Besov regularity for the stationary Navier–Stokes equation on bounded Lipschitz domains

Abstract

We use the scale B s τ (L τ (Ω)), 1/τ = s/d + 1/2, s > 0, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains Ω ⊂ R d , d ≥ 3, with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. By using Banach's fixed point theorem, we extend this analysis to the stationary Navier-Stokes equation with suitable Reynolds number and data, respectively.

On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

“…Similar results for (stochastic) evolution equations can be found, e.g., in [20,25]. At the same time, we know that the solution to most of the equations in the aforementioned references may have higher regularity α > s p in the scale ( * ), see, e.g., [3,5,7,8,10,11,12,15,16,21]. For instance, in the example above, it is known that S(Ω) ⊆ B α τ,τ (Ω),…”

“…and hence p z satisfies (16). Thus, Lemma 3.10 ensures that S u (Ω) ⊆ H s pz (Ω) = F s pz,2 (Ω) for all s < z := 1 + 1/p z .…”

“…Thus, the claim follows from Proposition 3.9 applied for A := F , q := 2, as well as 0 < s < 1 and p ∈ [2, ∞) restricted by (16), and Definition A.2.…”

“…Recently Eckhardt et al [16,Theorem 3.3] addressed the question of Besov regularity for dimensions d ≥ 3 under the additional conditions that the boundary of Ω is connected and g = 0. Rewritten in our notation they were able to show that for σ 1 = 1/2 and σ 3 = 0 we have for d ≥ 4…”

“…the scale ( * ) with p = 2. Moreover note that this b = b(s 2 ) is monotonically increasing in s 2 , where3 ≤ b(s 2 ) < b(3/2 + m) = ∞, s 2 ∈ [3/2, 3/2 + m).Recently Eckhardt et al[16, Theorem 3.3] addressed the question of Besov regularity for dimensions d ≥ 3 under the additional conditions that the boundary of Ω is connected and g = 0. Rewritten in our notation they were able to show that for σ 1 = 1/2 and σ 3 = 0 we have for d ≥ 4α 2 (S u (Ω)) ≥ 3 S π (Ω)) ≥ 1A Appendix: Basics from function space theoryIn this supplementary section we collect the main definitions and assertions concerning function spaces on domains which are needed throughout the paper.Here 'domain' always means 'non-empty, connected, open set'.…”