Local existence of solutions to 3D Prandtl equations with a special structure

TL;DR

This paper establishes the local existence and uniqueness of solutions to a special structured 3D Prandtl equation in a periodic domain, using energy methods without the Crocco transform, extending previous 2D results.

Contribution

It introduces a new approach for 3D Prandtl equations with a special structure, avoiding the Crocco transform and employing a cancellation mechanism.

Findings

Proves local existence and uniqueness of solutions.

Develops a new unknown function for analysis.

Extends 2D results to 3D with special structure.

Abstract

In this paper, we consider the 3D Prandtl equation in a periodic domain and prove the local existence and uniqueness of solutions by the energy method in a polynomial weighted Sobolev space. Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the Crocco transform has always been used with the general outer flow $U\neq\text{constant}$, this Crocco transform is not needed here for 3D Prandtl equations. We use the skill of cancellation mechanism and construct a new unknown function to show that the existence and uniqueness of solutions to 3D Prandtl equations (cf. Masmoudi and Wong, Comm. Pure Appl. Math., 68(10)(2015), 1683-1741) which extends from the two dimensional case in \cite {12} to the present three dimensional case with a special structure.