arXiv subjects

Xiuqing Wang

Publications and source records attributed to Xiuqing Wang.

3 recordsLinked to original sources

arXiv2024

Prandtl Equations and Related Boundary Layer Equations

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.

math.AP↗

arXiv2023

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

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.

math.AP↗

arXiv2023

Local well-posedness of solutions to 2D magnetic Prandtl model in the Prandtl-Hartmann regine

In this paper, we prove local existence and uniqueness for the 2D Prandtl-Hartmann regine in weighted Sobolev spaces. Our proof is based on using uniform estimates of the regularized parabolic equation and maximal principle under the Oleinik's monotonicity assumption. Compared with Nash-Moser iteration in \cite{[5]}, the data do not require high regularity.

math.AP↗