Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids Caidi Zhao, Yanjiao Li and Grzegorz L ukaszewicz Abstract. In this article, the authors investigate the non-autonomous magneto-micropolar fluids in a two-dimensional bounded domain. They first prove the existence of a pullback attractor for the associated process. Then, they construct a family of invariant Borel probability measures supported on the pullback attractor and prove that this family of probability measures is indeed a statistical solution for the magneto-micropolar fluids. Further, they establish that if some form of the Grashof number is small enough, then the pullback attractor degenerates to a single bounded complete trajectory, which implies the partial degenerate regularity of the statistical solution in the sense that it is supported on a set in which the weak solutions are in fact partially strong solutions. Mathematics Subject Classification. 35B41, 34D35, 76F20. Keywords. Statistical solution, Invariant Borel probability measure, Degenerate regularity, Magneto-micropolar fluid, Pullback attractor.
1. Introduction In this article, we investigate the following two-dimensional (2D) equations of magneto-micropolar fluids: ⎧ ∂u 1 ⎪ ⎪ − (ν + ν h · h = 2νr ∇ × ω + r(h · ∇)h + f, )Δu + (u · ∇)u + ∇ p + r ⎪ ⎪ ∂t 2 ⎪ ⎪ ⎪ ⎨ ∂ω − αΔω + 4νr ω + j(u · ∇)ω = 2νr ∇ × u + g, j (1.1) ∂t ⎪ ⎪ ∂h ⎪ ⎪ − μΔh + (u · ∇)h − (h · ∇)u = 0, ⎪ ⎪ ⎪ ⎩ ∂t div u = 0, div h = 0, with the initial and boundary conditions
where
u(x, τ ) = uτ (x), ω(x, τ ) = ωτ (x), h(x, τ ) = hτ (x), x ∈ Ω,
(1.2)
u(x, t) = ω(x, t) = h(x, t) = 0, (x, t) ∈ ∂Ω × [τ, +∞),
(1.3)
⎧ u = u(x, t) = (u1 (x, t), u2 (x, t)) ⎪ ⎪ ⎨ ω = ω(x, t) h = h(x, t) = (h1 (x, t), h2 (x, t)) ⎪ ⎪ ⎩ p = p(x, t)
Supported by NSF of China with Nos. 11971356, 11271290 and by NSF of Zhejiang Province with No. LY17A010011. Also by National Science Center (NCN) of Poland under Project No. DEC-2017/25/B/ST1/00302. 0123456789().: V,-vol
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are unknown functions and ∂u1 ∂u2 − , ∇×u= ∂x1 ∂x2
C. Zhao, Y. Li and G. L ukaszewicz
divu =
∂u1 ∂u2 + , ∂x1 ∂x2
ZAMP
∇×ω =
∂ω ∂ω ,− ∂x2 ∂x1
.
Equation (1.1) describes the motion of electrically conducting micropolar fluids (see [20]) in the presence of magnetic fields. Here, the vector functions u(x, t) and h(x, t) denote, respectively, the velocity field and magnetic field of the fluid at a physical point x = (x1 , x2 ) and at the moment of time t, the scalar functions ω(x, t) and p(x, t) stand for, respectively, the micro-rotational velocity and hydrostatic pressure of the fluid, the vector function f = f (x, t) = (f1 (x, t), f2 (x, t)) and the scalar function g = g(x, t) are given external force and angular momentum, respectively. In addition, ν, νr , r, j, α, μ are positive constants associated with the properties of the fluids material and we will take r = j = 1 for simplicity. In this article, we consider Eq. (1.1) in a 2D bounded domain Ω ⊂ R2 wi
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