Global tangentially analytical solutions of the 3D axially symmetric Prandtl equations

TL;DR

This paper proves the global existence of solutions to the 3D axially symmetric Prandtl equations with small initial data, extending previous results from 2D to 3D and from almost global to global solutions.

Contribution

It introduces a tangentially weighted analytic energy functional and a good unknown to establish global solutions, improving prior work in 2D and extending it to 3D axially symmetric cases.

Findings

Proves global existence for small initial data in 3D axially symmetric Prandtl equations.

Extends previous 2D results to 3D case.

Introduces a new energy functional and good unknown for analysis.

Abstract

In this paper, we will prove the global existence of solutions to the three dimensional axially symmetric Prandtl boundary layer equations with small initial data, which lies in $H^1$ Sobolev space with respect to the normal variable and is analytical with respect to the tangential variables. Proof of the main result relies on the construction of a tangentially weighted analytic energy functional, which acts on a specially designed good unknown. The constructed energy functional can find its two dimensional parallel in Ignatova-Vicol [2016ARMA] where no tangential weight is introduced and the specially good unknown is set to control the lower bound of the analytical radius, whose two dimensional similarity can be traced to Paicu-Zhang [2021ARMA]. Our result is an improvement of that in Ignatova-Vicol [2016ARMA] from the almost global existence to the global existence and an extension of that in Paicu-Zhang [2021ARMA] from the two dimensional case to the three dimensional axially symmetric case.