Low Regularity Well-Posedness for the 3D Viscous Non-resistive MHD System with Internal Surface Wave
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Journal of Mathematical Fluid Mechanics
Low Regularity Well-Posedness for the 3D Viscous Non-resistive MHD System with Internal Surface Wave Xiaoxia Ren and Zhaoyin Xiang Communicated by G. Seregin
Abstract. In this paper, we establish the local well-posedness of strong solutions to the three dimensional viscous non-resistive MHD system with internal surface wave in low regularity Sobolev spaces. Due to the absence of magnetic diffusion, the proof will be based on the new Stokes estimates involving in optimal surface regularity, the weighted energy estimates and the higher order horizontal regularity estimates. Mathematics Subject Classification. 35A01, 35Q35, 35Q60. Keywords. Viscous non-resistive MHD system, Internal surface wave, Low regularity, Local well-posedness.
1. Introduction 1.1. Presentation of the Problem In this paper, we investigate the local well-posedness of the three-dimensional magneto-hydrodynamical (MHD) system with zero magnetic diffusion ⎧ in Ω±t , ⎪ ⎨ ∂t u± + u± · ∇u± + ∇p± − Δu± = b± · ∇b± ∂t b± + u± · ∇b± = b± · ∇u± in Ω±t , (1.1) ⎪ ⎩ div u = div b = 0 in Ω , ±
in moving domains
and
±
±t
Ω+t := (y, t) ∈ R3 × R+ η(y , t) < y3 < 1 Ω−t := (y, t) ∈ R3 × R+ − 1 < y3 < η(y , t)
with y = (y1 , y2 , y3 ) = (y , y3 ), where u± = (u±1 , u±2 , u±3 ) are the fluid velocity fields, b± = (b±1 , b±2 , b±3 ) are the magnetic fields and p± are the pressures. Here η : R2 × R+ → R is an unknown function and denotes the free internal interface Σt := y ∈ R3 | y3 = η(y , t) , which is advected with the fluids according to the kinematic boundary condition ∂t η = u+3 − u+1 ∂y1 η − u+2 ∂y2 η Denoting
Σ± := y ∈ R3 | y3 = ±1 0123456789().: V,-vol
on
Σt .
(1.2)
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JMFM
by the upper and lower boundaries of Ω±t , we will prescribe the boundary conditions by assuming that ⎧ u =0 on Σ± × {t > 0}, ⎪ ⎨ ±
(p+ I − Du+ ) − (p− I − Du− ) n = −ηn on Σt , (1.3) ⎪ ⎩ u+ = u− and b+ · n = b− · n on Σt ,
where Du± ij = ∂yj u±i + ∂yi u±j are the symmetric gradients of u± and the unit normal n of Σt is pointing up and is given by
− ∂y1 η, −∂y2 η, 1 , n= 1 + (∂y1 η)2 + (∂y2 η)2 and the initial conditions will be that u± (x, 0) = u±0 , η(x, 0) = η0 (x)
b± (x, 0) = b±0 in
R . 2
in
Ω±0 ,
(1.4)
The MHD system of form (1.1) can be applied to model plasmas when the plasmas are strongly collisional, or the resistivity due to these collisions is extremely small. We refer to [2] for some detailed discussions on the relevant physical background of this system. 1.2. Some Previous Works For an MHD system, when the magnetic diffusion is absent, the question of whether smooth solution develops singularity in finite time has been a long-standing open problem, but there are some important progresses in recent years. For instance, Lin, Xu and Zhang [12] first proved that the 2D Cauchy problem admits a global smooth solution for a class of admissible perturbations around the equilibrium state (0, e1 ). Then Wu, Zhang and the authors [16] further established the global
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