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Advances in Nonlinear Analysis

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Maximal Lp -Lq regularity to the Stokes problem with Navier boundary conditions

Hind Al Baba
  • Corresponding author
  • Institute of Mathematics of the Czech Academy of Sciences, Z̆itná 25, 11567 Praha 1, Czech Republic; and Laboratoire de Mathématiques et de leurs applications Pau, UMR, CNRS 5142, Université de Pau et des pays de L’Adour, 64013 Pau cedex, France
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Published Online: 2017-08-15 | DOI: https://doi.org/10.1515/anona-2017-0012

Abstract

We prove in this paper some results on the complex and fractional powers of the Stokes operator with slip frictionless boundary conditions involving the stress tensor. This is fundamental and plays an important role in the associated parabolic problem and will be used to prove maximal Lp-Lq regularity results for the non-homogeneous Stokes problem.

Keywords: Non-homogeneous Stokes problem; slip boundary conditions; maximal regularity; complex and fractional powers of operators

MSC 2010: 35B65; 35D30; 35D35; 35K20; 35Q30; 76D05; 76D07; 76N10

1 Introduction

This paper studies maximal Lp-Lq regularity for the Stokes problem with Navier-slip boundary conditions

𝒖t-Δ𝒖+π=𝒇,div𝒖=0in Ω×(0,T),(1.1a)𝒖𝒏=0,[𝔻(𝒖)𝒏]𝝉=𝟎on Γ×(0,T),(1.1b)𝒖(0)=𝒖0in Ω,(1.1c)

where 𝔻(𝒖)=12(𝒖+𝒖T) is the stress tensor and Ω is a bounded domain of 3 of class C2,1. A unit normal vector to the boundary can be defined almost everywhere; it is denoted by 𝒏. Here 𝒖 and π denote respectively unknowns velocity field and the pressure of a fluid occupying the domain Ω, while 𝒖0 and 𝒇 represent respectively the given initial velocity and the external force.

In the opinion of engineers and physicists, systems of the form (1.1) play an important role in many real life situations, such as in aerodynamics, weather forecast, hemodynamics. Thus naturally the need arises to carry out a mathematical analysis of these systems which represent the underlying fluid dynamic phenomenology. The Navier boundary conditions (1.1b) have been used to simulate flows near rough walls as in [5, 17], perforated walls [10] and turbulent flows [12, 20]. We note that among the earliest works on the mathematical analysis of the Stokes and Navier–Stokes problems with the Navier-slip boundary conditions (1.1b) we can cite the work of Solonnikov and Ščadilov [30] who considered the stationary Stokes problem with the boundary conditions (1.1b) in bounded or unbounded domains of 3 and proved the existence and regularity of solutions to this problem.

The author together with Amrouche and Rejaiba [7] proved the analyticity of the Stokes semigroup with the boundary conditions (1.1b) in Lp-spaces, which guarantees the existence of complex and fractional powers of the Stokes operator with the corresponding boundary conditions. They also studied the homogeneous Stokes problem with Navier-slip boundary conditions (i.e. problem (1.1) with 𝒇=𝟎) and proved the existence of strong, weak and very weak solutions to this problem. In this paper we shall prove maximal Lp-Lq regularity for the non-homogeneous case (i.e. problem (1.1) with 𝒇𝟎). We shall also prove the existence of strong, weak and very weak solutions to this problem with maximal regularity. The key tool is the use of the complex and fractional powers of the Stokes operator with Navier-slip boundary conditions (1.1b). We note that the concept of very weak solution 𝒖𝑳p(Ω) to certain elliptic and parabolic problem with initial data of low regularity was introduced by Lions and Magenes [19] and is usually based on duality arguments for strong solutions. Therefore the boundary regularity required in this theory is the same as for strong solutions.

Concerning the maximal Lp regularity for the Stokes problem we can cite [29] by Solonnikov among the first works on this problem. He constructed a solution (𝒖,π) to the initial value Stokes problem with Dirichlet boundary conditions (𝒖=𝟎 on Γ×[0,T)). His proof was based on methods in the theory of potentials. However, when Ω is not bounded, the result in [29] was not global in time. Later on, Giga and Sohr [16] strengthened Solonnikov’s result in two directions. First, their result was global in time. Second, the integral norms that they used may have different exponents p,q in space and time. To derive such global maximal Lp-Lq regularity for the Stokes system with Dirichlet boundary conditions, Giga and Sohr used the boundedness of the pure imaginary power of the Stokes operator. More precisely, they used and extended an abstract perturbation result developed by Dore and Venni [14].

Saal considered in [24] the Stokes problem in spatial regions with moving boundary and proved maximal Lp-Lq regularity to this problem. The proof relies on a reduction of the problem to an equivalent non-autonomous system on a cylindrical space-time domain and the result includes bounded and unbounded regions. In [25], Saal proved maximal Lp-Lq regularity for the Stokes problem with homogeneous Robin boundary conditions in the half space +3. To this end, he proved that the associated Stokes operator is sectorial and admits a bounded -calculus on 𝑳σ,τp(Ω). Nevertheless, as shown by Shimada in [27] the same approach can not be applied to the Stokes problem with non-homogeneous Robin boundary condition. For this reason Shimada derived the maximal Lp-Lq regularity for the Stokes problem with non-homogeneous Robin boundary conditions by applying Weis’ operator-valued Fourier multiplier theorem to the concrete representation formulas of solutions to the Stokes problem as well as a localization procedure.

Geissert et al. [15] considered the Lp realization of the Hodge Laplacian operator defined by

ΔM:𝐃(ΔM)𝑳p(Ω)𝑳p(Ω),

where

𝐃(ΔM)={𝒖𝑾2,p(Ω):𝒖𝒏=0,𝐜𝐮𝐫𝐥𝒖×𝒏=𝟎 on Γ},ΔM𝒖=Δ𝒖in Ωfor all 𝒖𝐃(ΔM),

in a domain Ω3 with a suitably smooth boundary. Geissert et al. [15] proved that for all λ>0, the Lp realization of the operator λI-ΔM admits a bounded -calculus. They also showed that in the case where Ω is simply connected their result is true for λ=0. Since the class of operators having a bounded -calculus in their corresponding Banach spaces enjoys the property of bounded pure imaginary powers, they deduced the maximal Lp-Lq regularity to magneto-hydrodynamic equation with perfectly conducting wall condition.

Following the results in [15], the author together with Amrouche and Escobedo proved in [2] maximal Lp-Lq regularity to the solution of the inhomogeneous Stokes problem with slip frictionless boundary conditions involving the tangential component of the velocity vortex instead of the stress tensor in a domain Ω not necessarily simply connected. More precisely, the authors considered in [2] the Stokes problem (1.1a) with the following boundary conditions:

𝒖𝒏=0,𝐜𝐮𝐫𝐥𝒖×𝒏=𝟎on Γ×(0,T),(1.2)

which we call the Navier-type boundary conditions. It is known that the Navier-slip boundary (1.1b) differs from (1.2) only by a lower-order term, and this term is equal to zero in the case of the flat boundary (in this context we can cite the paper of Beirão da Veiga and Crispo [11]). This means that the Stokes operator with Navier-slip boundary conditions (1.1b) can be considered as perturbation of the Stokes operator with Navier-type boundary conditions (1.2) and the maximal regularity to problem (1.1) can be deduced from the results in [2]. However, the perturbation on the boundary can not be treated as a normal perturbation of operator because it changes the domains of definition of the operator. We follow [23] to overcome this difficulty. The idea is to consider the Stokes operator in weak sense and apply a perturbation argument to this weak operator. Using then “Amann interpolation-extrapolation” argument we can deduce the result for the Stokes operator with Navier-slip boundary conditions in Lp spaces. For more information on the “Amann interpolation-extrapolation scales” we refer to [4]. We note that Wilke and Prüss [23] studied the critical spaces for the Navier–Stokes equations with Navier boundary conditions. Their work is based on the theory of weighted Lp-maximal regularity for abstract semilinear evolution equations and the Amann interpolation-extrapolation scales.

The organization of the paper is as follows. In Section 2 we recall some properties of the Stokes operator with the boundary conditions (1.1b) that are crucial in our work. In Section 3 we study the complex and fractional powers of the Stokes operator with Navier-slip boundary conditions. We prove the boundedness of the complex powers of the Stokes operator with the above mentioned boundary conditions, we also characterize the domains of its fractional powers. The result for the pure imaginary powers can be deduced from [2] using a perturbation argument and the Amann interpolation-extrapolation argument. For the convenience of the reader we give an outline of the proof in Appendix A. The results of Section 3 will be used in Section 4 to study problem (1.1) and derive a maximal Lp-Lq regularity result to the inhomogeneous Stokes problem (1.1). In Appendix A we give a brief review on the Amann interpolation-extrapolation theory. We also give a proof of the boundedness of the pure imaginary powers of the Stokes operator with the Navier boundary conditions.

2 Preliminaries

In this section we review some properties of the Stokes operator with Navier-slip boundary conditions (1.1b). Throughout this paper, if we do not state otherwise, p will be a real number such that 1<p<.

2.1 Stokes operator

In this subsection we introduce the different Stokes operators with different regularities in order to obtain strong, weak and very weak solutions to the Stokes problem (1.1a).

First we consider the Stokes operator with the boundary conditions (1.1b) on the space 𝑳σ,τp(Ω) given by

𝑳σ,τp(Ω)={𝒇𝑳p(Ω):div𝒇=0 in Ω,𝒇𝒏=0 on Γ}(2.1)

and we denote it by Ap. The trace value in (2.1) is justified (see below). Thanks to [7, Section 3] we know that the operator Ap is a closed linear densely defined operator on 𝑳σ,τp(Ω) defined as follows:

𝐃(Ap)={𝒖𝑾2,p(Ω):div𝒖=0 in Ω,𝒖𝒏=0,[𝔻(𝒖)𝒏]𝝉=𝟎 on Γ},(2.2)Ap𝒖=-PΔ𝒖in Ωfor all 𝒖𝐃(Ap).(2.3)

The operator P in (2.3) is the Helmholtz projection P:𝑳p(Ω)𝑳σ,τp(Ω) defined by

P𝒇=𝒇-gradπfor all 𝒇𝑳p(Ω),(2.4)

where πW1,p(Ω)/ is the unique solution of the following weak Neumann problem (cf. [28]):

div(gradπ-𝒇)=0in Ω,(gradπ-𝒇)𝒏=0on Γ.(2.5)

An easy computation shows that

Ap𝒖=-Δ𝒖+gradπin Ωfor all 𝒖𝐃(Ap),

where πW1,p(Ω)/ is the unique solution up to an additive constant of the problem

div(gradπ-Δ𝒖)=0in Ω,(gradπ-Δ𝒖)𝒏=0on Γ.

Consider the space

𝑽σ,τp(Ω)={𝒖𝑾1,p(Ω):div𝒖=0 in Ω,𝒖𝒏=0 on Γ}.(2.6)

Observe that the Stokes operator Ap can be defined by the following weak formulation: for all 𝒖𝑽σ,τp(Ω) and all 𝒗𝑽σ,τp(Ω) we have

Ap𝒖,𝒗(𝑽σ,τp(Ω))×𝑽σ,τp(Ω)=Ω𝔻(𝒖):𝔻(𝒗¯)dx.

We note (cf. [7, 32]) that in the general case the Stokes operator Ap has a non-trivial kernel, we denote this kernel by 𝓣p(Ω). When the domain Ω is obtained by rotation around a vector 𝒃3, then

𝓣p(Ω)=Span{𝜷},𝜷(𝒙)=𝒃×𝒙for 𝒙Ω.

Otherwise

𝓣p(Ω)={𝟎}

(see [32] for more details). The kernel 𝓣p(Ω) can be characterized as follows (see [7]):

𝓣p(Ω)={𝒗𝑾1,p(Ω):𝔻(𝒗)=𝟎 in Ω,div𝒗=0 in Ω and 𝒗𝒏=0 on Γ}.

We also note (see [7, Theorem 3.9]) that the operator -Ap is sectorial and generates a bounded analytic semigroup on 𝑳σ,τp(Ω) for all 1<p<. We denote by e-tAp the analytic semigroup associated to the operator Ap in 𝑳σ,τp(Ω).

Next we consider the space

𝑯p(div,Ω)={𝒗𝑳p(Ω):div𝒗𝑳p(Ω)},

equipped with the graph norm. For every 1<p<, the space 𝓓(Ω¯) is dense in 𝑯p(div,Ω) (cf. [9, Section 2] and [6, Proposition 2.3]). In addition, for any function 𝒗 in 𝑯p(div,Ω) the normal trace 𝒗𝒏|Γ exists and belongs to W-1/p,p(Γ) and the closure of 𝓓(Ω) in 𝑯p(div,Ω) is equal to

𝑯0p(div,Ω)={𝒗𝑯p(div,Ω):𝒗𝒏=0 on Γ}.

We have denoted by 𝓓(Ω) the set of infinitely differentiable functions with compact support in Ω and by 𝓓(Ω¯) the restriction to Ω of infinitely differentiable functions with compact support in 3. The dual space [𝑯0p(div,Ω)] of 𝑯0p(div,Ω) can be characterized as follows (cf. [26, Proposition 1.0.4]): A distribution 𝒇 belongs to [𝑯0p(div,Ω)] if and only if there exist 𝝍𝑳p(Ω) and χLp(Ω) such that 𝒇=𝝍+gradχ and

𝒇[𝑯0p(div,Ω)]=inf𝒇=𝝍+gradχmax(𝝍𝑳p(Ω),χ𝑳p(Ω)).

We consider the following space:

[𝑯0p(div,Ω)]σ,τ={𝒇[𝑯0p(div,Ω)]:div𝒇=0 in Ω and 𝒇𝒏=0 on Γ}.

We note (cf. [2]) that for a function 𝒇 in the dual space [𝑯0p(div,Ω)] such that div𝒇Lp(Ω), the normal trace value 𝒇𝒏|Γ exists and belongs to the space 𝑾-1-1/p,p(Γ).

The Stokes operator Ap can be extended to the space [𝑯0p(div,Ω)]σ,τ (cf. [7, Section 3.2]). This extension is a closed linear densely defined operator

Bp:𝐃(Bp)[𝑯0p(div,Ω)]σ,τ[𝑯0p(div,Ω)]σ,τ,𝐃(Bp)={𝒖𝑾1,p(Ω):div𝒖=0 in Ω,𝒖𝒏=0,[𝔻(𝒖)𝒏]𝝉=𝟎 on Γ},(2.7)Bp𝒖=-Δ𝒖+gradπin Ωfor all 𝒖𝐃(Bp),

where πLp(Ω)/ is the unique solution up to an additive constant of the problem

div(gradπ-Δ𝒖)=0in Ω,(gradπ-Δ𝒖)𝒏=0on Γ.

The operator -Bp generates a bounded analytic semigroup on [𝑯0p(div,Ω)]σ,τ for all 1<p< (see [7, Theorem 3.10]). We note that the trace value [𝔻(𝒖)𝒏]𝝉 for a function 𝒖 in (2.7) is justified by the fact that for a function 𝒖𝑾1,p(Ω) such that Δ𝒖[𝑯0p(div,Ω)], the trace value [𝔻(𝒖)𝒏]𝝉 exists and belongs to 𝑾-1/p,p(Γ) (see [7, Lemma 2.4]).

Consider also the following space:

𝑻p(Ω)={𝒗𝑯0p(div,Ω):div𝒗W01,p(Ω)}.

Thanks to [8, Lemmas 4.11, 4.12] we know that 𝓓(Ω) is dense in 𝑻p(Ω) and a distribution 𝒇(𝑻p(Ω)) if and only if there exist 𝝍𝑳p(Ω) and f0W-1,p(Ω) such that 𝒇=𝝍+f0, with

𝒇(𝑻p(Ω))=inf𝒇=𝝍+f0max(𝝍𝑳p(Ω),f0W-1,p(Ω)).

We consider the subspace

[𝑻p(Ω)]σ,τ={𝒇(𝑻p(Ω)):div𝒇=0 in Ω and 𝒇𝒏=0 on Γ}.

We recall from [2] that for a function 𝒇 in the dual space [𝑻p(Ω)] such that div𝒇Lp(Ω) the normal trace 𝒇𝒏|Γ exists and belongs to 𝑾-2-1/p,p(Γ).

The Stokes operator with Navier-slip boundary condition can also be extended to the space [𝑻p(Ω)]σ,τ (see [7, Section 3.3]). This extension is a densely defined closed linear operator

Cp:𝐃(Cp)[𝑻p(Ω)]σ,τ[𝑻p(Ω)]σ,τ,

where

𝐃(Cp)={𝒖𝑳p(Ω):div𝒖=0 in Ω,𝒖𝒏=0,[𝔻(𝒖)𝒏]𝝉=𝟎 on Γ}(2.8)

and Cp𝒖=-Δ𝒖+gradπ in Ω for all 𝒖𝐃(Cp), with πW-1,p(Ω)/ the unique solution up to an additive constant of the problem

div(gradπ-Δ𝒖)=0in Ω,(gradπ-Δ𝒖)𝒏=0on Γ.

The operator -Cp generates a bounded analytic semigroup on [𝑻p(Ω)]σ,τ for all 1<p< (see [7, Theorem 3.12]). We recall from [7, Lemma 5.4] that for a function 𝒖𝑳p(Ω) such that Δ𝒖(𝑻p(Ω)), the trace value [𝔻(𝒖)𝒏]𝝉 exists and belongs to 𝑾-1-1/p,p(Γ). This give a meaning to the trace [𝔻(𝒖)𝒏]𝝉 of a function 𝒖 in (2.8).

In the sequel we need a relation between the boundary conditions (1.1b) and (1.2). To this end we introduce some notation to describe a boundary. Let us consider any point P on Γ and choose an open neighborhood W of P in Γ small enough to allow the existence of two families of 𝒞2 curves on W with these properties: a curve of each family passes through every point of W and the unit tangent vectors to these curves form an orthonormal system (which we assume to have the direct orientation) at every point of W. The lengths s1,s2 along each family of curves, respectively, are a possible system of coordinates in W. We denote by 𝝉1,𝝉2 the unit tangent vectors to the boundary. With this notation, we have

𝒗=k=12vk𝝉k+(𝒗𝒏)𝒏,

where 𝝉kT=(τk1,τk2,τk3) and vk=𝒗𝝉k. As a result for any 𝒗𝓓(Ω¯) the following formulas hold (see [7]):

[D(𝒗)𝒏]𝝉=𝝉(𝒗𝒏)+(𝒗𝒏)𝝉-𝚲𝒗on Γ

and

𝐜𝐮𝐫𝐥𝒗×𝒏=𝝉(𝒗𝒏)-(𝒗𝒏)𝝉-𝚲𝒗on Γ,

where

𝚲𝒘=k=12(𝒘𝝉𝒏sk)𝝉k.

In the particular case 𝒗𝒏=0 on Γ, the following equality holds:

[2𝐃(𝒗)𝒏]𝝉=-𝐜𝐮𝐫𝐥𝒗×𝒏-2𝚲𝒗in 𝑾1/p,p(Γ).(2.9)

It can be seen from (2.9) that the Navier-slip boundary (1.1b) differs from (1.2) only by the term -2𝚲𝒗 which is a lower-order term. The following corollary shows that relation (2.9) can be obtained in weak sense (see [7, Corollary 2.5] for the proof).

Corollary 2.1.

For any vector 𝐯𝐖1,p(Ω) such that 𝐯[𝐇0p(div,Ω)] and 𝐯𝐧=0 on Γ, we have

[2𝐃(𝒗)𝒏]𝝉=-𝒄𝒖𝒓𝒍𝒗×𝒏-2𝚲𝒗in 𝑾-1/p,p(Γ).

3 Fractional powers of the Stokes operator

This section is devoted to the study of the complex and the fractional powers of the Stokes operators Ap on 𝑳σ,τp(Ω). Since the Stokes operator Ap in 𝑳σ,τp(Ω) generates a bounded analytic semigroup, it is in particular a non-negative operator. It follows from the results in [18, 31] that its complex and fractional powers Apα, α, are well, densely defined and closed linear operators on 𝑳σ,τp(Ω) with domain 𝐃(Apα). Furthermore,

𝓓σ(Ω)𝐃(Apα)𝑳σ,τp(Ω)for all α.

We denote by the set of complex number, ={0}, and by 𝓓σ(Ω) the set of divergence free infinitely differentiable functions with compact support in Ω

Nevertheless, as described above, since the Stokes operator Ap is not invertible with bounded inverse, its complex powers can not be expressed through an integral formula and it is not easy in general to compute the calculus inequality involving these powers. To avoid this difficulty we prove the desired results for the operator (I+Ap). We start by the following proposition.

Proposition 3.1.

There exists an angle 0<θ0<π2 such that the resolvent set of the operator -(I+Ap) contains the sector

Σθ0={λ:|argλ|π-θ0}.

Moreover, the following estimate holds:

(λI+I+Ap)-1(𝑳σ,τp(Ω))κ(Ω,p)|λ|for all λΣθ0,λ0,(3.1)

with a constant κ independent of λ.

Proof.

Since the operator -Ap generates a bounded analytic semigroup on 𝑳σ,τp(Ω), the operator I+Ap is an isomorphism from 𝐃(Ap)𝑳σ,τp(Ω) in 𝑳σ,τp(Ω). We recall that 𝐃(Ap) is given by (2.2). Let λ such that Reλ0. It is clear that the operator λI+I+Ap is an isomorphism from 𝐃(Ap) to 𝑳σ,τp(Ω). Using [7, Theorem 3.8], one has, since Reλ0,

(λI+I+Ap)-1(𝑳σ,τp(Ω))κ(Ω,p)|λ+1|κ(Ω,p)|λ|,(3.2)

where the constant κ(Ω,p) is independent of λ. This means that the resolvent set of the operator -(I+Ap) contains the set {λ:Reλ0} where the estimate (3.2) is satisfied. Using the result of [33, Chapter VIII, Theorem 1], we deduce that there exists an angle 0<θ0<π2, such that the resolvent set of -(I+Ap) contains the sector Σθ0. In addition for every λΣθ0 such that λ0 estimate (3.1) holds. ∎

Corollary 3.2.

Let θ0 be as in Proposition 3.1.

  • (i)

    The estimate

    (λI+I+Ap)-1(𝑳σ,τp(Ω))C(Ω,p)(3.3)

    holds for all λΣθ0 with a constant C independent of λ.

  • (ii)

    Let 0<α<1 be fixed and let λΣθ0 such that λ0 and |λ|12κ(Ω,p) . Then one has

    (λI+I+Ap)-1(𝑳σ,τp(Ω))2ακα(Ω,p)|λ|α-1,(3.4)

    with a constant κ independent of λ.

Proof.

Let 𝒇𝑳σ,τp(Ω) and 𝒖𝐃(Ap) such that (λI+I+Ap)-1𝒇=𝒖. Observe that 𝒖 satisfies

{𝒖-Δ𝒖+π=𝒇-λ𝒖,div𝒖=0in Ω,𝒖𝒏=0,[𝔻(𝒖)𝒏]𝝉=𝟎on Γ.

Using [7, Theorem 3.8], we have

𝒖𝑳p(Ω)κ(Ω,p)𝒇-λ𝒖𝑳p(Ω)κ(Ω,p)[𝒇𝑳p(Ω)+|λ|𝒖𝑳p(Ω)].

Thus using (3.1), we get (3.3).

Next let 0<α<1 be fixed. Then for all λΣθ0 such that λ0 and |λ|12κ(Ω,p) we have

𝒖𝑳p(Ω)2κ(Ω,p)𝒇𝑳p(Ω)2ακα(Ω,p)|λ|α-1𝒇𝑳p(Ω).

Observe that estimate (3.4) holds for all 0<α<1. ∎

Remark 3.3.

One may say that it is superfluous to prove an estimate of type (3.4) for the operator (I+Ap) since 0ρ(I+Ap). Estimate (3.4) is maybe not optimal but it may be used in the sequel in the computations of the complex powers of the operator (I+Ap).

Next we state and prove our results on the complex and pure imaginary powers of the operator I+Ap. We start by the following proposition.

Proposition 3.4.

Let θ0 be as in Proposition 3.1. For all zC with -1<Rez<0 we have

(I+Ap)z(𝑳σ,τp(Ω))C(Rez,Ω,p)e|Imz|θ0,(3.5)

with a constant C(Rez,Ω,p) depending on Rez,Ω,p.

Proof.

Let z such that -1<Rez<0. Thanks to Proposition 3.1 we know that the operator I+Ap is a non-negative bounded operator with bounded inverse. As a result its complex powers can be expressed through the following Dunford integral formula (cf. [18]):

(I+Ap)z=12πiΓθ0(-λ)z(λI+I+Ap)-1dλ,(3.6)

where

Γθ0={ρei(π-θ0):0ρ}{-ρei(θ0-π):-ρ0}.

This means that

(I+Ap)z=12πi[0+(-ρei(π-θ0))z(ρei(π-θ0)I+I+Ap)-1ei(π-θ0)dρ-0+(-ρei(θ0-π))z(ρei(θ0-π)I+I+Ap)-1ei(θ0-π)dρ].

In addition, we know that (-λ)z=ez(ln|λ|+iArg(-λ)), where Arg(-λ) is the principal argument of -λ. An easy computation shows that

|(-λ)|zρReze|Imz|θ0.

As a result we have

(I+Ap)z(𝑳σ,τp(Ω))e|Imz|θ02π[I1+I2],(3.7)

with

I1=0+ρRez(ρei(π-θ0)I+I+Ap)-1(𝑳σ,τp(Ω))dρ,I2=0+ρRez(ρei(θ0-π)I+I+Ap)-1(𝑳σ,τp(Ω))dρ.

Next, we write I1 in the form

I1=01ρRez(ρei(π-θ0)I+I+Ap)-1(𝑳σ,τp(Ω))dρ+1+ρRez(ρei(π-θ0)I+I+Ap)-1(𝑳σ,τp(Ω))dρ,

In other words

Γθ0=[Γθ0{λ:|λ|1}][Γθ0{λ:|λ|>1}].

As a consequence, thanks to Proposition 3.1 and Corollary 3.2 we have

I1C(Ω,p)01dρρ-Rez+κ(Ω,p)1+dρρ1-Rez.

Thanks to our assumption on z we can verify that the improper integrals I1 and I2 are convergent and satisfy

I1<C(Rez,Ω,p),I2<C(Rez,Ω,p),

with a constant C(Rez,Ω,p) depending on Rez,Ω,p.

Finally, substituting in (3.7), we have for all z with -1<Rez<0,

(I+Ap)z(𝑳σ,τp(Ω))C(Ω,p)[11+Rez+1Rez]e|Imz|θ0.(3.8)

This completes the proof of the proposition. ∎

Remark 3.5.

(i) We recall from [18, Propositions 4.7, 4.10] that for all 𝒇𝐃(Ap) the operator (I+Ap)z𝒇 is analytic in z for -1<Rez<1.

(ii) Observe that if we replace Γθ0 in (3.6) by

Γ=Γθ0{λ:|λ|c}

with some constant c>0, we obtain for all z with -1<Rez<0,

(I+Ap)z(𝑳σ,τp(Ω))C(Ω,p)[c1+Rez1+Rez]e|Imz|θ0.

Let 𝒇𝐃(Ap), taking the limit as Rez tends to 0, we obtain that for all s,

(I+Ap)is𝒇𝑳p(Ω)=limRez0(I+Ap)Rez+is𝒇𝑳p(Ω)C(Ω,p)e|s|θ0𝒇𝑳p(Ω).(3.9)

Using then the density of 𝐃(Ap) in 𝑳σ,τp(Ω), we obtain estimate (3.9) for all 𝒇𝑳σ,τp(Ω).

Corollary 3.6.

Let sR and 0<α<1. The operator (I+Ap)is is bounded from D((I+Ap)α) to 𝐋σ,τp(Ω). Furthermore, there exists an angle 0<θ0<π2 and a constant C>0 such that

(I+Ap)is𝒖𝑳p(Ω)Ce|s|θ0𝒖𝐃((I+Ap)α).(3.10)

Proof.

Let 0<α<1, 𝒇𝑳σ,τp(Ω) and let 𝒖𝐃((I+Ap)α) such that (I+Ap)α𝒖=𝒇 in Ω. Using Proposition 3.4, we deduce that there exists an angle 0<θ0<π2 and a constant C>0 such that

(I+Ap)-α+is𝒇𝑳p(Ω)Ce|s|θ0𝒇𝑳p(Ω).

Next observe that

(I+Ap)is𝒖𝑳p(Ω)=(I+Ap)-α+is(I+Ap)α𝒖𝑳p(Ω)=(I+Ap)-α+is𝒇𝑳p(Ω)Ce|s|θ0𝒇𝑳p(Ω)Ce|s|θ0(I+Ap)α𝒖𝑳p(Ω).

Therefore one has estimate (3.10). This means that for all s, the operator (I+Ap)is is bounded from 𝐃((I+Ap)α) to 𝑳σ,τp(Ω) and satisfies estimate (3.10). We recall that the operator I+Ap has a bounded inverse and thus for all α with Reα>0, the operator (I+Ap)α is an isomorphism from 𝐃(Apα) to 𝑳σ,τp(Ω) (cf. [31, Theorem 1.15.2, part (e)]). Thus the above analysis is true. ∎

The estimates obtained in Proposition 3.4, Remark 3.5 and Corollary 3.6 are not sufficient to extend Theorem 3.4 to the case where Rez=0 in 𝑳σ,τp(Ω), since the second term on the right-hand side of (3.8) blows up as Rez tends to zero. Nevertheless, this attempt looks interesting by itself and we want to present it to the reader.

The following theorem extends estimate (3.5) to the case where Rez=0. We can see from Corollary 2.1 that the Stokes operator with Navier-slip boundary conditions (1.1b) can be considered as a lower-order perturbation of the Stokes operator with Navier-type boundary conditions (1.2). Thus we can deduce the boundedness of the pure imaginary power of our operator using the result in [2]. However, the perturbation on the boundary can not be treated directly as a normal perturbation of operator because it changes the domains of definition of the operator. The proof is done using a perturbation in weak sense as well as the Amann interpolation-extrapolation theory developed in [4] and it will be done in Appendix A.

Theorem 3.7.

There exist an angle θ0(0,π2) and a constant M>0 such that for all sR and all λ>0,

(λI+Ap)is(𝑳σ,τp(Ω))Me|s|θ0,(3.11)

where M is independent of λ.

The following theorem extends Theorem 3.7 to the operators (λI+Bp) and (λI+Cp), λ>0, on [𝑯0p(div,Ω)]σ,τ and [𝑻p(Ω)]σ,τ respectively. This result will be used in Section 4 in order to obtain weak and very weak solutions to problem (1.1).

Theorem 3.8.

There exist 0<θ0<π2 and a constant C>0 such that for all sR and all λ>0,

(λI+Bp)is([𝑯0p(div,Ω)]σ,τ)Ce|s|θ0,(3.12)(λI+Cp)is([𝑻p(Ω)]σ,τ)Ce|s|θ0,(3.13)

where the constant C in (3.12) and (3.13) is independent of λ.

Proof.

It suffices to prove estimate (3.12), estimate (3.13) follows in the same way. Using Theorem 3.7, one has for all λ>0 and all 𝒇𝑳σ,τp(Ω),

(λI+Bp)is𝒇[𝑯0p(div,Ω)]=(λI+Ap)is𝒇[𝑯0p(div,Ω)]Ce|s|θ0𝒇𝑳p(Ω).

Using that 𝒇[𝑯0p(div,Ω)]=𝒇𝑳σ,τp(Ω), we deduce that

(λI+Bp)is𝒇[𝑯0p(div,Ω)]Ce|s|θ0𝒇[𝑯0p(div,Ω)].

Next, using the density of 𝑳σ,τp(Ω) in [𝑯0p(div,Ω)]σ,τ (see [2, Proposition 3.9]), we can extend (λI+Bp)is to a bounded linear operator on [𝑯0p(div,Ω)]σ,τ and we deduce estimate (3.12). ∎

In the case where the domain Ω is not obtained by rotation around a vector 𝒃3, the Stokes operator with Navier-slip boundary conditions is invertible with bounded inverse. In this case we can pass to the limit in (3.11), (3.12) and (3.13) as λ tends to zero (cf. [16, Lemma A2]). As a result we deduce the following theorem.

Theorem 3.9.

Suppose that the domain Ω is not obtained by rotation around a vector 𝐛R3. There exist 0<θ0<π2 and a constant C>0 such that for all sR,

(Ap)is(𝑳σ,τp(Ω))Ce|s|θ0,(Bp)is([𝑯0p(div,Ω)]σ,τ)Ce|s|θ0,(Cp)is([𝑻p(Ω)]σ,τ)Ce|s|θ0.

Next we study the domains of fractional powers of the operator Ap on 𝑳σ,τp(Ω). Since the Stokes operator with the boundary conditions (1.1b) does not have bounded inverse, attention should be paid in the calculus of the domains 𝐃(Apα) and their norms. It follows from [18] that for Reα>0, the domain 𝐃(νI+Apα) does not depend on ν0 and coincides with 𝐃(μI+Apα) for μ0, that is,

𝐃(Apα)=𝐃(μI+Apα)=𝐃(νI+Apα)for all μ,ν>0.

We also know from [31, Theorem 1.15.3] that the boundedness of the pure imaginary powers of the operator (I+Ap) allows us to determine the domain of definition of 𝐃(I+Ap)α), and then of 𝐃(Apα) for any complex number α satisfying Reα>0 using complex interpolation theory. In addition for all α>0, the map 𝒗(I+Ap)α𝒗𝑳p(Ω) is a norm on 𝐃(Apα). This is due to the fact (cf. [31, Theorem 1.15.2, part (e)]) that the operator I+Ap has a bounded inverse, and thus for all α with Reα>0, the operator (I+Ap)α is an isomorphism from 𝐃(Apα) to 𝑳σ,τp(Ω).

The following theorem characterizes the domain of Ap1/2.

Theorem 3.10.

For all 1<p<, D(Ap1/2)=𝐕σ,τp(Ω) (given by (2.6)) with equivalent norms.

Proof.

Since the pure imaginary powers of the operator (I+Ap) are bounded and satisfy estimates (3.11), thanks to [31, Theorem 1.15.3] we have

𝐃(Ap1/2)=𝐃((I+Ap)1/2)=[𝐃(I+Ap);𝑳σ,τp(Ω)]1/2=[𝐃(Ap);𝑳σ,τp(Ω)]1/2.

Consider a function 𝒖𝐃(Ap) (see (2.2) for the definition of 𝐃(Ap)) and set 𝒛=𝔻(𝒖) and 𝑼=(𝒖,𝒛). It is clear that if 𝒖𝐃(Ap), then 𝒛𝑾1,p(Ω) and 𝑼𝑳σ,τp(Ω)×𝑾1,p(Ω). In addition, if 𝒖𝑳σ,τp(Ω), then 𝒛𝑾-1,p(Ω) and 𝑼𝑳σ,τp(Ω)×𝑾-1,p(Ω). Now, let 𝒖𝐃(Ap1/2), then

𝑼𝑳σ,τp(Ω)×[𝑾1,p(Ω);𝑾-1,p(Ω)]1/2=𝑳σ,τp(Ω)×𝑳p(Ω).

As a result, 𝒖𝑳p(Ω), 𝒛=𝔻(𝒖)𝑳p(Ω), div𝒖=0 in Ω and 𝒖𝒏=0 on Γ. Thanks to [7], we know that for all 𝒖𝑽σ,τp(Ω), the norm 𝒖𝑾1,p(Ω) is equivalent to 𝒖𝑳p(Ω)+𝔻(𝒖)𝑳p(Ω). As a result 𝒖𝑽σ,τp(Ω) and

𝐃(Ap1/2)𝑽σ,τp(Ω).

Next we prove the second inclusion. Since I+Ap has a bounded inverse, the operator (I+Ap)1/2 is an isomorphism from 𝐃((I+Ap)1/2) to 𝑳σ,τp(Ω) for all 1<p<. This means that for all 𝑭𝑳σ,τp(Ω) there exists a unique 𝒗𝐃((I+Ap)1/2) solution of

(I+Ap)1/2𝒗=𝑭.(3.14)

Let 𝒖𝐃(Ap) and observe that

(I+Ap)1/2𝒖𝑳p(Ω)=sup𝑭𝑳σ,τp(Ω),𝑭𝟎|(I+Ap)1/2𝒖,𝑭𝑳σ,τp(Ω)×𝑳σ,τp(Ω)|𝑭𝑳p(Ω)=sup𝒗𝐃(Ap1/2),𝒗𝟎|(I+Ap)𝒖,𝒗𝑳σ,τp(Ω)×𝑳σ,τp(Ω)|(I+Ap)1/2𝒗𝑳p(Ω)=sup𝒗𝐃(Ap1/2),𝒗𝟎|Ω𝒖𝒗¯dx+Ω𝔻(𝒖):𝔻(𝒗¯)dx|(I+Ap)1/2𝒗𝑳p(Ω)C(Ω,p)𝒖𝑾1,p(Ω).(3.15)

We recall that 𝒗 is the unique solution of problem (3.14) and that the adjoint operator ((I+Ap)1/2) of (I+Ap)1/2 is equal to the operator (I+Ap)1/2. We also recall that the dual of 𝑳σ,τp(Ω) is equal to 𝑳σ,τp(Ω). Using the density of 𝐃(Ap) in 𝑽σ,τp(Ω), we obtain estimate (3.15) for all 𝒖𝑽σ,τp(Ω) and then

𝑽σ,τp(Ω)𝐃(Ap1/2).

Remark 3.11.

In the case where the domain Ω is not obtained by rotation around a vector 𝒃3, the Stokes operator Ap is invertible with bounded inverse and the following equivalence holds:

Ap1/2𝒖𝑳p(Ω)𝔻(𝒖)𝑳p(Ω)for all 𝒖𝐃(Ap1/2).

The following proposition gives us an embeddings of Sobolev type for the domains of fractional powers of the Stokes operator Ap.

Proposition 3.12.

For all 1<p< and all αR such that 0<α<32p the following Sobolev embedding holds:

𝐃(Apα)𝑳q(Ω),1q=1p-2α3.(3.16)

Moreover, for all 𝐮D(Apα) the following estimate holds:

𝒖𝑳q(Ω)C(Ω,p)(I+Ap)α𝒖𝑳p(Ω).(3.17)

In the particular case where the domain Ω is not obtained by rotation around a vector 𝐛R3, the following estimate holds:

𝒖𝑳q(Ω)C(Ω,p)Apα𝒖𝑳p(Ω).(3.18)

Proof.

Consider first the case where 0<α<1 and recall that

𝐃(Apα)=𝐃((I+Ap)α)=[𝐃(I+Ap);𝑳σ,τp(Ω)]α=[𝐃(Ap);𝑳σ,τp(Ω)]α.

The embedding (3.16) is obtained using the classical Sobolev embedding as in [1, Theorem 7.57]. To extend (3.16) to any real α such that 0<α<32p, we proceed as in the proof of [2, Corollary 6.11]. This result is similar to the result of Borchers and Miyakawa [13] who proved the result for the Stokes operator with Dirichlet boundary conditions in exterior domains for 1<p<3.

Estimate (3.17) is a direct consequence of (3.16) since the domain 𝐃(Apα) is equipped with the graph norm of the operator (I+Ap)α.

In the particular case where the domain Ω is not obtained by rotation around a vector 𝒃3, the operator Apα is an isomorphism from 𝐃(Apα) to 𝑳σ,τp(Ω). Thus one has estimate (3.18). ∎

4 Applications to the Stokes problem

In this section we shall apply the results of Sections 2 and 3 in order to prove maximal Lp-Lq for the inhomogeneous Stokes problem (1.1).

Consider first the two problems

{𝒖t+Ap𝒖=𝒇,div𝒖=0in Ω×(0,T),𝒖𝒏=0,[𝔻(𝒖)𝒏]𝝉=𝟎on Γ×(0,T),𝒖(0)=𝒖0in Ω(4.1)

and

{𝒖t-Δ𝒖+π=𝒇,div𝒖=0in Ω×(0,T),𝒖𝒏=0,[𝔻(𝒖)𝒏]𝝉=𝟎on Γ×(0,T),𝒖(0)=𝒖0in Ω,(4.2)

where 𝒖0𝑳σ,τp(Ω)), 𝒇C1(0,T;𝑳p(Ω) and 1<p,q<. Notice that a function

𝒖C(]0,+[,𝐃(Ap))C1(]0,+[,𝑳σ,τp(Ω))

solves (4.1) if and only if there exists a function πC(]0,[;W1,p(Ω)/) such that (𝒖,π) solves (4.2). Indeed, let 𝒖 be a solution to (4.1). Thus

Ap𝒖=-PΔ𝒖=𝒇-𝒖t,

where P is the Helmholtz projection defined by (2.4) and (2.5). Since (𝒖,𝒇-𝒖t)𝐃(Ap)×𝑳σ,τp(Ω), due to [7, Theorem 4.1] there exists πW1,p(Ω)/ such that

Ap𝒖=-Δ𝒖+π=-𝒇-𝒖t.

Moreover, we have the estimate

𝒖𝑾2,p(Ω)/𝓣p(Ω)+πW1,p(Ω)/C(Ω,p)(𝒇𝑳p(Ω)+𝒖t𝑳p(Ω)),

where 𝓣p(Ω) is the kernel of the Stokes operator with Navier-slip boundary condition described above. This means the mapping 𝒇-𝒖tπ is continuous from 𝑳σ,τp(Ω) to W1,p(Ω). As a result, πC(]0,[;W1,p(Ω)/) and (𝒖,π) solves (4.2). Conversely, let (𝒖,π) be a solution of (4.2). Applying the Helmholtz projection P to the first equation of problem (4.2), one gets directly that 𝒖 solves (4.1).

For the homogeneous problem (i.e. 𝒇=𝟎), the analyticity of the semigroup gives us a unique solution satisfying all the regularity desired. As stated in [7], when the initial data 𝒖0𝑳σ,τp(Ω) and when 𝒇=𝟎, problem (1.1) has a unique solution (𝒖,π) satisfying

𝒖C([0,+[,𝑳σ,τp(Ω))C(]0,+[,𝐃(Ap))C1(]0,+[,𝑳σ,τp(Ω)),𝒖Ck(]0,+[,𝐃(Ap))for all k,,πC(]0,[;W1,p(Ω)/).

Moreover, the following estimates hold:

𝒖(t)𝑳p(Ω)C(Ω,p)𝒖0𝑳p(Ω),(4.3)𝒖(t)t𝑳p(Ω)C(Ω,p)t𝒖0𝑳p(Ω),(4.4)𝔻(𝒖)𝑳p(Ω)C(Ω,p)t𝒖0𝑳p(Ω).(4.5)

Remark 4.1.

In the case where the domain Ω is not obtained by rotation around a vector b3, the Stokes semigroup decays exponentially and we can extend estimates (4.3)–(4.5) to the following Lp-Lq estimates. More precisely, for every p and q such that 1<pq<, for every 𝒖0𝑳σ,τp(Ω) and 𝒇=𝟎, there exists a constant δ>0 such that the unique solution 𝒖(t) to problem (1.1) belongs to 𝑳σ,τq(Ω) and satisfies

𝒖(t)𝑳q(Ω)Ce-δtt-3/2(1/p-1/q)𝒖0𝑳p(Ω),(4.6)𝔻(𝒖(t))𝑳q(Ω)Ce-δtt-1/2t-3/2(1/p-1/q)𝒖0𝑳p(Ω),(4.7)mtmApn𝒖(t)𝑳q(Ω)Ce-δtt-(m+n)t-3/2(1/p-1/q)𝒖0𝑳p(Ω)for all m,n.(4.8)

Estimates (4.6)–(4.8) are obtained using the embedding of Sobolev type (3.16), estimate (3.17) and the fact that for all α we have

Aα𝒖(t)𝑳p(Ω)=Aαe-tAp𝒖0𝑳p(Ω)Ce-δtt-α𝒖0𝑳p(Ω).

Consider now the non-homogeneous case, where 𝒖0=𝟎 and 𝒇Lq(0,T;𝑳σ,τp(Ω)), with 1<p,q< and 0<T. It is well known (cf. [21]) that for such 𝒇, problem (1.1) has a unique solution 𝒖C(0,T;𝑳σ,τp(Ω)). It is also known that for such 𝒇 the analyticity of the Stokes semigroup is not enough to obtain a unique solution (𝒖,π) satisfying the following maximal Lp-Lq regularity:

𝒖Lq(0,T;𝐃(Ap)),𝒖tLq(0,T;Lσ,τp(Ω)),πLq(0,T;W1,p(Ω)/).

In what follows we prove maximal Lp-Lq regularity of the solution to the Stokes problem (1.1) using Theorem 3.7 and [16, Theorem 2.1].

Theorem 4.2 (Strong solution to the Stokes problem).

Let 0<T, 1<p,q<, 𝐟𝐋q(0,T;𝐋σ,τp(Ω)) and 𝐮0=0. Problem (1.1) has a unique solution (𝐮,π) such that

𝒖Lq(0,T0;𝑾2,p(Ω)),T0T if T<,T0<T if T=,(4.9)πLq(0,T;W1,p(Ω)/),𝒖tLq(0,T;𝑳p(Ω)),0T𝒖t𝑳p(Ω)qdt+0TAp𝒖(t)𝑳p(Ω)qdt+0Tπ(t)W1,p(Ω)/qdtC(p,q,Ω)0T𝒇(t)𝑳p(Ω)qdt.(4.10)

Proof.

As stated above, problem (1.1) has a unique solution 𝒖C(0,T;𝑳σ,τp(Ω)). Let us prove that this solution satisfies the maximal Lp-Lq regularity (4.9). Indeed, let μ>0 and set 𝒖μ(t)=e-1/μt𝒖(t). The function 𝒖μ(t) is a solution to the following problem:

{𝒖μt+(1μI+Ap)𝒖μ(t)=e-1μt𝒇,div𝒖μ(t)=0in Ω×(0,T),𝒖μ(t)𝒏=0,[𝔻(𝒖μ(t))𝒏]τ=𝟎on Γ×(0,T),𝒖μ(0)=𝒖(0)=𝟎in Ω.(4.11)

Since the pure imaginary powers of the operator (1μI+Ap) are bounded in 𝑳σ,τp(Ω) (see Theorem 3.7) and since for all 1<p<, 𝑳σ,τp(Ω) is ζ-convex, we can apply the result of [16, Theorem 2.1] to the operator (1μI+Ap). Thus, the solution 𝒖μ(t) to problem (4.11) satisfies the following maximal Lp-Lq regularity:

𝒖μLq(0,T0;𝐃(Ap))W1,q(0,T;𝑳σ,τp(Ω)),

with T0T if T< and T0<T if T=. Furthermore, 𝒖μ(t) satisfies the following estimate:

0T𝒖μt𝑳p(Ω)qdt+0T(1μI+Ap)𝒖μ(t)𝑳p(Ω)qdtC(p,q,Ω)0Te-1μt𝒇(t)𝑳p(Ω)qdtC(p,q,Ω)0T𝒇(t)𝑳p(Ω)qdt,(4.12)

where the constant C(p,q,Ω) is independent of μ. In addition,

𝒖μ𝒖in Lq(0,T0;𝐃(Ap))W1,q(0,T;𝑳σ,τp(Ω))as μ.

Thus the solution 𝒖 to problem (1.1) satisfies

𝒖Lq(0,T0;𝐃(Ap))W1,q(0,T;𝑳σ,τp(Ω)),(4.13)

with T0T if T< and T0<T if T=.

Using now the fact that Ap𝒖=-Δ𝒖+π=𝒇-𝒖t, thanks to [7, Theorem 4.1] we have

𝒖(t)𝑾2,p(Ω)/𝓣p(Ω)+πW1,p(Ω)/C(𝒖t𝑳p(Ω)+𝒇𝑳p(Ω)).(4.14)

As a result we deduce that πLq(0,T;W1,p(Ω)/).

It remains to prove estimate (4.10). We recall first the following equivalence of norms:

𝒗𝑾2,p(Ω)𝒗𝐃(Ap)(1μI+Ap)𝒗𝑳p(Ω)for all 𝒗𝐃(Ap),

independently of μ>0. Then, substituting in (4.12), we have

0T𝒖μt𝑳p(Ω)qdt+0T𝒖μ(t)𝐃(Ap)qdtC(p,q,Ω)0T𝒇(t)𝑳p(Ω)qdt,(4.15)

where the constant C(p,q,Ω) is independent of μ. Using then the dominated convergence theorem and passing to the limit as μ tends to infinity in (4.15), we obtain

0T𝒖t𝑳p(Ω)qdt+0T𝒖(t)𝐃(Ap)qdtC(p,q,Ω)0T𝒇(t)𝑳p(Ω)qdt.

Finally, using (4.14) and the fact that 𝒖𝐃(Ap) is equivalent to 𝒖𝑳p(Ω)+Ap𝒖𝑳p(Ω), estimate (4.10) follows directly. We also note that the regularity (4.13) allows us to deduce that the operator t+Ap is an isomorphism from Lq(0,T0;𝐃(Ap))W1,q(0,T;𝑳σ,τp(Ω)) to Lq(0,T;𝑳σ,τp(Ω)). Thus estimate (4.10) follows directly. ∎

The boundedness of the pure imaginary powers of the operators λI+Bp and λI+Cp, with λ>0 on the spaces [𝑯0p(div,Ω)]σ,τ and [𝑻p(Ω)]σ,τ, respectively (see Theorem 3.8), allows us to obtain weak and very weak solutions to problem (1.1). Indeed, using [2, Proposition 2.16] we know that the spaces [𝑯0p(div,Ω)]σ,τ and [𝑻p(Ω)]σ,τ are ζ-convex Banach spaces. As a result, proceeding as in the proof of Theorem 4.2, we obtain the following two theorems.

Theorem 4.3 (Weak solution to the Stokes problem).

Let 1<p,q<, 0<T, 𝐮0=0 and let 𝐟Lq(0,T;[𝐇0p(div,Ω)]σ,τ). The evolutionary Stokes problem (1.1) has a unique solution (𝐮,π) satisfying

𝒖Lq(0,T0;𝑾1,p(Ω)),T0T if T<,T0<T if T=,πLq(0,T;Lp(Ω)/),𝒖tLq(0,T;[𝑯0p(divΩ)]σ,τ),0T𝒖t[𝑯0p(divΩ)]qdt+0TBp𝒖(t)[𝑯0p(divΩ)]qdt+0Tπ(t)Lp(Ω)/qdtC(p,q,Ω)0T𝒇(t)[𝑯0p(divΩ)]qdt.(4.16)

Proof.

Proceeding in the same way as in the proof of Theorem 4.2, using the boundedness of the pure imaginary powers of the operators 1μI+Bp, μ>0, on [𝑯0p(div,Ω)]σ,τ and the change of variable 𝒖μ(t)=e-1/μt𝒖(t), we obtain that problem (1.1) has a unique solution satisfying

𝒖Lq(0,T0;𝑾1,p(Ω))W1,q(0,T;[𝑯0p(divΩ)]σ,τ),

with T0T if T< and T0<T if T=. Next, using that Bp𝒖=-Δ𝒖+π=𝒇-𝒖t, thanks to [7, Theorems 3.7, 3.9] we have

𝒖(t)𝑾1,p(Ω)/𝓣p(Ω)+πLp(Ω)/C(𝒖t[𝑯0p(divΩ)]+𝒇[𝑯0p(divΩ)]).

As a result we deduce that πLq(0,T;Lp(Ω)/) and estimate (4.16) follows. ∎

Theorem 4.4 (Very weak solution to the Stokes problem).

Let 1<p,q<, 0<T, 𝐮0=0 and let 𝐟Lq(0,T;[𝐓p(Ω)]σ,τ). The evolutionary Stokes problem (1.1) has a unique solution (𝐮,π) satisfying

𝒖Lq(0,T0;𝑳p(Ω)),T0T if T<,T0<T if T=,πLq(0,T;W-1,p(Ω)/),𝒖tLq(0,T;[𝑻p(Ω)]σ,τ),0T𝒖t[𝑻p(Ω)]qdt+0TCp𝒖(t)[𝑻p(Ω)]qdt+0Tπ(t)W-1,p(Ω)/qdtC(p,q,Ω)0T𝒇(t)[𝑻p(Ω)]qdt.

Proof.

The proof is similar to the proof of Theorem 4.2. First we use the boundedness of the pure imaginary powers of 1μI+Cp, with μ>0 on the space [𝑻p(Ω)]σ,τ and the change of variable 𝒖μ(t)=e-1/μt𝒖(t). We obtain then a unique solution to problem (1.1) satisfying

𝒖Lq(0,T0;𝑳p(Ω))W1,q(0,T;[𝑻p(Ω)]σ,τ),

with T0T if T< and T0<T if T=. Next, using the fact Cp𝒖=-Δ𝒖+π=𝒇-𝒖t, thanks to [7, Theorem 5.5] we have

𝒖(t)𝑳p(Ω)/𝓣p(Ω)+πW-1,p(Ω)/C(𝒖t[𝑻p(Ω)]+𝒇[𝑻p(Ω)]).

As a result we deduce that πLq(0,T;W-1,p(Ω)/) and estimate (4.16) follows. ∎

A Appendix

A.1 Amann interpolation-extrapolation scales

In this subsection we give a brief review on the Amann interpolation-extrapolation technique that is crucial in our work and has many applications to the quasilinear parabolic evolution equations. We refer to [4, Chapter V] for a detailed description of this technique and its applications.

We will denote by (X1,X0) the set of all A(X1,X0) such that -A considered as a linear operator in X0 with domain X1 is the infinitesimal generator of a strongly continuous analytic semigroup (e-tA)t0 on X0.

Let us fix a negative generator 𝔸 of a strongly continuous analytic semigroup on an arbitrary Banach space X0:=(X0,0) and assume for simplicity that 𝔸 has a bounded inverse on X0. Thus 𝔸 is a closed densely defined operator in X0 having a non-empty resolvent set. If we denote by X1:=D(𝔸) the domain of 𝔸, then uu1:=𝔸u0 defines a norm on X1 which is equivalent to the graph norm. Thus X1:=(D(𝔸),1) is a Banach space such that

X1𝑑X0.

The couple (X0,X1) is called a densely injected Banach couple and 𝔸(X1,X0). Observe that the Banach space X0 is the completion of X1 in the norm u𝔸1-1u1=u0, with 𝔸1 the X1-realization of 𝔸. Notice that if 𝔸 is not invertible, it suffices to replace 𝔸 by λI+𝔸 with λ>0.

The above description allows to introduce a superspace X-1:=(X-1,-1) of X0 by choosing for X-1 a completion of X0 in the norm uu-1:=𝔸-1u0. Then (X-1,X0) is a densely injected Banach space as well and it is not difficult to show that 𝔸0:=𝔸 extends continuously to an operator 𝔸-1(X0,X-1).

Next, given θ(0,1), we choose an interpolation functor (,)θ of exponent θ with the property that X1 is dense in (X0,X1)θ whenever (X0,X1) is a densely injected Banach couple, an “admissible interpolation functor”. Then we put Xθ:=(X0,X1)θ and Xθ-1:=(X-1,X0)θ. This defines a scale of Banach spaces

X1𝑑Xα𝑑Xβ𝑑X-1,-1<β<α<1.

Furthermore, denoting by 𝔸α-1 with α[0,1] the Xα-1-realization of the 𝔸-1, it follows that

𝔸α-1(Xα,Xα-1),0α1.

These extensions are natural in the sense that e-t𝔸α-1 is the restriction to Xα-1 of e-t𝔸β-1 for 0βα1.

This interpolation-extrapolation technique is rather flexible, has many applications and is crucial for our work. Indeed, let (X0,𝔸) be as above and let (X0,𝔸)α be the interpolation-extrapolation scale generated by (X0,𝔸) and [,]θ, 0<θ<1. It is well known that the complex power 𝔸z can be defined for every z. Similarly we can define (𝔸α)z and

(𝔸α)zx=𝔸zxfor all xX1Xα+1.

Moreover, if 𝔸 belongs to the class 𝒫(X0) of operators with bounded imaginary powers on X0, then the interpolation-extrapolation scale (X0,𝔸)α is equivalent to the fraction power scale generated by (X0,𝔸) (see [4, Chapter V, Theorem 1.5.4]). If in addition 𝔸 belongs to the class 𝒫(X0,M,ϑ) of operators with uniformly bounded imaginary powers on X0 with constant M and angle ϑ, then 𝔸α𝒫(X0,M,ϑ) (see [4, Chapter V, Theorem 1.5.5]).

A.2 Proof of Theorem 3.7

The first proof of Theorem 3.7 has been given by Prüss and Wilke [23] in another context of functional framework in the study of the critical spaces for the Navier–Stokes equations subject to the following boundary conditions:

𝒖𝒏=0,𝑷Γ(𝔻(𝒖)𝒏)+α𝒖=𝟎on Γ×(0,T),(A.1)

where

𝑷Γ=I-𝒏𝒏

and α is the coefficient of friction. Their work is based on the theory of weighted Lp-maximal regularity for abstract semilinear evolution equations and the Amann interpolation-extrapolation scales. They also treated (A.1) as a perturbation of the following boundary condition:

𝒖𝒏=0,𝑷Γ((𝒖)𝒏)=𝟎on Γ×(0,T),

with

(𝒖)=𝒖-T𝒖.

For the convenience of the reader and to make the paper self-contained we give an outline of the proof of Theorem 3.7 based on the Amann interpolation-extrapolation theory and the result in [2].

Let ANT be the Stokes operator with the Navier-type boundary conditions

𝒖𝒏=0,𝐜𝐮𝐫𝐥𝒖×𝒏=𝟎on Γ×(0,T)

in 𝑳σ,τp(Ω). Using [2, Proposition 6.4], we know that

(λI+ANT)𝒫(𝑳σ,τp(Ω),M,θNT),(λI+ANT)is(𝑳σ,τp(Ω))Me|s|θNT

for λ>0 with a constant M being independent of λ and 0<θNT<π2.

Let us employ the Amann interpolation-extrapolation theory to the pair (X0,A0) where

A0=ANT,X0=𝑳σ,τp(Ω),X1=𝐃(ANT).

This yields the scale (Xα,Aα)α with

Aα=A0α=ANTα,Xα=𝐃(A0α),α>0

and

X-α=(Xα),α>0.

For a Banach space E we denote by E the dual space of E, i.e. E:=E. For α=-12, we obtain the isomorphism (λI+ANT)-1/2:X1/2X-1/2, where

X1/2=𝑽σ,τp(Ω)andX-1/2=[𝑽σ,τp(Ω)].

(We recall that 𝑽σ,τp(Ω) is given by (2.6)). Using Amann theory (see [4, Chapter V, Theorem 1.5.5]), we deduce that for all λ>0,

(λI+ANT)-1/2𝒫(X-1/2,M,θ-1/2)

and

[(λI+ANT)-1/2]is([𝑽σ,τp(Ω)])Me|s|θ-1/2,

with a constant M being independent of λ and θ-1/2=θNT(0,π2). In addition, the Stokes operator with the Navier-type boundary conditions (1.2) has the following natural weak formulation: For all 𝒖𝑽σ,τp(Ω) and all 𝒗𝑽σ,τp(Ω),

(λI+ANT)W𝒖,𝒗Ω=λΩ𝒖𝒗dx+Ω𝐜𝐮𝐫𝐥𝒖𝐜𝐮𝐫𝐥𝒗dx,

with ,Ω=,(𝑽σ,τp(Ω))×𝑽σ,τp(Ω) and ANT,W the weak Stokes operator in [𝑽σ,τp(Ω)] (the dual of 𝑽σ,τp(Ω)). It can be shown that

ANT,W:=A-1/2and(λI+ANT)W:=(λI+ANT)-1/2.

In addition (λI+ANT)-1/2(𝑽σ,τp(Ω),[𝑽σ,τp(Ω)]). Observe that

X1W=X1/2=𝑽σ,τp(Ω)=𝐃(ANT,W)andX0W=X-1/2=[𝑽σ,τp(Ω)].

Next let us denote by AN the Stokes operator with the Navier-slip boundary conditions (1.1b) and by AN,W the weak Stokes operator with the same boundary conditions (1.1b). By using Corollary 2.1, an easy computation shows that for all 𝒖𝑽σ,τp(Ω) and all 𝒗𝑽σ,τp(Ω),

AN,W𝒖,𝒗Ω=Ω𝐜𝐮𝐫𝐥𝒖𝐜𝐮𝐫𝐥𝒗dx+2𝚲𝒖,𝒗Γ=ANT,W𝒖,𝒗Ω+2𝚲𝒖,𝒗Γ,

with ,Ω=,(𝑽σ,τp(Ω))×𝑽σ,τp(Ω) and ,Γ=,𝑾-1/p,p(Γ)×𝑾1/p,p(Γ).

Similarly, for any λ>0, for any 𝒖𝑽σ,τp(Ω) and 𝒗𝑽σ,τp(Ω) we have

(λI+AN,W)𝒖,𝒗Ω=(λI+ANT,W)𝒖,𝒗Ω+2𝚲𝒖,𝒗Γ.

As described in Section 2, the operator 𝚲 is a lower-order perturbation of ANT,W, hence applying [22, Proposition 3.3.9], we obtain that for all λ>0,

(λI+AN,W)𝒫(X0W,M,θNW)

and

(λI+AN,W)is([𝑽σ,τp(Ω)])Me|s|θNW,

where M is independent of λ and 0<θNW<π2.

Consider now the interpolation-extrapolation scale (XαW,𝒜α,W)α generated by (X0W,𝒜0,W) where

X0W=X-1/2=[𝑽σ,τp(Ω)],𝒜0,W=AN,W,X1W=X1/2=𝑽σ,τp(Ω)=𝐃(ANT,W).

It can be shown that

X1/2W=𝑳σ,τp(Ω),𝒜1/2,W=AN,𝐃(𝒜1/2,W)=X3/2W=𝐃(AN).

Since 𝒜1/2,W is the restriction of 𝒜0,W to X3/2W, it follows that for all λ>0, (λI+AN)𝒫(𝑳σ,τp(Ω)) and that it satisfies (3.11).

Acknowledgements

The author wishes to thank Prof. R. Farwig and Prof. M. Hieber for their remarks that helped to improve the manuscript. The author wishes also to thank Prof. M. Wilke and Prof. J. Prüss for their remark, during the Japanese–German international workshop on Mathematical Fluid Dynamics 2016, on the perturbation of operator in weak sense and on the Amann extrapolation-interpolation argument.

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About the article

Received: 2017-01-22

Revised: 2017-04-04

Accepted: 2017-07-23

Published Online: 2017-08-15


The author acknowledges the support of the GAČR (Czech Science Foundation) project 16-03230S in the framework of RVO: 67985840.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 743–761, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0012.

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