Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay

TL;DR

This paper investigates the ill-posedness of the Prandtl equations in Sobolev spaces around shear flows with general decay, extending previous results from exponential decay to more general decay conditions.

Contribution

It extends the known ill-posedness results of the Prandtl equations to shear flows with general decay, using an approximate solution capturing initial layers.

Findings

Demonstrates ill-posedness for shear flows with general decay

Constructs an approximate solution capturing initial layers

Extends previous exponential decay results to broader decay conditions

Abstract

Motivated by the paper by D. Gerard-Varet and E. Dormy [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in \cite{GV-D} to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.