Several non-standard problems for the stationary Stokes system

Dagmar Medková

doi:10.1515/anly-2018-0035

Published by Oldenbourg Wissenschaftsverlag January 17, 2020

Several non-standard problems for the stationary Stokes system

Dagmar Medková

From the journal Analysis

https://doi.org/10.1515/anly-2018-0035

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Abstract

This paper studies the Stokes system -Δ⁢𝐮+∇⁡ρ=𝐟, ∇⋅𝐮=χ in Ω with three boundary conditions:

𝐧⋅𝐮=𝐧⋅𝐠,𝐧×(∇×𝐮)=𝐧×𝐡on ⁢∂⁡Ω,

𝐧⋅𝐮=𝐧⋅𝐠,𝝉⋅[∂⁡𝐮∂⁡𝐧-ρ⁢𝐧+b⁢𝐮]=𝐡⋅τon ⁢∂⁡Ω,

𝐧⋅𝐮=𝐧⋅𝐠,[T⁢(𝐮,ρ)⁢𝐧+b⁢𝐮]⋅τ=𝐡⋅τon ⁢∂⁡Ω.

Here Ω is a bounded simply connected planar domain. We find a necessary and sufficient condition for the existence of a solution in Sobolev spaces Ws,q⁢(Ω;ℝ2)×Ws-1,q⁢(Ω), with 1+1/q<s<∞, in Besov spaces Bsq,r⁢(Ω;ℝ2)×Bs-1q,r⁢(Ω), with 1+1/q<s<∞, and classical solutions in 𝒞k,α⁢(Ω¯,ℝ2)×𝒞k-1,α⁢(Ω¯), with 0<α<1, k∈ℕ.

Keywords: Stokes system; Navier type boundary conditions

MSC 2010: 35Q35

Funding source: Grantová Agentura České Republiky

Award Identifier / Grant number: GA17-01747S

Funding statement: The research was supported by RVO: 67985840 and GAČR grant No. 17-01747S.

A Appendix

A.1 Holomorphic functions

Lemma A.1.

Let ξ+i⁢η be a holomorphic function in Ω. If η∈Ck,α⁢(Ω¯) with k∈N0 and 0<α<1, then ξ∈Ck,α⁢(Ω¯).

Proof.

Suppose first that k=0. The domain with Lipschitz boundary satisfies the property Pα by [40, p. 394]. Therefore, ξ∈𝒞0,α⁢(Ω¯), by [40, Corollary 3.7].

If k≠0, we use that fact that ∂β⁡ξ+i⁢∂β⁡η is a holomorphic function for arbitrary multiindex β. ∎

Lemma A.2.

Let ξ+i⁢η be a holomorphic function in Ω, 1<q<∞, 0<s<∞. If η∈Ws,q⁢(Ω), then ξ∈Ws,q⁢(Ω).

Proof.

For the case s∉ℕ, see [40, Proposition 9.2]. Let now s∈ℕ. Since Ws,q⁢(Ω)⊂W1/2,q⁢(Ω), one has ξ∈W1/2,q⁢(Ω)⊂Lq⁢(Ω). The rest is a consequence of the fact that ∂1⁡ξ=∂2⁡η, ∂2⁡ξ=-∂1⁡η. ∎

Lemma A.3.

Let ξ+i⁢η be a holomorphic function in Ω. If η∈Bsq,r⁢(Ω), with 1<q,r<∞, 0<s<∞, then ξ∈Bsq,r⁢(Ω).

Proof.

We can suppose that η∈Bsq,r⁢(ℝ2). Then ∂j⁡η∈Bs-1q,r⁢(ℝ2) by [47, Chapter 3, Theorem 9]. Thus, ∂1⁡ξ=∂2⁡η∈Bs-1q,r⁢(ℝ2), ∂2⁡ξ=-∂1⁡η∈Bs-1q,r⁢(ℝ2). Proposition 7.6 of [41] gives ξ∈Bsq,r⁢(Ω). ∎

Lemma A.4.

Let ξ+i⁢η be a holomorphic function in Ω and 1<q<∞. If Ma⁢(η)∈Lq⁢(∂⁡Ω), then Ma⁢(ξ)∈Lq⁢(∂⁡Ω) and there exist non-tangential limits of ξ and η at almost all points of ∂⁡Ω.

Proof.

If u is a harmonic function in Ω, then Ma⁢(u)∈Lq⁢(∂⁡Ω) if and only if Aa⁢(u)∈Lq⁢(∂⁡Ω), where

Aa(u)(𝐱)=[∫Γa⁢(𝐱)|∇u(𝐲)|2d𝐱]1/2.

(See [17, Theorem] or [28, Theorem 1.5.10].) Since ∂1⁡ξ=∂2⁡η, ∂2⁡ξ=-∂1⁡η and therefore Aa⁢(ξ)=Aa⁢(η), we infer Ma⁢(ξ)∈Lq⁢(∂⁡Ω). Since Ma⁢(ξ),Ma⁢(η)∈Lq⁢(∂⁡Ω), there exist non-tangential limits of ξ and η at almost all points of ∂⁡Ω by [23] and [24, Theorem 1]. ∎

A.2 The Dirichlet problem for the Laplace equation

Theorem A.5.

Let ∂⁡Ω be of class Ck,1 with k∈N, and 1<p,q<∞, 1/p<s≤k+1.

If f∈Ws-2,p⁢(Ω), g∈Ws-1/p,p⁢(∂⁡Ω) and s-1/p∉ℕ, then there exists a unique solution u∈Ws,p⁢(Ω) of the Dirichlet problem
(A.1)Δ⁢u=f in ⁢Ω,u=g on ⁢∂⁡Ω.
If f∈Bs-2p,q⁢(Ω), g∈Bs-1/pp,q⁢(∂⁡Ω) and s<k+1, then there exists a unique solution u∈Bsp,q⁢(Ω) of the Dirichlet problem (A.1).

Proof.

If s<1+1/p, then (a) is a consequence of [39, Corollary 4.2]. If s∈ℕ, s>1, then (a) follows from [22, Theorem 2.4.2.5 and Theorem 2.5.1.1]. We obtain the rest by real interpolation. (See [53, Lemma 22.3], [56, Corollary 1.111, Theorem 1.122] and [19, Theorem 6.7].) ∎

A.3 Besov spaces

Proposition A.6.

Let -∞<t≤τ<∞ and 1<p,q,r,β<∞. Suppose that one from the following conditions holds:

τ>t, τ-2/q>t-2/p.
τ>t, τ-2/q=t-2/p, r≤β.
τ=t, p≤q, r≤β.

Then Bτq,r⁢(Ω)↪Btp,β⁢(Ω).

Proof.

If condition (a) holds, then Bτq,r⁢(Ω)↪Btp,β⁢(Ω) by [56, Theorem 1.97]. If condition (b) holds, then Bτq,r⁢(Ω)↪Btp,β⁢(Ω) by [56, pp. 78–79]. If condition (c) holds, then Bτq,r⁢(Ω)↪Btp,r⁢(Ω)↪Btp,β⁢(Ω) by [55, Section 3.3.1, Theorem] and [54, Section 4.6.1, Theorem]. ∎

A.4 Stokes system with prescribed pressure

Proposition A.7.

Let k∈N, 1<p,q<∞, ∂⁡Ω be of class Ck,1, 1/q<s≤k+1, s-1/q∉N0, 1/p<t≤k, t-1/p∉N0, and t≤s+1. Suppose that s+1-2/q≥t-2/p. Let g∈Wt-1/p,p⁢(∂⁡Ω), h∈Ws-1/q,q⁢(∂⁡Ω). Then there exists a unique solution (u,ρ)∈Wt,p⁢(Ω,R2)×Ws,q⁢(Ω) of the problem

-Δ⁢𝐮+∇⁡ρ=0,∇⋅𝐮=0 in ⁢Ω,𝐮⋅𝝉Ω=g,ρ=h on ⁢∂⁡Ω,

where 𝛕Ω=(n2Ω,-n1Ω) is the tangential vector on ∂⁡Ω. Moreover, Ma⁢(u)∈Lp⁢(∂⁡Ω), Ma⁢(ρ)∈Lq⁢(∂⁡Ω), and there exist non-tangential limits of u and ρ at almost all points of ∂⁡Ω.

Proof.

For s<k+1, see [36, Theorem 5.2]. The proof for s=k+1 is the same but we use the new embedding result Ws+1,q⁢(Ω)↪Wt,p⁢(Ω) (see [12, Theorem 3.8]). ∎

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Received: 2018-06-06

Revised: 2019-05-30

Accepted: 2019-11-25

Published Online: 2020-01-17

Published in Print: 2020-03-01

Several non-standard problems for the stationary Stokes system

Abstract

A Appendix

A.1 Holomorphic functions

Lemma A.1.

Proof.

Lemma A.2.

Proof.

Lemma A.3.

Proof.

Lemma A.4.

Proof.

A.2 The Dirichlet problem for the Laplace equation

Theorem A.5.

Proof.

A.3 Besov spaces

Proposition A.6.

Proof.

A.4 Stokes system with prescribed pressure

Proposition A.7.

Proof.

References

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