Global Well-Posedness for Navier–Stokes Equations with Small Initial Value in $${B^{0}_{n,\infty}(\Omega)}$$ B n , ∞ 0 ( Ω )

Existence of Incompressible and Immiscible Flows in Critical Function Spaces on Bounded Domains

“…Proof: First let us prove the theorem for ν = 1. The case g = 0 has already been proved by [27], Corollary 4.14, (i) as a particular case.…”

“…In order to prove the main result, beforehand, we consider momentum equations with a prescribed variable density and transport equations in Section 2 and 3, respectively. In Section 2, existence of a solution to instationary Stokes system with nonzero divergence is proved (Theorem 2.6) relying on a consideration of the divergence problem (Subsection 2.1) and on maximal L ∞ γ -regularity of the Stokes operator in b α q,∞ (Ω), α ∈ R, exploited in [27]. Then, unique solvability for the nonlinear momentum equations with prescribed density…”

“…Lemma 2.7 (see [27], Lemma 5.1) Let Ω be a bounded domain of R n , n ≥ 3, of C 2 -class, then for q ∈ [n, ∞) and s ∈ (0, n q )…”

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We study global existence and uniqueness of solutions to instationary inhomogeneous Navier-Stokes equations on bounded domains of R n , n ≥ 3, with initial velocity in B 0 q,∞ (Ω), q ≥ n, and piecewise constant initial density. To this end, first, existence for momentum equations with prescribed density is obtained based on maximal L ∞ γ -regularity of the Stokes operator in little Nicolskii space b s q,∞ (Ω), s ∈ R, exploited in [27] and existence for divergence problem in b −s q,∞ (Ω), s > 0. Then, we obtain an existence result for transport equations in the space of pointwise multipliers for b −s q,∞ (Ω), s > 0.

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Global in time solvability of the Navier-Stokes equations in the half-space

“…x u, ∇p ∈ L q (0, ∞; L p (R n + )), t β ∇u ∈ L q 2 (0, ∞; L p 2 (R n + )), t γ u ∈ L q 3 (0, ∞; L p 3 (R n + )) for some β, γ > 0, 1 < p 2 , p 3 , q 2 , q 3 < ∞ with α = β + γ, 1 p = 1 p 2 + 1 p 3 and 1 q = 1 q 2 + 1 q 3 . The limiting case q = ∞ has been studied by M. Cannone, F. Planchon, and M. Schonbek [5] for h ∈ L 3 (R 3 + )(⊂ B −1+ 3 p p∞ (R 3 + )), by H. Amann [3] for h ∈ b −1+ n p p,∞ (R n + ), p > n 3 , p = n, and by M. Ri, P. Zhang and Z. Zhang [26] for h ∈ b 0 n∞ (R n + ), where b s pq (Ω) denotes the completion of the generalized Sobolev space H s p (Ω) in B s pq (Ω). In particular, in [5], the solution exists globally in time when h…”