Abstract
We prove global-in-time existence of weak solutions to a pde-model for the motion of dilute superparamagnetic nanoparticles in fluids influenced by quasi-stationary magnetic fields. This model has recently been derived in Grün and Weiß(On the field-induced transport of magnetic nanoparticles in incompressible flow: modeling and numerics, Mathematical Models and Methods in the Applied Sciences, in press). It couples evolution equations for particle density and magnetization to the hydrodynamic and magnetostatic equations. Suggested by physical arguments, we consider no-flux-type boundary conditions for the magnetization equation which entails H({text {div}},{text {curl}})-regularity for magnetization and magnetic field. By a subtle approximation procedure, we nevertheless succeed to give a meaning to the Kelvin force (mathbf {m}cdot nabla )mathbf {h} and to establish existence of solutions in the sense of distributions in two space dimensions. For the three-dimensional case, we suggest two regularizations of the system which each guarantee existence of solutions, too.
Highlights
Given two domains Ω ⊂⊂ Ω ⊂⊂ Rd, d ∈ {2, 3}, we are concerned with existence results for the model ρ0ut + ρ0(u · ∇)u + ∇p − div(2ηDu) = μ0(m · ∇)(α1h +β 2 ha) μ0 2 curl(m × (α1h β 2 ha )),=:hdiv u = 0, ct + u · ∇c + div(cVpart) = 0, Vpart
(C2) The magnetic field h is a gradient field on Ω, satisfying h|Ω \Ω ∈ H(div0)(Ω \Ω) due to the magnetostatic equations curl h = 0, div(h + m) = 0, and h|Ω ∈ H(div)(Ω) due to the formal energy estimate (2.2). (C3) Approximation functions for the magnetization should satisfy the boundary conditions curl m × ν|∂Ω = 0 if d = 3, curl m|∂Ω = 0 if d = 2 (3.1)
We can deduce that div ∇Rn −(div ∇Rn)Ω can be written in terms of the basis functions ψ4Ri, see (3.2.13), (3.2.5), (3.1.11), which just extend from their constant trace on ∂Ω constantly to Ω \Ω—cf. (3.2.15)—just as div ∇Rn −Ω does
Summary
To cope with the nonlinearities (m·∇)h and (∇m)T m in the convective term of the evolution equation for the particle density c, we refrain to local arguments – which are needed already in the first limit passage Discrete to Regularized Continuous This requires a subtle choice of Galerkin approximation spaces as in general the projections ΠXn φ, n ∈ N, of C0∞-functions onto ansatz spaces Xn do not have compact support. 5, we introduce a (T)ransport and (M)obility (R)egularized model – replacing the usual c log centropy by a strictly convex approximation with quadratic growth, and using a further density cut-off in the transport velocity Vpart, see (5.2) and (5.3) For this TMR-model, global existence of discrete solutions is established, and Sections 6 and 7 provide compactness results as well as the limit passage Discrete to Regularized Continuous. For the reader’s convenience, the Appendix A.5 explains the notation and provides references on definitions and further properties
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