Abstract
We consider an instationary generalized Stokes system with nonhomogeneous divergence data under a periodic condition in only some directions. The problem is set in the whole space, the half space or in (after an identification of the periodic directions with a torus) bounded domains with sufficiently regular boundary. We show unique solvability for all times in Muckenhoupt weighted Lebesgue spaces. The divergence condition is dealt with by analyzing the associated reduced Stokes system and in particular by showing maximal regularity of the partially periodic reduced Stokes operator.
Highlights
Consider the partially periodic instationary generalized Stokes problem ⎧ ⎪⎪⎪⎨ ∂tu − Δu + ∇p div u = =f g in (0, T ) × Ω, in (0, T ) × Ω, ⎪⎪⎪⎩u|∂Ω = 0 on (0, T ) × ∂Ω, u|t=0 = u0 in Ω, (1.1)
The paper is structured as follows: Firstly, in Sect. 2 we prove that Theorem 1.3 can be deduced from Theorem 1.4 by arguments similar to the ones in [1,2]
Since the topology of G is inherited by Rn, we can talk in virtue of the canonical quotient mapping about the space of smooth functions C∞(G) and the Schwartz–Bruhat space S(G) [4,19]
Summary
Where u : (0, T ) × Ω → Rn is the fluid velocity and p : (0, T ) × Ω → R is the pressure. Theorem 3.5 in [23] shows that the partially periodic Stokes operator admits maximal Lp regularity in Lqω,σ(G) for all q ∈ (1, ∞) and all ω ∈ Aq(G) for n ≥ 3 with an Aq-consistent estimate. By Theorem 1.4, there is a solution (u, p) ∈ Wω2,q(Ω)n × Wω1,q(Ω) to (1.2) with data (fr, g), where g ∈ Wω1,q(Ω) with λg ∈ W0−,ω1,q(Ω) is the unique solution to λ(g, φ) + (∇g, ∇φ) = (fr, ∇φ), φ ∈ Wω1,q (Ω), which exists due to Lemma 7.1, and where λ = 0 is allowed in the case of a bounded domain. Since Arqe,ωd is invertible on bounded domains by Lemma 2.1, the additional remark follows from the Theorem of Weis.
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