Global Solutions of Nonlinear Magnetohydrodynamics with Large Initial Data

Abstract

A free boundary problem for nonlinear magnetohydrodynamics with general large initial data is investigated. The existence, uniqueness, and regularity of global solutions are established with large initial data in H 1 : It is shown that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. An existence theorem of global solutions with large discontinuous initial data is also established. # 2002 Elsevier Science (USA)

Global Existence and Large-Time Behavior of Solutions to the Three-Dimensional Equations of Compressible Magnetohydrodynamic Flows

“…In this paper, we are interested in the global existence and large-time behavior of solutions to the three-dimensional MHD equations (1.1) in a bounded domain Ω ⊂ R 3 with the following initial-boundary conditions: There have been a lot of studies on MHD by physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathematical challenges; see [3,4,5,7,12,15,19,25] and the references cited therein. In particular, the one-dimensional problem has been studied in many papers, for examples, [3,4,7,15,17,22,25] and so on.…”

“…In particular, the one-dimensional problem has been studied in many papers, for examples, [3,4,7,15,17,22,25] and so on. However, many fundamental problems for MHD are still open.…”

Existence of Global Solutions for Nonlinear Magnetohydrodynamics with Finite Larmor Radius Corrections

“…Recently, Ramos [5] discussed the heat flux terms for the gyrofluid, and Passot and Sulem [6] discussed the closure relations for the gyrofluid. For the existence of solutions to MHD, Chen and Wang [7] showed the existence of global solutions to the piston problem. Hu and Wang [8] obtained the existence of global weak solutions in three dimensions for MHD equations, by using the Faedo-Galerkin method.…”

Regularity of large solutions for the compressible magnetohydrodynamic equations

“…For the non-radiative and non self-gravitation magnetohydrodynamic flows, there have been a number of studies under various conditions by several authors (see, e.g., [2,[15][16][17][18][19][20][21][22]). The existence and uniqueness of local smooth solutions were first obtained in [21]; moreover, the existence of global smooth solutions with small smooth initial data was shown in [20].…”

“…The existence and uniqueness of local smooth solutions were first obtained in [21]; moreover, the existence of global smooth solutions with small smooth initial data was shown in [20]. Chen and Wang [15] investigated a free boundary problem with general large initial data with exponents r [0, 1], q ≥ 2r + 2. Under the technical condition that (r, θ) satisfies…”