linear ill-posedness of the Prandtl equation in Sobolev spaces and constructs unstable quasimodes, highlighting viscosity's role in this instability.

Findings

The Prandtl equation is linearly ill-posed in Sobolev spaces.

Unstable quasimodes exist at high tangential frequencies.

Viscosity contributes to the strong instability observed.

Abstract

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data, or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with non-degenerate critical points. Interestingly, the strong instability is due to vicosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.