Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and
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https://doi.org/10.1007/s11425-019-1694-0
Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data Kunquan Li1 , Zilai Li2 & Yaobin Ou1,∗
2School
1School of Mathematics, Renmin University of China, Beijing 100872, China; of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
Email: [email protected], [email protected], [email protected] Received November 22, 2019; accepted May 20, 2020
Abstract
In this paper, the global existence of the classical solution to the vacuum free boundary problem
of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied. The solutions to the system (1.6)–(1.8) are in the class of radius-dependent solutions, i.e., independent of the axial variable and the angular variable. In particular, the expanding rate of the moving boundary is obtained. The main difficulty of this problem lies in the strong coupling of the magnetic field, velocity, temperature and the degenerate density near the free boundary. We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates, and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variable rx by weighted estimates, and also the uniform-in-time weighted estimates of the higher order derivatives of solutions by delicate analysis. Keywords
compressible magnetohydrodynamic equations, vacuum free boundary, global axisymmetric clas-
sical solutions, large initial data MSC(2010)
35B07, 35R35, 76N10, 76W05
Citation: Li K Q, Li Z L, Ou Y B. Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-019-1694-0
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Introduction
Magnetohydrodynamics concerns the motion of fluids in a magnetic field, and it has a wide range of applications. Examples of such fluids include astrophysics, geophysics, high-speed aerodynamics, and plasma physics, etc. The mechanisms which describe magnetohydrodynamics are governed by a combination of the equations of fluid dynamics and the equations of magnetic fields. Due to the strongly coupling interaction of fluids with magnetic fields, the mathematical analysis is much more complicated in the compressible case. When the solution is sufficiently regular, the full system of partial differential * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Li K Q et al.
2
Sci China Math
equations for the three-dimensional viscous compressible magnetohydrodynamic (MHD) flows takes the form [13, 21, 22, 24]: ρt + div(ρu) = 0, ρ(u + u · ∇u) + ∇p + 1 ∇(|H|2 ) − H · ∇H = µ∆u + (µ + λ)∇divu, t 2 ρ(e + u · ∇e) + p div u = div(κ∇T ) + ν|∇ × H|2 + 2µ|D(u)|2 + λ(divu)2 , t Ht + u · ∇H − H · ∇u + Hdivu = ν∆H, divH = 0,
(1.1)
where D(u)
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