Strong Solutions for Ferrofluid Equations in Exterior Domains
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Strong Solutions for Ferrofluid Equations in Exterior Domains Jáuber C. Oliveira1
Received: 1 November 2016 / Accepted: 21 December 2017 © Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract We prove the global existence strong solutions for the system of partial differential equations corresponding to the Shliomis model for magnetic fluids in exterior domains without regularization terms in the magnetization equation under the assumption of small data and also small coupling parameter. Keywords Navier-Stokes equation · Magnetization equation · Strong solution · Exterior domain · Magnetic fluids · Ferrofluids
1 Introduction Ferrofluids (also called magnetic fluids) are suspensions of ferromagnetic particles in a nonmagnetic carrier fluid (see [5, 9]). Recent developments in theory and applications of ferrofluids are described in [8] and the references therein. As a small sample of applications, we mention biomedical applications where the nanoparticles are used as drug carriers and in producing magnetic forces on individual tumor cells to separate them from healthy cells. Technical applications include actuators, use in electrical drives, in adaptive bearing and dampers. Recently, two mathematical models for ferrofluids have been the subject of contributions in the matter of existence of solutions (see [2–4, 13, 14] and references therein). Nonlinear heat transfer effects have been recently considered in [1]. We will follow the terminology used in these papers, in which these models are called the Rosensweig model and the Shliomis model. The subject of this paper is the Shliomis model, so in this introduction we briefly review the mathematical contributions on this model in the matter of well-posedness. We consider the following system of PDEs (Shliomis model, see [11, 12]) for ferrofluids. μ0 ∂u + (u · ∇)u − ηu + ∇p = μ0 M · ∇H + curl(M × H) ∂t 2
in Q
B J.C. Oliveira
[email protected]
1
Departamento de Matemática, Universidade Federal de Santa Catarina, CEP 88040-900 Florianópolis, SC, Brazil
(1)
J.C. Oliveira
div u = 0
in Q
(2)
∂M 1 + (u · ∇)M + (M − χ0 H) ∂t δ 1 = [curl u] × M − β0 M × (M × H) 2 div(H + M) = F in Q
in Q
(4)
curl H = 0 in Q u(0) = u0 , u = 0,
(3)
(5)
∇ · u0 = 0,
M(0) = M0
in Ω
H · n = −M · n in Σ
(6) (7)
where Ω ⊂ R3 is an exterior domain, simply-connected with regular boundary, Q = Ω × (0, ∞), Σ = ∂Ω × (0, ∞), and n denotes the unit vector normal to the boundary and pointing outwards. The third equation was derived in 1972 by Shliomis [10]. u and p denote the fluid velocity and pressure, respectively. M and H is the magnetization and the magnetic field inside Ω, respectively. Equations (4), (5) correspond to the magnetostatic equations. Amirat and Hamdache [2] investigated for bounded domains the existence of weak solutions globally in time and with finite energy using a model which included a regularization term σ M in the equation for the magnetization field. In [3], Amirat and Hamdache established the local existence of strong solutions i
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