Time-decay solutions of the initial-boundary value problem of rotating magnetohydrodynamic fluids

Weiwei Wang,Youyi Zhao

doi:10.1186/s13661-017-0845-2

Abstract

We have investigated an initial-boundary problem for the perturbation equations of rotating, incompressible, and viscous magnetohydrodynamic (MHD) fluids with zero resistivity in a horizontally periodic domain. The velocity of the fluid in the domain is non-slip on both upper and lower flat boundaries. We switch the analysis of the initial-boundary problem from Euler coordinates to Lagrangian coordinates under proper initial data, and get a so-called transformed MHD problem. Then, we exploit the two-tiers energy method. We deduce the time-decay estimates for the transformed MHD problem which, together with a local well-posedness result, implies that there exists a unique time-decay solution to the transformed MHD problem. By an inverse transformation of coordinates, we also obtain the existence of a unique time-decay solution to the original initial-boundary problem with proper initial data.

Highlights

2.2 Main results Before stating our first main result on the transformed MHD problem in detail, we introduce some simplified notations that shall be used throughout this paper: R+ := [, ∞)
We have proved the existence of a unique time-decay solution to the initial-boundary problem ( . )-( . ) of rotating MHD fluids in Lagrangian coordinates, which, together with the inverse transformation of coordinates, implies the existence of a time-decay solution to the original initial-boundary problem ( . )-( . ) with proper initial data in H ( )
Our result holds for the case ω =, it improves Tan and Wang’s result in [ ], in which the sufficiently small initial data at least belongs to H ( )

Summary

Introduction

The three-dimensional ( D) rotating, incompressible and viscous magnetohydrodynamic (MHD) equations with zero resistivity in a domain ⊂ R read as follows:. Α +α =m a b means that a ≤ cb, where, and in what follows, the letter c denotes a generic constant which may depend on the domain and some physical parameters, such as λ , M , g, μ and ρ in the MHD equations There is a sufficiently small constant δ > , such that for any (η , u ) ∈ H × H satisfying the following conditions:. ). More precisely, there is a sufficiently small constant δ > , such that, for any (v , N ) ∈ H satisfying the following conditions:. Before deriving the lower-order energy inequality defined on The last three integrals I L, . . . , I L can be estimated as follows:

L A At u ut

Horizontal spatial estimates

Stokes problem and stability condition

D L for d dt cL DL

H DH where the norm η is equivalent to η

Conclusion