On the ill-posedness of the Prandtl equations in three space dimensions

TL;DR

This paper establishes an instability criterion for the three-dimensional Prandtl equations, revealing that classical monotonicity conditions are insufficient for well-posedness, and highlights the importance of velocity field invariance.

Contribution

It provides the first instability criterion for 3D Prandtl equations, showing the limitations of monotonicity assumptions and extending the understanding of stability conditions.

Findings

Monotonicity of tangential velocity is not sufficient for well-posedness in 3D.

Linear stability requires velocity invariance with respect to the normal variable.

The results complement previous work on special structured cases.

Abstract

In this paper, we give an instability criterion for the Prandtl equations in three space variables, which shows that the monotonicity condition of tangential velocity fields is not sufficient for the well-posedness of the three dimensional Prandtl equations, in contrast to the classical well-posedness theory of the Prandtl equations in two space variables under the Oleinik monotonicity assumption of the tangential velocity. Both of linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linear stable for the three dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work \cite{LWY} on the well-posedness theory for the three dimensional Prandtl equations with special structure.