Global Solvability of 2D MHD Boundary Layer Equations in Analytic Function Spaces

TL;DR

This paper proves the global well-posedness of 2D MHD boundary layer equations in analytic spaces for small perturbations, even when initial far-field states are large, by reformulating the problem and analyzing decay properties.

Contribution

It establishes the global existence and uniqueness of solutions to 2D MHD boundary layer equations in analytic spaces under broader conditions than previous work.

Findings

Global existence of solutions proven for small perturbations.

Solutions decay to zero at infinity over time.

Long-time existence with explicit lifespan bounds.

Abstract

In this paper we are concerned with the global well-posedness of solutions to magnetohydrodynamics (MHD) boundary layer equations in analytic function spaces. When the initial data is a small perturbation around a selected profile, and such a profile is governed by an one dimensional heat equation with a source term, we establish the global in time existence and uniqueness of analytic solutions to the two dimensional (2D) MHD boundary layer equations. It is noted that the far-field state of velocity is not required to be small initially, but decays to zero as time tends to infinity with suitable decay rates. The whole analysis is divided into two parts: When the initial far-field states are not small, we reformulate the original problem into a small perturbation problem by extracting a suitable background profile which is governed by a heat equation, then we prove a long-time existence of solutions, and the lower bound of lifespan of solutions is given in term of the small parameter of initial perturbation; After establishing the long-time existence, combining the decay properties of far-field state and the properties of solutions to the heat equation, it is shown that both the solution obtained in the first part and the background profile indeed satisfy some smallness requirements when time goes to the lower bound of lifespan of solutions. Then we can show the global existence of solutions after this moment.