Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum Yang Liu and Xin Zhong Abstract. This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided that the quantity ρ0 L∞ + b0 L3 is suitably small and the viscosity coefficients satisfy 3μ > λ. Here, the initial velocity and initial temperature could be large. The assumption on the initial density does not exclude that the initial density may vanish in a subset of R3 and that it can be of a nontrivially compact support. Our result is an extension of the works of Fan and Yu (Nonlinear Anal Real World Appl 10:392–409, 2009) and Li et al. (SIAM J Math Anal 45:1356–1387, 2013), where the local strong solutions in three dimensions and the global strong solutions for isentropic case were obtained, respectively. The analysis is based on some new mathematical techniques and some new useful energy estimates. This paper can be viewed as the first result concerning the global existence of strong solutions with vacuum at infinity in some classes of large data in higher dimension. Mathematics Subject Classification. 76W05, 76N10. Keywords. Full compressible magnetohydrodynamic equations, Global well-posedness, Vacuum.
1. Introduction Let Ω ⊂ R3 be a domain, and the motion of a viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flow in Ω can be described by full compressible MHD equations (see [20, Chapter 3]): ⎧ ρt + div(ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ρut + ρu · ∇u − μΔu − (λ + μ)∇ div u + ∇p = curl b × b, cv ρ(θt + u · ∇θ) + p div u − κΔθ = Q(∇u) + ν| curl b|2 , (1.1) ⎪ ⎪ − b · ∇u + u · ∇b + b div u = νΔb, b ⎪ t ⎪ ⎩ div b = 0, where the unknowns ρ ≥ 0, u ∈ R3 , θ ≥ 0, and b ∈ R3 are the density, velocity, absolute temperature, and magnetic field, respectively; p = Rρθ, with positive constant R, is the pressure, and μ (1.2) Q(∇u) = |∇u + (∇u) |2 + λ(div u)2 , 2 with (∇u) being the transpose of ∇u. The constant viscosity coefficients μ and λ satisfy the physical restrictions μ > 0,
2μ + 3λ ≥ 0.
(1.3)
Corresponding author: Xin Zhong. Yang Liu was partially supported by National Natural Science Foundation of China (No. 11901288). Xin Zhong was partially supported by National Natural Science Foundation of China (Nos. 11901474, 12071359). 0123456789().: V,-vol
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Y. Liu and X. Zhong
ZAMP
Positive constants cν , κ, and ν are the heat capacity, the ratio of the heat conductivity coefficient over the heat capacity, and the magnetic diffusive coefficient, respectively. Let Ω = R3 , and we consider the Cauchy problem of (1.1) with (ρ, u, θ, b) vanishing at infinity (in some weak sense) with given initial data ρ0 , u0 , θ0 , and b0 , as (ρ, u, θ, b)|t=0 = (ρ0 , u0 , θ0 , b0 ),
x ∈ R3 .
(1.4)
The compressible MHD equations govern the motion of electrically conducting fluids such as plas
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