Long-time existence of Gevrey-2 solutions to the 3D Prandtl boundary layer equations

TL;DR

This paper proves that solutions to the 3D Prandtl boundary layer equations with initial data in Gevrey-2 spaces exist for a long time, specifically at least of size ^{-M} for small  and arbitrary M.

Contribution

It establishes long-time existence of Gevrey-2 solutions to the 3D Prandtl equations with small initial data, extending previous local existence results.

Findings

Gevrey-2 solutions exist for at least ^{-M} time for small initial data

The lifespan depends inversely on the size of initial data in Gevrey-2 spaces

Provides a framework for analyzing long-time behavior of boundary layer solutions

Abstract

For the three dimensional Prandtl boundary layer equations, we will show that for arbitrary $M$ and sufficiently small $\epsilon$, the lifespan of the Gevrey-2 solution is at least of size $\epsilon^{-M}$ if the initial data lies in suitable Gevrey-2 spaces with size of $\epsilon$.