Abstract
A logarithmically improved regularity criterion for the 3D nematic liquid crystal flows is established.
Highlights
We consider the following hydrodynamical systems modeling the flow of nematic liquid crystal materials 1, 2 : ut u · ∇u ∇π − μΔu −λ∇ · ∇d ∇d Δd − f d ⊗ d, 1.1 dt u · ∇d − d · ∇u γ Δd − f d, 1.2 div u 0, 1.3 v, d |t 0 v0, d0 in R3
The 3 × 3 matrix is defined by ∇ ∇d ij ∂id · ∂j d . ⊗ is the usual Kronecker multiplication, for example, a ⊗ b ij aibj for a, b ∈ R3
Condition 1.5 only involves the velocity field u, which plays a dominant role in regularity theorem
Summary
We consider the following hydrodynamical systems modeling the flow of nematic liquid crystal materials 1, 2 : ut u · ∇u ∇π − μΔu −λ∇ · ∇d ∇d Δd − f d ⊗ d , 1.1 dt u · ∇d − d · ∇u γ Δd − f d , 1.2 div u 0, 1.3 v, d |t 0 v0, d0 in R3.1.4 u x, t ∈ R3 is the velocity field of the flow. d x, t ∈ R3 is the averaged macroscopic/ continuum molecular orientations vector in R3. π x, t is a scalar function representing the pressure including both the hydrostatic part and the induced elastic part from the orientation field . μ is a positive viscosity constant. A logarithmically improved regularity criterion for the 3D nematic liquid crystal flows is established. The constant γ is the microscopic elastic relaxation time Deborah number for the molecular orientation field. Equation 1.5 can be regarded as a logarithmically improved regularity criterion of the form ∇u ∈ Lr 0, T ; Lp R3 with 2/r 3/p 2.
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