Lp-Estimates for Solutions to the Initial Boundary-Value Problem for the Generalized Stokes System in a Bounded Domain

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Vanishing Viscous Limits for 3D Navier–Stokes Equations with a Navier-Slip Boundary Condition (2012, methodology): “See [29] For simplicity, we set P = PJ and decompose R = PR + (I − P)R .”
Strong solutions of the compressible nematic liquid crystal flow (2011, unknown): “ p p−1, k∇(divu)kLp ≤ C sup nZ ∇(divu)· gdx : g ∈ C∞(Ω,R3), kgk Lp′ = 1 o . For any g ∈ C∞(Ω,R3), with kgk Lp′ = 1, by the Helmholtz’s decomposition Theorem (see Fujiwara-Morimoto [11] and Solonnikov [33]), there exist G ∈ C∞(Ω)∩W1,p′(Ω) and H ∈ C∞(Ω)∩ Lp′(Ω,R3) such that g = ∇G+H, divH = 0 in Ω, ∂G”
The L∞-Stokes semigroup in exterior domains (2012, background): “1) is studied in the literature [53], [55], [8] to construct a solution for the Dirichlet-boundary value problem to the Stokes equations.”
Liquid Crystal Flows in Two Dimensions (2010, unknown): “(4.16) Since f ≡ −u·∇u−∇·(∇d ∇d) ∈ L 4 3 (ΩT ), the Lp-theory [20] of non homogeneous, non-stationary Stokes equations to (4.14)-(4.16) implies that ∇P ∈ L 4 3 (ΩT ) and ‖∇P‖ L 4 3 (ΩT ) .”