On the Cauchy problem of 3D nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and vacuum
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
On the Cauchy problem of 3D nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and vacuum Mingyu Zhang Abstract. This paper concerns the Cauchy problem of the three-dimensional nonhomogeneous incompressible magnetohydrodynamic (MHD) equations with density-dependent viscosity and vacuum. We first establish some key a priori algebraic decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence and large time asymptotic behavior of strong solutions in the whole three-dimensional space, provided that the initial velocity and magnetic field are suitable small in the H˙ β -norm for some β ∈ (1/2, 1]. Note that any smallness and compatibility conditions assumed on the initial data are not used in this result. Moreover, the density can contain vacuum states and even have compact support initially. Mathematics Subject Classification. 35Q35, 76D03, 76W05. Keywords. Incompressible magnetohydrodynamic equations, Density-dependent viscosity, Vacuum, Global strong solutions.
1. Introduction The nonhomogeneous incompressible magnetohydrodynamic (MHD) equations with density-dependent viscosity read as follows (cf. [22,29]): ⎧ ρt + div(ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨(ρu)t + div(ρu ⊗ u) + ∇P − div(2μ(ρ)d) = B · ∇B, (1.1) ⎪ ⎪ Bt + u · ∇B − B · ∇u − νΔB = 0, ⎪ ⎪ ⎪ ⎩ divu = 0, divB = 0, with t 0 and x = (x1 , x2 , x3 ) ∈ R3 . The unknown functions ρ = ρ(x, t), u = (u1 , u2 , u3 )(x, t), B = (B 1 , B 2 , B 3 )(x, t) and P = P (x, t) denote the fluid density, velocity, magnetic field and pressure, respectively. The deformation tensor is defined by 1 d = [∇u + (∇u)T ], (1.2) 2 the viscosity μ(ρ) satisfies the following hypothesis: μ ∈ C 1 [0, ∞),
μ(ρ) > 0.
(1.3)
The constant ν > 0 is the resistivity coefficient which is inversely proportional to the electrical conductivity constant and acts as the magnetic diffusivity of magnetic fields. We consider the Cauchy problem of (1.1) with (ρ, u, B) vanishing at infinity and the initial conditions: ρ(x, 0) = ρ0 (x),
ρu(x, 0) = m0 (x),
for given initial data ρ0 , m0 and B0 . 0123456789().: V,-vol
B(x, 0) = B0 (x),
x ∈ R3 ,
(1.4)
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M. Zhang
ZAMP
The mathematical model of magnetohydrodynamics (MHD) is used to simulate the motion of a conducting fluid under the effect of the electromagnetic field, and there are many literatures on the mathematical study of nonhomogeneous incompressible MHD flow by many physicists and mathematicians due to its physical importance, complexity, rich phenomena and mathematical challenges, see for example, [7,9,11,14,16–18]. If there is no electromagnetic effect, that is B = 0, the MHD system reduces to the Navier–Stokes equations, which have been discussed in numerous studies, see [4,5,10,19,24,33] and the references therein. The issues of well-posedness and dynamical behaviors of MHD system are rather complicated to investigate because of the strong coupling and interplay interaction between the fluid motion and the magnetic field
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