Abstract
In this paper, the author establishes a blow-up criterion of strong solutions to 3D compressible viscous magneto-micropolar fluids. It is shown that if the density and the velocity satisfy , where and , then the strong solutions to the Cauchy problem can exist globally over . The initial density may vanish on open sets, that is, the initial vacuum is allowed. MSC:76N10, 35B44, 35B45.
Highlights
In this paper, we consider the following D compressible viscous magneto-micropolar fluids: ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨(ρρt+ div(ρu) =, u)t + div(ρu ⊗ u) – (μ = ξ ∇ × w + (∇ × H) + × ξ) H, u – (μ + λ – ξ )∇ div u + ∇P⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩(dHρitvw–H)t∇=+× di,v(u(ρ×u ⊗ w) H) = –μ –∇ × w
We introduce the following notations
We can take (ρ, u, w, H)(x, T∗) as the initial data and apply the local existence theorem to extend the local strong solutions beyond T∗. This contradicts the assumption that T∗ is the maximal time of existence
Summary
We consider the following D compressible viscous magneto-micropolar fluids:. Chen [ ] established the local existence and uniqueness of strong solutions under the assumption that the initial density may vanish, and in [ ] Chen et al proved a blow-up criterion that. To present the main result, we first give the following local existence and uniqueness of strong solutions to the Cauchy problem There is no any additional growth condition on the micro-rotational velocity w and magnetic field H This reveals that the density and the linear velocity play a more important role compared to the angular velocity of rotation of particles and the magnetic field in the regularity theory of solutions to D compressible magneto-micropolar fluid flows. ≤ Cδ which, together ( . ) and ( . ), choosing δ > suitably small, gives d μ|∇u| + (μ + λ)(div u) + μ |∇w| + μ + λ (div w) + σ |∇H| dx dt d +
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