Gevrey class smoothing effect for the Prandtl equation

TL;DR

This paper investigates the Gevrey class smoothing effects for solutions to the Prandtl equation, showing that solutions become Gevrey regular in tangential variables over time under certain conditions.

Contribution

It establishes the Gevrey regularity of solutions to the Prandtl equation in the context of monotonic initial data, extending understanding of solution regularity.

Findings

Solutions gain Gevrey regularity in tangential variables over time.

The result applies to solutions with initial data satisfying Oleinik's monotonicity condition.

Provides new insights into the smoothing effects for the Prandtl boundary layer equation.

Abstract

It is well known that the Prandtl boundary layer equation is instable, and the well-posedness in Sobolev space for the Cauchy problem is an open problem. Recently, under the Oleinik's monotonicity assumption for the initial datum, [1] have proved the local well-posedness of Cauchy problem in Sobolev space (see also [21]). In this work, we study the Gevrey smoothing effects of the local solution obtained in [1]. We prove that the Sobolev's class solution belongs to some Gevrey class with respect to tangential variables at any positive time.