Remarks on the Prandtl Equation

TL;DR

This paper investigates the ill-posedness of the linearized Prandtl equation around general stationary solutions with non-degenerate critical points, extending previous results to more complex solution structures.

Contribution

It extends the understanding of ill-posedness in the Prandtl equation to solutions with non-degenerate critical points depending on x.

Findings

Ill-posedness occurs around stationary solutions with non-degenerate critical points.

The results generalize previous ill-posedness findings to more complex solution types.

The study highlights the sensitivity of the Prandtl equation to critical point structures.

Abstract

In a recent result of Gerard-Varet and Dormy [5], they established ill-posedness for the Cauchy problem of the linearized Prandtl equation around non-monotic special solution which is independent of x and satisfies the heat equation. In [6] and [7], some nonlinear ill-posedness were established with this counterexample. Then it is natural to consider the problem that does this linear ill-posedness happen whenever the non-degenerate critical points appear. In this paper, we concern the linearized Prandtl equation around general stationary solutions with non-degenerate critical points depending on x which could be considered as the time-periodic solutions and show some ill-posdness.