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Pré-Publication, Document De Travail Année : 2020

A new regularity criterion for weak solutions to the 3D micropolar fluid flows in terms of the pressure

Résumé

This note aims to giving a new regularity criterion for weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the field of pressure. Keywords Micropolar fluid flows; · weak solutions; · Pressure criterion · Besov spaces.. Mathematics Subject Classification (2000) 35Q35 · 35B65 · 76D05 In this note we consider the following Cauchy problem (1) for the incompressible micropolar fluid equations in R 3 : ∂ t u − ∆ u + (u · ∇) u + ∇π − ∇ × ω = 0, (1a) ∂ t ω − ∆ ω − ∇∇ · ω + 2ω + (u · ∇)ω − ∇ × u = 0, (1b) ∇ · u = 0, (1c) u(x, 0) = u 0 (x), ω(x, 0) = ω 0 (x), (1d) where u = u(x,t) ∈ R 3 , ω = ω(x,t) ∈ R 3 and π = π (x,t) denote the unknown velocity of the fluid, the micro-rotational velocity of the fluid particles and the unknown scalar pressure of the fluid at the point (x,t) ∈ R 3 × (0, T), respectively, while u 0 , ω 0 are given initial data satisfying ∇ · u = 0 in the sense of distributions. This model for micropolar fluid flows proposed by Eringen [6] enables to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations for the viscous incompressible fluids, such as for example, the motion of animal blood, muddy fluids, liquid crystals and dilute aqueous polymer solutions, colloidal suspensions, etc. When the micro-rotation effects are neglected or ω = 0, (1) reduces to the in-compressible Navier-Stokes equations, and it is well known that regularity criteria
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Dates et versions

hal-02651131 , version 1 (29-05-2020)

Identifiants

  • HAL Id : hal-02651131 , version 1

Citer

Sadek Gala, Maria Alessandra Ragusa, Michel Thera. A new regularity criterion for weak solutions to the 3D micropolar fluid flows in terms of the pressure. 2020. ⟨hal-02651131⟩

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