Suitable Weak Solutions of the Incompressible Magnetohydrodynamic Equations in Time Varying Domains
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Suitable Weak Solutions of the Incompressible Magnetohydrodynamic Equations in Time Varying Domains Yunsoo Jang1 · Dugyu Kim2
Received: 29 January 2020 / Accepted: 18 August 2020 © Springer Nature B.V. 2020
Abstract The purpose of this paper is to study the three-dimensional system of magnetohydrodynamic (MHD equations) for a viscous incompressible resistive fluid. We are interested in the existence of suitable weak solutions to the system in time varying domains. To do this, we consider the approximate equations related to the MHD equations and we apply the Leray-Schauder fixed point theorem to the solutions of the equations over the moving boundary domains. Existence of suitable weak solutions is established by the energy estimates and the compactness results in Lebesgue and Sobolev spaces. Keywords Suitable weak solution · MHD equations · Localized energy inequality · Moving boundary · Schauder theory Mathematics Subject Classification (2010) 35D30 · 76D05 · 76W05
1 Introduction We study the three-dimensional incompressible magnetohydrodynamic equations in time varying domains which means that the boundary of the domain moves as time varies. For this, we let Ut be non-empty bounded open sets in R3 with connected smooth boundary ∂ Ut , and denote t = Ut × {t} and D = ∪t∈[0,T ] t ⊂ R3 × R+ . Additionally, we consider the
B D. Kim
[email protected] Y. Jang [email protected]
1
Department of Mathematics Education, Kangwon National University, Chuncheon, 24341, Korea
2
Center for Mathematical Analysis and Computation (CMAC), Yonsei University, Seoul, 03722, Korea
Y. Jang, D. Kim
incompressible MHD equations with unit viscosity and zero external force: ⎧ ∂u ⎪ ⎪ − u + (u · ∇)u − (∇ × B) × B + ∇π = 0 ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂B + ∇ × ∇ × B − ∇ × (u × B) = 0 ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎩ div u = 0, divB = 0
(1)
in time varying domains D with the initial and boundary conditions in 0 u = u0 , B = B 0 u = U,
B · n = 0,
(∇ × B) × n = 0
on t := ∂t = ∂ Ut × {t},
where n is the unit outward normal vector along the boundary t . Here u, π , and B are quantities corresponding to the velocity of the fluid, its pressure, and the magnetic filed, respectively. We assume that the initial data u0 and B0 are solenoidal vector fields in L2 (0 ). The boundary condition u = U means that the velocity field u has no slip boundary condition in the time varying domain since we define U is to be the boundary velocity in the next section. The MHD equations describe the motion of an electrically conducting fluid interacting with a magnetic field, e.g. in plasma and liquid metals; see [6] for detailed background. Over the past few decades, there have been significant contributions to the existence and regularity of the solutions to the MHD equations. Due to the work of Duvaut and Lions [8], the initial value problem on R3 of the system (1) admits a global weak solutions with finite energy and global strong solutions for small initial data. Sermange and Temam [27] showed that if (u, B) is a weak solution belongs to L∞ (0, T : H 1 (R3 )), t
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