Prandtl Equations and Related Boundary Layer Equations

TL;DR

This book reviews recent advances in the mathematical analysis of Prandtl and MHD boundary layer equations, focusing on well-posedness and existence of solutions in various settings and dimensions.

Contribution

It provides a comprehensive survey up to 2020 and presents new results on local and global well-posedness of solutions to boundary layer equations in different mathematical frameworks.

Findings

Global well-posedness of 2D Prandtl-Hartmann equations in analytic framework

Local existence of solutions to 2D Prandtl equations in weighted Sobolev space

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

Abstract

This book aims to present some recent results on Prandtl equations and MHD boundary layer equations. This book is essentially divided into two parts. Chapter 1 as the first part systematically surveys the results till 2020 on Prandtl equations and MHD boundary layer equations. Chapter 2 to 6 are the main part of the book, which presents the local and the global well-posedness of solutions to the Prandtl equations and MHD boundary layer equations. In detail, Chapter 2 is concerned with global well-posedness of solutions to the 2D Prandtl-Hartmann equations in an analytic framework. Chapter 3 investigates the local existence of solutions to the 2D Prandtl equations in a weighted Sobolev space. Chapter 4 studies the local well-posedness of solutions to the 2D mixed Prandtl equations in a Sobolev space without monotonicity and lower bound. Chapter 5 is concerned with global existence of solutions to the 2D magnetic Prandtl equations in the Prandtl-Hartmann regime. Chapter 6 proves the local existence of solutions to the 3D Prandtl equations with a special structure. Mathematicians and physicists who are interested in fluid dynamics will find this book helpful.