A Regularity Criterion for the 2D Full Compressible MHD Equations with Zero Heat Conductivity
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A Regularity Criterion for the 2D Full Compressible MHD Equations with Zero Heat Conductivity Xiuhui Yang1
Received: 13 February 2019 / Accepted: 3 January 2020 © Springer Nature B.V. 2020
Abstract In this paper we establish a regularity criterion for the 2D full compressible MHD equations with zero heat conductivity and initial vacuum in a bounded domain. Keywords 2D full compressible MHD equations · Bounded domain · Regularity criterion Mathematics Subject Classification (2000) 35Q60 · 35B44 · 76W05
1 Introduction Let Ω ⊂ R2 be a bounded and simply connected domain with smooth boundary ∂Ω and n be the unit outward normal vector to ∂Ω. We consider the 2D full compressible MHD equations: ∂t ρ + div (ρu) = 0,
(1.1)
∂t (ρu) + div (ρu ⊗ u) + ∇p − μu − (λ + μ)∇div u 1 = b · ∇b − ∇|b|2 , 2 CV ∂t p + div (pu) − kθ + pdiv u R 2 μ = ∇u + ∇ut + λ(div u)2 + η|rot b|2 , 2 ∂t b + u · ∇b − b · ∇u + bdiv u = ηb,
div b = 0,
B X. Yang
[email protected]
1
Department of Mathematics, College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, P.R. China
(1.2)
(1.3) (1.4)
X. Yang
with the initial and boundary conditions u = 0,
∂θ = 0, ∂n
b · n = 0,
rot b = 0
(ρ, u, p, b)(·, 0) = (ρ0 , u0 , p0 , b0 )(·)
in Ω ⊂ R2 .
k
on ∂Ω × (0, ∞),
(1.5) (1.6)
Here the unknowns ρ, u, p := Rρθ, θ and b stand for the density, velocity, pressure, and temperature of the fluid, and the magnetic field, respectively. R > 0 is the generic gas constant, μ and λ are the shear viscosity and bulk viscosity of the fluid, respectively, and satisfy μ > 0 and λ + μ ≥ 0. CV > 0 is the specific heat at constant volume, k > 0 is the heat conductivity coefficient, and η > 0 is the magnetic diffusivity coefficient. ∇ut denotes the transpose of the matrix ∇u. We will use rot b := ∂1 b2 − ∂2 b1 for the 2D vector b = (b1 , b2 )t and rot φ := (∂2 φ, −∂1 φ)t for the scalar φ. Due to the physical importance of the MHD, there are a lot of works on the system (1.1)–(1.4), among others, we mention [5] on the local strong solutions, [3, 4, 9] on the global weak solutions, [15, 16] on the low Mach number limit, and [19] on the time decay of small smooth solutions. In [10], Huang and Li proved the following regularity criterion: ρ ∈ L∞ 0, T ; L∞ ,
u ∈ Ls 0, T ; Lr
for
2 3 + = 1, 3 < r ≤ ∞, s r
(1.7)
with b satisfying the homogeneous Dirichlet boundary condition b = 0 on ∂Ω × (0, ∞). Later, this result was generalized in [6] to the case when b satisfies the Navier boundary condition, i.e., b · n = 0,
rot b × n = 0
on ∂Ω × (0, ∞).
(1.8)
We remark that the model (1.1)–(1.4) with k = 0, i.e. without heat conductivity, is corresponding to some important physical process in thermo-nonequilibrium [22] and magnetohydrodynamics [18], and there are many mathematical results on it, among others, we mention [2, 7, 12] on regularity criterion of solutions, [8, 21] on global existence and time decay of small smooth solutions, and [14, 24] on the thermal instability of solutions. Recently, Fan-Li-Nakamura [7] show the
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