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

Hind Al Baba

doi:10.1515/anona-2017-0012

Open Access Published by De Gruyter August 15, 2017

Maximal L^p -L^q regularity to the Stokes problem with Navier boundary conditions

Hind Al Baba

From the journal Advances in Nonlinear Analysis

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 L p - L q 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 L p - L q regularity for the Stokes problem with Navier-slip boundary conditions

(1.1)

(1.1a) ∂ ⁡ 𝒖 ∂ ⁡ t - Δ ⁢ 𝒖 + ∇ ⁡ π = 𝒇 , div ⁡ 𝒖 = 0 ⁢ in ⁢ Ω × ( 0 , T ) ,

(1.1b) 𝒖 ⋅ 𝒏 = 0 , [ 𝔻 ⁢ ( 𝒖 ) ⁢ 𝒏 ] 𝝉 = 𝟎 ⁢ on ⁢ Γ × ( 0 , T ) ,

(1.1c) 𝒖 ⁢ ( 0 ) = 𝒖 0 ⁢ in ⁢ Ω ,

where 𝔻 ⁢ ( 𝒖 ) = 1 2 ⁢ ( ∇ ⁡ 𝒖 + ∇ ⁡ 𝒖 T ) is the stress tensor and Ω is a bounded domain of ℝ 3 of class C 2 , 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 𝒖 0 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 ℝ 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 L p -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 L p - L q regularity for the non-homogeneous case (i.e. problem (1.1) with 𝒇 ≠ 𝟎 ). 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 𝒖 ∈ 𝑳 p ⁢ ( 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 L p 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 L p - L q 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 L p - L q 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 L p - L q regularity for the Stokes problem with homogeneous Robin boundary conditions in the half space ℝ + 3 . To this end, he proved that the associated Stokes operator is sectorial and admits a bounded ℋ ∞ -calculus on 𝑳 σ , τ p ⁢ ( 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 L p - L q 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 L p realization of the Hodge Laplacian operator defined by

Δ M : 𝐃 ⁢ ( Δ M ) ⊂ 𝑳 p ⁢ ( Ω ) → 𝑳 p ⁢ ( Ω ) ,

where

𝐃 ⁢ ( Δ M ) = { 𝒖 ∈ 𝑾 2 , p ⁢ ( Ω ) : 𝒖 ⋅ 𝒏 = 0 , 𝐜𝐮𝐫𝐥 ⁡ 𝒖 × 𝒏 = 𝟎 ⁢ on ⁢ Γ } ,

Δ M ⁢ 𝒖 = Δ ⁢ 𝒖 in ⁢ Ω for all ⁢ 𝒖 ∈ 𝐃 ⁢ ( Δ M ) ,

in a domain Ω ⊂ ℝ 3 with a suitably smooth boundary. Geissert et al. [15] proved that for all λ > 0 , the L p realization of the operator λ ⁢ I - Δ M admits a bounded ℋ ∞ -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 ℋ ∞ -calculus in their corresponding Banach spaces enjoys the property of bounded pure imaginary powers, they deduced the maximal L p - L q 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 L p - L q 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:

(1.2) 𝒖 ⋅ 𝒏 = 0 , 𝐜𝐮𝐫𝐥 ⁡ 𝒖 × 𝒏 = 𝟎 on ⁢ Γ × ( 0 , T ) ,

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 L p 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 L p -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 L p - L q 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 𝑳 σ , τ p ⁢ ( Ω ) given by

(2.1) 𝑳 σ , τ p ⁢ ( Ω ) = { 𝒇 ∈ 𝑳 p ⁢ ( Ω ) : div ⁡ 𝒇 = 0 ⁢ in ⁢ Ω , 𝒇 ⋅ 𝒏 = 0 ⁢ on ⁢ Γ }

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

(2.2) 𝐃 ⁢ ( A p ) = { 𝒖 ∈ 𝑾 2 , p ⁢ ( Ω ) : div ⁡ 𝒖 = 0 ⁢ in ⁢ Ω , 𝒖 ⋅ 𝒏 = 0 , [ 𝔻 ⁢ ( 𝒖 ) ⁢ 𝒏 ] 𝝉 = 𝟎 ⁢ on ⁢ Γ } ,

(2.3) A p ⁢ 𝒖 = - P ⁢ Δ ⁢ 𝒖 in ⁢ Ω for all ⁢ 𝒖 ∈ 𝐃 ⁢ ( A p ) .

The operator P in (2.3) is the Helmholtz projection P : 𝑳 p ⁢ ( Ω ) → 𝑳 σ , τ p ⁢ ( Ω ) defined by

(2.4) P ⁢ 𝒇 = 𝒇 - grad ⁡ π for all ⁢ 𝒇 ∈ 𝑳 p ⁢ ( Ω ) ,

where π ∈ W 1 , p ⁢ ( Ω ) / ℝ is the unique solution of the following weak Neumann problem (cf. [28]):

(2.5) div ⁡ ( grad ⁡ π - 𝒇 ) = 0 in ⁢ Ω , ( grad ⁡ π - 𝒇 ) ⋅ 𝒏 = 0 on ⁢ Γ .

An easy computation shows that

A p ⁢ 𝒖 = - Δ ⁢ 𝒖 + grad ⁡ π in ⁢ Ω for all ⁢ 𝒖 ∈ 𝐃 ⁢ ( A p ) ,

where π ∈ W 1 , p ⁢ ( Ω ) / ℝ is the unique solution up to an additive constant of the problem

div ⁡ ( grad ⁡ π - Δ ⁢ 𝒖 ) = 0 in ⁢ Ω , ( grad ⁡ π - Δ ⁢ 𝒖 ) ⋅ 𝒏 = 0 on ⁢ Γ .

Consider the space

(2.6) 𝑽 σ , τ p ⁢ ( Ω ) = { 𝒖 ∈ 𝑾 1 , p ⁢ ( Ω ) : div ⁡ 𝒖 = 0 ⁢ in ⁢ Ω , 𝒖 ⋅ 𝒏 = 0 ⁢ on ⁢ Γ } .

Observe that the Stokes operator A p can be defined by the following weak formulation: for all 𝒖 ∈ 𝑽 σ , τ p ⁢ ( Ω ) and all 𝒗 ∈ 𝑽 σ , τ p ′ ⁢ ( Ω ) we have

〈 A p ⁢ 𝒖 , 𝒗 〉 ( 𝑽 σ , τ p ′ ⁢ ( Ω ) ) ′ × 𝑽 σ , τ p ′ ⁢ ( Ω ) = ∫ Ω 𝔻 ⁢ ( 𝒖 ) : 𝔻 ⁢ ( 𝒗 ¯ ) ⁢ d ⁢ x .

We note (cf. [7, 32]) that in the general case the Stokes operator A p has a non-trivial kernel, we denote this kernel by 𝓣 p ⁢ ( Ω ) . When the domain Ω is obtained by rotation around a vector 𝒃 ∈ ℝ 3 , then

𝓣 p ⁢ ( Ω ) = Span ⁡ { 𝜷 } , 𝜷 ⁢ ( 𝒙 ) = 𝒃 × 𝒙 for ⁢ 𝒙 ∈ Ω .

Otherwise

𝓣 p ⁢ ( Ω ) = { 𝟎 }

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

𝓣 p ⁢ ( Ω ) = { 𝒗 ∈ 𝑾 1 , p ⁢ ( Ω ) : 𝔻 ⁢ ( 𝒗 ) = 𝟎 ⁢ in ⁢ Ω , div ⁡ 𝒗 = 0 ⁢ in ⁢ Ω ⁢ and ⁢ 𝒗 ⋅ 𝒏 = 0 ⁢ on ⁢ Γ } .

We also note (see [7, Theorem 3.9]) that the operator - A p is sectorial and generates a bounded analytic semigroup on 𝑳 σ , τ p ⁢ ( Ω ) for all 1 < p < ∞ . We denote by e - t ⁢ A p the analytic semigroup associated to the operator A p in 𝑳 σ , τ p ⁢ ( Ω ) .

Next we consider the space

𝑯 p ⁢ ( div , Ω ) = { 𝒗 ∈ 𝑳 p ⁢ ( Ω ) : div ⁡ 𝒗 ∈ 𝑳 p ⁢ ( Ω ) } ,

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

𝑯 0 p ⁢ ( div , Ω ) = { 𝒗 ∈ 𝑯 p ⁢ ( div , Ω ) : 𝒗 ⋅ 𝒏 = 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 ℝ 3 . The dual space [ 𝑯 0 p ⁢ ( div , Ω ) ] ′ of 𝑯 0 p ⁢ ( div , Ω ) can be characterized as follows (cf. [26, Proposition 1.0.4]): A distribution 𝒇 belongs to [ 𝑯 0 p ⁢ ( div , Ω ) ] ′ if and only if there exist 𝝍 ∈ 𝑳 p ′ ⁢ ( Ω ) and χ ∈ L p ′ ⁢ ( Ω ) such that 𝒇 = 𝝍 + grad ⁡ χ and

∥ 𝒇 ∥ [ 𝑯 0 p ⁢ ( div , Ω ) ] ′ = inf 𝒇 = 𝝍 + grad ⁡ χ ⁡ max ⁡ ( ∥ 𝝍 ∥ 𝑳 p ′ ⁢ ( Ω ) , ∥ χ ∥ 𝑳 p ′ ⁢ ( Ω ) ) .

We consider the following space:

[ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ = { 𝒇 ∈ [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] ′ : div ⁡ 𝒇 = 0 ⁢ in ⁢ Ω ⁢ and ⁢ 𝒇 ⋅ 𝒏 = 0 ⁢ on ⁢ Γ } .

We note (cf. [2]) that for a function 𝒇 in the dual space [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] ′ such that div ⁡ 𝒇 ∈ L p ⁢ ( Ω ) , the normal trace value 𝒇 ⋅ 𝒏 | Γ exists and belongs to the space 𝑾 - 1 - 1 / p , p ⁢ ( Γ ) .

The Stokes operator A p can be extended to the space [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ (cf. [7, Section 3.2]). This extension is a closed linear densely defined operator

B p : 𝐃 ⁢ ( B p ) ⊂ [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ → [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ ,

(2.7) 𝐃 ⁢ ( B p ) = { 𝒖 ∈ 𝑾 1 , p ⁢ ( Ω ) : div ⁡ 𝒖 = 0 ⁢ in ⁢ Ω , 𝒖 ⋅ 𝒏 = 0 , [ 𝔻 ⁢ ( 𝒖 ) ⁢ 𝒏 ] 𝝉 = 𝟎 ⁢ on ⁢ Γ } ,

B p ⁢ 𝒖 = - Δ ⁢ 𝒖 + grad ⁡ π in ⁢ Ω for all ⁢ 𝒖 ∈ 𝐃 ⁢ ( B p ) ,

where π ∈ L p ⁢ ( Ω ) / ℝ is the unique solution up to an additive constant of the problem

div ⁡ ( grad ⁡ π - Δ ⁢ 𝒖 ) = 0 in ⁢ Ω , ( grad ⁡ π - Δ ⁢ 𝒖 ) ⋅ 𝒏 = 0 on ⁢ Γ .

The operator - B p generates a bounded analytic semigroup on [ 𝑯 0 p ′ ⁢ ( 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 𝒖 ∈ 𝑾 1 , p ⁢ ( Ω ) such that Δ ⁢ 𝒖 ∈ [ 𝑯 0 p ′ ⁢ ( 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:

𝑻 p ⁢ ( Ω ) = { 𝒗 ∈ 𝑯 0 p ⁢ ( div , Ω ) : div ⁡ 𝒗 ∈ W 0 1 , p ⁢ ( Ω ) } .

Thanks to [8, Lemmas 4.11, 4.12] we know that 𝓓 ⁢ ( Ω ) is dense in 𝑻 p ⁢ ( Ω ) and a distribution 𝒇 ∈ ( 𝑻 p ⁢ ( Ω ) ) ′ if and only if there exist 𝝍 ∈ 𝑳 p ′ ⁢ ( Ω ) and f 0 ∈ W - 1 , p ′ ⁢ ( Ω ) such that 𝒇 = 𝝍 + ∇ ⁡ f 0 , with

∥ 𝒇 ∥ ( 𝑻 p ⁢ ( Ω ) ) ′ = inf 𝒇 = 𝝍 + ∇ ⁡ f 0 ⁡ max ⁡ ( ∥ 𝝍 ∥ 𝑳 p ′ ⁢ ( Ω ) , ∥ f 0 ∥ W - 1 , p ′ ⁢ ( Ω ) ) .

We consider the subspace

[ 𝑻 p ′ ⁢ ( Ω ) ] σ , τ ′ = { 𝒇 ∈ ( 𝑻 p ′ ⁢ ( Ω ) ) ′ : div ⁡ 𝒇 = 0 ⁢ in ⁢ Ω ⁢ and ⁢ 𝒇 ⋅ 𝒏 = 0 ⁢ on ⁢ Γ } .

We recall from [2] that for a function 𝒇 in the dual space [ 𝑻 p ′ ⁢ ( Ω ) ] ′ such that div ⁡ 𝒇 ∈ L p ⁢ ( Ω ) the normal trace 𝒇 ⋅ 𝒏 | 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 [ 𝑻 p ′ ⁢ ( Ω ) ] σ , τ ′ (see [7, Section 3.3]). This extension is a densely defined closed linear operator

C p : 𝐃 ⁢ ( C p ) ⊂ [ 𝑻 p ′ ⁢ ( Ω ) ] σ , τ ′ → [ 𝑻 p ′ ⁢ ( Ω ) ] σ , τ ′ ,

where

(2.8) 𝐃 ⁢ ( C p ) = { 𝒖 ∈ 𝑳 p ⁢ ( Ω ) : div ⁡ 𝒖 = 0 ⁢ in ⁢ Ω , 𝒖 ⋅ 𝒏 = 0 , [ 𝔻 ⁢ ( 𝒖 ) ⁢ 𝒏 ] 𝝉 = 𝟎 ⁢ on ⁢ Γ }

and C p ⁢ 𝒖 = - Δ ⁢ 𝒖 + grad ⁡ π in Ω for all 𝒖 ∈ 𝐃 ⁢ ( C p ) , with π ∈ W - 1 , p ⁢ ( Ω ) / ℝ the unique solution up to an additive constant of the problem

div ⁡ ( grad ⁡ π - Δ ⁢ 𝒖 ) = 0 in ⁢ Ω , ( grad ⁡ π - Δ ⁢ 𝒖 ) ⋅ 𝒏 = 0 on ⁢ Γ .

The operator - C p generates a bounded analytic semigroup on [ 𝑻 p ′ ⁢ ( Ω ) ] σ , τ ′ for all 1 < p < ∞ (see [7, Theorem 3.12]). We recall from [7, Lemma 5.4] that for a function 𝒖 ∈ 𝑳 p ⁢ ( Ω ) such that Δ ⁢ 𝒖 ∈ ( 𝑻 p ′ ⁢ ( 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 𝒞 2 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 s 1 , s 2 along each family of curves, respectively, are a possible system of coordinates in W. We denote by 𝝉 1 , 𝝉 2 the unit tangent vectors to the boundary. With this notation, we have

𝒗 = ∑ k = 1 2 v k ⁢ 𝝉 k + ( 𝒗 ⋅ 𝒏 ) ⁢ 𝒏 ,

where 𝝉 k T = ( τ k ⁢ 1 , τ k ⁢ 2 , τ k ⁢ 3 ) and v k = 𝒗 ⋅ 𝝉 k . As a result for any 𝒗 ∈ 𝓓 ⁢ ( Ω ¯ ) the following formulas hold (see [7]):

[ D ⁢ ( 𝒗 ) ⁢ 𝒏 ] 𝝉 = ∇ 𝝉 ⁡ ( 𝒗 ⋅ 𝒏 ) + ( ∂ ⁡ 𝒗 ∂ ⁡ 𝒏 ) 𝝉 - 𝚲 ⁢ 𝒗 on ⁢ Γ

and

𝐜𝐮𝐫𝐥 ⁡ 𝒗 × 𝒏 = ∇ 𝝉 ⁡ ( 𝒗 ⋅ 𝒏 ) - ( ∂ ⁡ 𝒗 ∂ ⁡ 𝒏 ) 𝝉 - 𝚲 ⁢ 𝒗 on ⁢ Γ ,

where

𝚲 ⁢ 𝒘 = ∑ k = 1 2 ( 𝒘 𝝉 ⋅ ∂ ⁡ 𝒏 ∂ ⁡ s k ) ⁢ 𝝉 k .

In the particular case 𝒗 ⋅ 𝒏 = 0 on Γ, the following equality holds:

(2.9) [ 2 ⁢ 𝐃 ⁢ ( 𝒗 ) ⁢ 𝒏 ] 𝝉 = - 𝐜𝐮𝐫𝐥 ⁡ 𝒗 × 𝒏 - 2 ⁢ 𝚲 ⁢ 𝒗 in ⁢ 𝑾 1 / p , p ⁢ ( Γ ) .

It can be seen from (2.9) that the Navier-slip boundary (1.1b) differs from (1.2) only by the term - 2 ⁢ 𝚲 ⁢ 𝒗 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 𝐯 ∈ 𝐖 1 , p ⁢ ( Ω ) such that 𝐯 ∈ [ 𝐇 0 p ′ ⁢ ( div , Ω ) ] ′ and 𝐯 ⋅ 𝐧 = 0 on Γ, we have

[ 2 ⁢ 𝐃 ⁢ ( 𝒗 ) ⁢ 𝒏 ] 𝝉 = - 𝒄𝒖𝒓𝒍 ⁡ 𝒗 × 𝒏 - 2 ⁢ 𝚲 ⁢ 𝒗 in ⁢ 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 A p on 𝑳 σ , τ p ⁢ ( Ω ) . Since the Stokes operator A p in 𝑳 σ , τ p ⁢ ( 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 A p α , α ∈ ℂ , are well, densely defined and closed linear operators on 𝑳 σ , τ p ⁢ ( Ω ) with domain 𝐃 ⁢ ( A p α ) . Furthermore,

𝓓 σ ⁢ ( Ω ) ↪ 𝐃 ⁢ ( A p α ) ↪ 𝑳 σ , τ p ⁢ ( Ω ) for all ⁢ α ∈ ℂ .

We denote by ℂ the set of complex number, ℂ ∗ = ℂ ∖ { 0 } , and by 𝓓 σ ⁢ ( Ω ) the set of divergence free infinitely differentiable functions with compact support in Ω

Nevertheless, as described above, since the Stokes operator A p 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 + A p ) . We start by the following proposition.

Proposition 3.1.

There exists an angle 0 < θ 0 < π 2 such that the resolvent set of the operator - ( I + A p ) contains the sector

Σ θ 0 = { λ ∈ ℂ : | arg ⁡ λ | ≤ π - θ 0 } .

Moreover, the following estimate holds:

(3.1) ∥ ( λ ⁢ I + I + A p ) - 1 ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ κ ⁢ ( Ω , p ) | λ | for all ⁢ λ ∈ Σ θ 0 , λ ≠ 0 ,

with a constant κ independent of λ.

Proof.

Since the operator - A p generates a bounded analytic semigroup on 𝑳 σ , τ p ⁢ ( Ω ) , the operator I + A p is an isomorphism from 𝐃 ⁢ ( A p ) ⊂ 𝑳 σ , τ p ⁢ ( Ω ) in 𝑳 σ , τ p ⁢ ( Ω ) . We recall that 𝐃 ⁢ ( A p ) is given by (2.2). Let λ ∈ ℂ ∗ such that Re ⁡ λ ≥ 0 . It is clear that the operator λ ⁢ I + I + A p is an isomorphism from 𝐃 ⁢ ( A p ) to 𝑳 σ , τ p ⁢ ( Ω ) . Using [7, Theorem 3.8], one has, since Re ⁡ λ ≥ 0 ,

(3.2) ∥ ( λ ⁢ I + I + A p ) - 1 ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ κ ⁢ ( Ω , p ) | λ + 1 | ≤ κ ⁢ ( Ω , p ) | λ | ,

where the constant κ ⁢ ( Ω , p ) is independent of λ. This means that the resolvent set of the operator - ( I + A p ) contains the set { λ ∈ ℂ ∗ : Re ⁡ λ ≥ 0 } where the estimate (3.2) is satisfied. Using the result of [33, Chapter VIII, Theorem 1], we deduce that there exists an angle 0 < θ 0 < π 2 , such that the resolvent set of - ( I + A p ) contains the sector Σ θ 0 . In addition for every λ ∈ Σ θ 0 such that λ ≠ 0 estimate (3.1) holds. ∎

Corollary 3.2.

Let θ 0 be as in Proposition 3.1.

The estimate

(3.3) ∥ ( λ ⁢ I + I + A p ) - 1 ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ C ⁢ ( Ω , p )

holds for all λ ∈ Σ θ 0 with a constant C independent of λ.
Let 0 < α < 1 be fixed and let λ ∈ Σ θ 0 such that λ ≠ 0 and | λ | ≤ 1 2 ⁢ κ ⁢ ( Ω , p ) . Then one has

(3.4) ∥ ( λ ⁢ I + I + A p ) - 1 ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ 2 α ⁢ κ α ⁢ ( Ω , p ) ⁢ | λ | α - 1 ,

with a constant κ independent of λ.

Proof.

Let 𝒇 ∈ 𝑳 σ , τ p ⁢ ( Ω ) and 𝒖 ∈ 𝐃 ⁢ ( A p ) such that ( λ ⁢ I + I + A p ) - 1 ⁢ 𝒇 = 𝒖 . Observe that 𝒖 satisfies

{ 𝒖 - Δ ⁢ 𝒖 + ∇ ⁡ π = 𝒇 - λ ⁢ 𝒖 , div ⁡ 𝒖 = 0 in ⁢ Ω , 𝒖 ⋅ 𝒏 = 0 , [ 𝔻 ⁢ ( 𝒖 ) ⁢ 𝒏 ] 𝝉 = 𝟎 on ⁢ Γ .

Using [7, Theorem 3.8], we have

∥ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) ≤ κ ⁢ ( Ω , p ) ⁢ ∥ 𝒇 - λ ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) ≤ κ ⁢ ( Ω , p ) ⁢ [ ∥ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) + | λ | ⁢ ∥ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) ] .

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

Next let 0 < α < 1 be fixed. Then for all λ ∈ Σ θ 0 such that λ ≠ 0 and | λ | ≤ 1 2 ⁢ κ ⁢ ( Ω , p ) we have

∥ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) ≤ 2 ⁢ κ ⁢ ( Ω , p ) ⁢ ∥ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) ≤ 2 α ⁢ κ α ⁢ ( Ω , p ) ⁢ | λ | α - 1 ⁢ ∥ 𝒇 ∥ 𝑳 p ⁢ ( 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 + A p ) since 0 ∈ ρ ⁢ ( I + A p ) . 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 + A p ) .

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

Proposition 3.4.

Let θ 0 be as in Proposition 3.1. For all z ∈ C with - 1 < Re ⁡ z < 0 we have

(3.5) ∥ ( I + A p ) z ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ C ⁢ ( Re ⁡ z , Ω , p ) ⁢ e | Im ⁡ z | ⁢ θ 0 ,

with a constant C ⁢ ( Re ⁡ z , Ω , p ) depending on Re ⁡ z , Ω , p .

Proof.

Let z ∈ ℂ such that - 1 < Re ⁡ z < 0 . Thanks to Proposition 3.1 we know that the operator I + A p 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]):

(3.6) ( I + A p ) z = 1 2 ⁢ π ⁢ i ⁢ ∫ Γ θ 0 ( - λ ) z ⁢ ( λ ⁢ I + I + A p ) - 1 ⁢ d λ ,

where

Γ θ 0 = { ρ ⁢ e i ⁢ ( π - θ 0 ) : 0 ≤ ρ ≤ ∞ } ∪ { - ρ ⁢ e i ⁢ ( θ 0 - π ) : - ∞ ≤ ρ ≤ 0 } .

This means that

( I + A p ) z = 1 2 ⁢ π ⁢ i [ ∫ 0 + ∞ ( - ρ e i ⁢ ( π - θ 0 ) ) z ( ρ e i ⁢ ( π - θ 0 ) I + I + A p ) - 1 e i ⁢ ( π - θ 0 ) d ρ

- ∫ 0 + ∞ ( - ρ e i ⁢ ( θ 0 - π ) ) z ( ρ e i ⁢ ( θ 0 - π ) I + I + A p ) - 1 e i ⁢ ( θ 0 - π ) d ρ ] .

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

| ( - λ ) | z ≤ ρ Re ⁡ z ⁢ e | Im ⁡ z | ⁢ θ 0 .

As a result we have

(3.7) ∥ ( I + A p ) z ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ e | Im ⁡ z | ⁢ θ 0 2 ⁢ π ⁢ [ I 1 + I 2 ] ,

with

I 1 = ∫ 0 + ∞ ρ Re ⁡ z ⁢ ∥ ( ρ ⁢ e i ⁢ ( π - θ 0 ) ⁢ I + I + A p ) - 1 ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ⁢ d ρ ,

I 2 = ∫ 0 + ∞ ρ Re ⁡ z ⁢ ∥ ( ρ ⁢ e i ⁢ ( θ 0 - π ) ⁢ I + I + A p ) - 1 ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ⁢ d ρ .

Next, we write I 1 in the form

I 1 = ∫ 0 1 ρ Re ⁡ z ⁢ ∥ ( ρ ⁢ e i ⁢ ( π - θ 0 ) ⁢ I + I + A p ) - 1 ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ⁢ d ρ + ∫ 1 + ∞ ρ Re ⁡ z ⁢ ∥ ( ρ ⁢ e i ⁢ ( π - θ 0 ) ⁢ I + I + A p ) - 1 ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ⁢ d ρ ,

In other words

Γ θ 0 = [ Γ θ 0 ∩ { λ ∈ ℂ : | λ | ≤ 1 } ] ∪ [ Γ θ 0 ∩ { λ ∈ ℂ : | λ | > 1 } ] .

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

I 1 ≤ C ⁢ ( Ω , p ) ⁢ ∫ 0 1 d ⁢ ρ ρ - Re ⁡ z + κ ⁢ ( Ω , p ) ⁢ ∫ 1 + ∞ d ⁢ ρ ρ 1 - Re ⁡ z .

Thanks to our assumption on z we can verify that the improper integrals I 1 and I 2 are convergent and satisfy

I 1 < C ⁢ ( Re ⁡ z , Ω , p ) , I 2 < C ⁢ ( Re ⁡ z , Ω , p ) ,

with a constant C ⁢ ( Re ⁡ z , Ω , p ) depending on Re ⁡ z , Ω , p .

Finally, substituting in (3.7), we have for all z ∈ ℂ with - 1 < Re ⁡ z < 0 ,

(3.8) ∥ ( I + A p ) z ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ C ⁢ ( Ω , p ) ⁢ [ 1 1 + Re ⁡ z + 1 Re ⁡ z ] ⁢ e | Im ⁡ z | ⁢ θ 0 .

This completes the proof of the proposition. ∎

Remark 3.5.

(i) We recall from [18, Propositions 4.7, 4.10] that for all 𝒇 ∈ 𝐃 ⁢ ( A p ) the operator ( I + A p ) z ⁢ 𝒇 is analytic in z for - 1 < Re ⁡ z < 1 .

(ii) Observe that if we replace Γ θ 0 in (3.6) by

Γ = Γ θ 0 ∩ { λ ∈ ℂ : | λ | ≤ c }

with some constant c > 0 , we obtain for all z ∈ ℂ with - 1 < Re ⁡ z < 0 ,

∥ ( I + A p ) z ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ C ⁢ ( Ω , p ) ⁢ [ c 1 + Re ⁡ z 1 + Re ⁡ z ] ⁢ e | Im ⁡ z | ⁢ θ 0 .

Let 𝒇 ∈ 𝐃 ⁢ ( A p ) , taking the limit as Re ⁡ z tends to 0, we obtain that for all s ∈ ℝ ,

(3.9) ∥ ( I + A p ) i ⁢ s ⁢ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) = lim Re ⁡ z → 0 ⁡ ∥ ( I + A p ) Re ⁡ z + i ⁢ s ⁢ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) ≤ C ⁢ ( Ω , p ) ⁢ e | s | ⁢ θ 0 ⁢ ∥ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) .

Using then the density of 𝐃 ⁢ ( A p ) in 𝑳 σ , τ p ⁢ ( Ω ) , we obtain estimate (3.9) for all 𝒇 ∈ 𝑳 σ , τ p ⁢ ( Ω ) .

Corollary 3.6.

Let s ∈ R and 0 < α < 1 . The operator ( I + A p ) i ⁢ s is bounded from D ⁢ ( ( I + A p ) α ) to 𝐋 σ , τ p ⁢ ( Ω ) . Furthermore, there exists an angle 0 < θ 0 < π 2 and a constant C > 0 such that

(3.10) ∥ ( I + A p ) i ⁢ s ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) ≤ C ⁢ e | s | ⁢ θ 0 ⁢ ∥ 𝒖 ∥ 𝐃 ⁢ ( ( I + A p ) α ) .

Proof.

Let 0 < α < 1 , 𝒇 ∈ 𝑳 σ , τ p ⁢ ( Ω ) and let 𝒖 ∈ 𝐃 ⁢ ( ( I + A p ) α ) such that ( I + A p ) α ⁢ 𝒖 = 𝒇 in Ω. Using Proposition 3.4, we deduce that there exists an angle 0 < θ 0 < π 2 and a constant C > 0 such that

∥ ( I + A p ) - α + i ⁢ s ⁢ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) ≤ C ⁢ e | s | ⁢ θ 0 ⁢ ∥ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) .

Next observe that

∥ ( I + A p ) i ⁢ s ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) = ∥ ( I + A p ) - α + i ⁢ s ⁢ ( I + A p ) α ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω )

= ∥ ( I + A p ) - α + i ⁢ s ⁢ 𝒇 ∥ 𝑳 p ⁢ ( Ω )

≤ C ⁢ e | s | ⁢ θ 0 ⁢ ∥ 𝒇 ∥ 𝑳 p ⁢ ( Ω )

≤ C ⁢ e | s | ⁢ θ 0 ⁢ ∥ ( I + A p ) α ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) .

Therefore one has estimate (3.10). This means that for all s ∈ ℝ , the operator ( I + A p ) i ⁢ s is bounded from 𝐃 ⁢ ( ( I + A p ) α ) to 𝑳 σ , τ p ⁢ ( Ω ) and satisfies estimate (3.10). We recall that the operator I + A p has a bounded inverse and thus for all α ∈ ℂ with Re ⁡ α > 0 , the operator ( I + A p ) α is an isomorphism from 𝐃 ⁢ ( A p α ) to 𝑳 σ , τ p ⁢ ( 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 Re ⁡ z = 0 in 𝑳 σ , τ p ⁢ ( Ω ) , since the second term on the right-hand side of (3.8) blows up as Re ⁡ z 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 Re ⁡ z = 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 θ 0 ∈ ( 0 , π 2 ) and a constant M > 0 such that for all s ∈ R and all λ > 0 ,

(3.11) ∥ ( λ ⁢ I + A p ) i ⁢ s ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ M ⁢ e | s | ⁢ θ 0 ,

where M is independent of λ.

The following theorem extends Theorem 3.7 to the operators ( λ ⁢ I + B p ) and ( λ ⁢ I + C p ) , λ > 0 , on [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ and [ 𝑻 p ′ ⁢ ( 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 < θ 0 < π 2 and a constant C > 0 such that for all s ∈ R and all λ > 0 ,

(3.12) ∥ ( λ ⁢ I + B p ) i ⁢ s ∥ ℒ ⁢ ( [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ ) ≤ C ⁢ e | s | ⁢ θ 0 ,

(3.13) ∥ ( λ ⁢ I + C p ) i ⁢ s ∥ ℒ ⁢ ( [ 𝑻 p ′ ⁢ ( Ω ) ] σ , τ ′ ) ≤ C ⁢ e | s | ⁢ θ 0 ,

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 𝒇 ∈ 𝑳 σ , τ p ⁢ ( Ω ) ,

∥ ( λ ⁢ I + B p ) i ⁢ s ⁢ 𝒇 ∥ [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] ′ = ∥ ( λ ⁢ I + A p ) i ⁢ s ⁢ 𝒇 ∥ [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] ′ ≤ C ⁢ e | s | ⁢ θ 0 ⁢ ∥ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) .

Using that ∥ 𝒇 ∥ [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] ′ = ∥ 𝒇 ∥ 𝑳 σ , τ p ⁢ ( Ω ) , we deduce that

∥ ( λ ⁢ I + B p ) i ⁢ s ⁢ 𝒇 ∥ [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] ′ ≤ C ⁢ e | s | ⁢ θ 0 ⁢ ∥ 𝒇 ∥ [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] ′ .

Next, using the density of 𝑳 σ , τ p ⁢ ( Ω ) in [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ (see [2, Proposition 3.9]), we can extend ( λ ⁢ I + B p ) i ⁢ s to a bounded linear operator on [ 𝑯 0 p ′ ⁢ ( 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 𝒃 ∈ ℝ 3 , 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 𝐛 ∈ R 3 . There exist 0 < θ 0 < π 2 and a constant C > 0 such that for all s ∈ R ,

∥ ( A p ) i ⁢ s ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ C ⁢ e | s | ⁢ θ 0 ,

∥ ( B p ) i ⁢ s ∥ ℒ ⁢ ( [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ ) ≤ C ⁢ e | s | ⁢ θ 0 ,

∥ ( C p ) i ⁢ s ∥ ℒ ⁢ ( [ 𝑻 p ′ ⁢ ( Ω ) ] σ , τ ′ ) ≤ C ⁢ e | s | ⁢ θ 0 .

Next we study the domains of fractional powers of the operator A p on 𝑳 σ , τ p ⁢ ( 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 ⁢ ( A p α ) and their norms. It follows from [18] that for Re ⁡ α > 0 , the domain 𝐃 ⁢ ( ν ⁢ I + A p α ) does not depend on ν ≥ 0 and coincides with 𝐃 ⁢ ( μ ⁢ I + A p α ) for μ ≥ 0 , that is,

𝐃 ⁢ ( A p α ) = 𝐃 ⁢ ( μ ⁢ I + A p α ) = 𝐃 ⁢ ( ν ⁢ I + A p α ) 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 + A p ) allows us to determine the domain of definition of 𝐃 ( I + A p ) α ) , and then of 𝐃 ⁢ ( A p α ) for any complex number α satisfying Re ⁡ α > 0 using complex interpolation theory. In addition for all α > 0 , the map 𝒗 ↦ ∥ ( I + A p ) α ⁢ 𝒗 ∥ 𝑳 p ⁢ ( Ω ) is a norm on 𝐃 ⁢ ( A p α ) . This is due to the fact (cf. [31, Theorem 1.15.2, part (e)]) that the operator I + A p has a bounded inverse, and thus for all α ∈ ℂ with Re ⁡ α > 0 , the operator ( I + A p ) α is an isomorphism from 𝐃 ⁢ ( A p α ) to 𝑳 σ , τ p ⁢ ( Ω ) .

The following theorem characterizes the domain of A p 1 / 2 .

Theorem 3.10.

For all 1 < p < ∞ , D ⁢ ( A p 1 / 2 ) = 𝐕 σ , τ p ⁢ ( Ω ) (given by (2.6)) with equivalent norms.

Proof.

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

𝐃 ⁢ ( A p 1 / 2 ) = 𝐃 ⁢ ( ( I + A p ) 1 / 2 ) = [ 𝐃 ⁢ ( I + A p ) ; 𝑳 σ , τ p ⁢ ( Ω ) ] 1 / 2 = [ 𝐃 ⁢ ( A p ) ; 𝑳 σ , τ p ⁢ ( Ω ) ] 1 / 2 .

Consider a function 𝒖 ∈ 𝐃 ⁢ ( A p ) (see (2.2) for the definition of 𝐃 ⁢ ( A p ) ) and set 𝒛 = 𝔻 ⁢ ( 𝒖 ) and 𝑼 = ( 𝒖 , 𝒛 ) . It is clear that if 𝒖 ∈ 𝐃 ⁢ ( A p ) , then 𝒛 ∈ 𝑾 1 , p ⁢ ( Ω ) and 𝑼 ∈ 𝑳 σ , τ p ⁢ ( Ω ) × 𝑾 1 , p ⁢ ( Ω ) . In addition, if 𝒖 ∈ 𝑳 σ , τ p ⁢ ( Ω ) , then 𝒛 ∈ 𝑾 - 1 , p ⁢ ( Ω ) and 𝑼 ∈ 𝑳 σ , τ p ⁢ ( Ω ) × 𝑾 - 1 , p ⁢ ( Ω ) . Now, let 𝒖 ∈ 𝐃 ⁢ ( A p 1 / 2 ) , then

𝑼 ∈ 𝑳 σ , τ p ⁢ ( Ω ) × [ 𝑾 1 , p ⁢ ( Ω ) ; 𝑾 - 1 , p ⁢ ( Ω ) ] 1 / 2 = 𝑳 σ , τ p ⁢ ( Ω ) × 𝑳 p ⁢ ( Ω ) .

As a result, 𝒖 ∈ 𝑳 p ⁢ ( Ω ) , 𝒛 = 𝔻 ⁢ ( 𝒖 ) ∈ 𝑳 p ⁢ ( Ω ) , div ⁡ 𝒖 = 0 in Ω and 𝒖 ⋅ 𝒏 = 0 on Γ. Thanks to [7], we know that for all 𝒖 ∈ 𝑽 σ , τ p ⁢ ( Ω ) , the norm ∥ 𝒖 ∥ 𝑾 1 , p ⁢ ( Ω ) is equivalent to ∥ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) + ∥ 𝔻 ⁢ ( 𝒖 ) ∥ 𝑳 p ⁢ ( Ω ) . As a result 𝒖 ∈ 𝑽 σ , τ p ⁢ ( Ω ) and

𝐃 ⁢ ( A p 1 / 2 ) ↪ 𝑽 σ , τ p ⁢ ( Ω ) .

Next we prove the second inclusion. Since I + A p has a bounded inverse, the operator ( I + A p ) 1 / 2 is an isomorphism from 𝐃 ⁢ ( ( I + A p ) 1 / 2 ) to 𝑳 σ , τ p ⁢ ( Ω ) for all 1 < p < ∞ . This means that for all 𝑭 ∈ 𝑳 σ , τ p ′ ⁢ ( Ω ) there exists a unique 𝒗 ∈ 𝐃 ⁢ ( ( I + A p ′ ) 1 / 2 ) solution of

(3.14) ( I + A p ′ ) 1 / 2 ⁢ 𝒗 = 𝑭 .

Let 𝒖 ∈ 𝐃 ⁢ ( A p ) and observe that

∥ ( I + A p ) 1 / 2 ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) = sup 𝑭 ∈ 𝑳 σ , τ p ′ ⁢ ( Ω ) , 𝑭 ≠ 𝟎 ⁡ | 〈 ( I + A p ) 1 / 2 ⁢ 𝒖 , 𝑭 〉 𝑳 σ , τ p ⁢ ( Ω ) × 𝑳 σ , τ p ′ ⁢ ( Ω ) | ∥ 𝑭 ∥ 𝑳 p ′ ⁢ ( Ω )

= sup 𝒗 ∈ 𝐃 ⁢ ( A p ′ 1 / 2 ) , 𝒗 ≠ 𝟎 ⁡ | 〈 ( I + A p ) ⁢ 𝒖 , 𝒗 〉 𝑳 σ , τ p ⁢ ( Ω ) × 𝑳 σ , τ p ′ ⁢ ( Ω ) | ∥ ( I + A p ′ ) 1 / 2 ⁢ 𝒗 ∥ 𝑳 p ′ ⁢ ( Ω )

= sup 𝒗 ∈ 𝐃 ⁢ ( A p ′ 1 / 2 ) , 𝒗 ≠ 𝟎 ⁡ | ∫ Ω 𝒖 ⋅ 𝒗 ¯ ⁢ d x + ∫ Ω 𝔻 ⁢ ( 𝒖 ) : 𝔻 ⁢ ( 𝒗 ¯ ) ⁢ d ⁢ x | ∥ ( I + A p ′ ) 1 / 2 ⁢ 𝒗 ∥ 𝑳 p ′ ⁢ ( Ω )

(3.15) ≤ C ⁢ ( Ω , p ) ⁢ ∥ 𝒖 ∥ 𝑾 1 , p ⁢ ( Ω ) .

We recall that 𝒗 is the unique solution of problem (3.14) and that the adjoint operator ( ( I + A p ) 1 / 2 ) ′ of ( I + A p ) 1 / 2 is equal to the operator ( I + A p ′ ) 1 / 2 . We also recall that the dual of 𝑳 σ , τ p ⁢ ( Ω ) is equal to 𝑳 σ , τ p ′ ⁢ ( Ω ) . Using the density of 𝐃 ⁢ ( A p ) in 𝑽 σ , τ p ⁢ ( Ω ) , we obtain estimate (3.15) for all 𝒖 ∈ 𝑽 σ , τ p ⁢ ( Ω ) and then

𝑽 σ , τ p ⁢ ( Ω ) ↪ 𝐃 ⁢ ( A p 1 / 2 ) . ∎

Remark 3.11.

In the case where the domain Ω is not obtained by rotation around a vector 𝒃 ∈ ℝ 3 , the Stokes operator A p is invertible with bounded inverse and the following equivalence holds:

∥ A p 1 / 2 ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) ≃ ∥ 𝔻 ⁢ ( 𝒖 ) ∥ 𝑳 p ⁢ ( Ω ) for all ⁢ 𝒖 ∈ 𝐃 ⁢ ( A p 1 / 2 ) .

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

Proposition 3.12.

For all 1 < p < ∞ and all α ∈ R such that 0 < α < 3 2 ⁢ p the following Sobolev embedding holds:

(3.16) 𝐃 ⁢ ( A p α ) ↪ 𝑳 q ⁢ ( Ω ) , 1 q = 1 p - 2 ⁢ α 3 .

Moreover, for all 𝐮 ∈ D ⁢ ( A p α ) the following estimate holds:

(3.17) ∥ 𝒖 ∥ 𝑳 q ⁢ ( Ω ) ≤ C ⁢ ( Ω , p ) ⁢ ∥ ( I + A p ) α ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) .

In the particular case where the domain Ω is not obtained by rotation around a vector 𝐛 ∈ R 3 , the following estimate holds:

(3.18) ∥ 𝒖 ∥ 𝑳 q ⁢ ( Ω ) ≤ C ⁢ ( Ω , p ) ⁢ ∥ A p α ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) .

Proof.

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

𝐃 ⁢ ( A p α ) = 𝐃 ⁢ ( ( I + A p ) α ) = [ 𝐃 ⁢ ( I + A p ) ; 𝑳 σ , τ p ⁢ ( Ω ) ] α = [ 𝐃 ⁢ ( A p ) ; 𝑳 σ , τ p ⁢ ( 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 < 3 2 ⁢ p , 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 ⁢ ( A p α ) is equipped with the graph norm of the operator ( I + A p ) α .

In the particular case where the domain Ω is not obtained by rotation around a vector 𝒃 ∈ ℝ 3 , the operator A p α is an isomorphism from 𝐃 ⁢ ( A p α ) to 𝑳 σ , τ p ⁢ ( 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 L p - L q for the inhomogeneous Stokes problem (1.1).

Consider first the two problems

(4.1) { ∂ ⁡ 𝒖 ∂ ⁡ t + A p ⁢ 𝒖 = 𝒇 , div ⁡ 𝒖 = 0 in ⁢ Ω × ( 0 , T ) , 𝒖 ⋅ 𝒏 = 0 , [ 𝔻 ⁢ ( 𝒖 ) ⁢ 𝒏 ] 𝝉 = 𝟎 on ⁢ Γ × ( 0 , T ) , 𝒖 ⁢ ( 0 ) = 𝒖 0 in ⁢ Ω

and

(4.2) { ∂ ⁡ 𝒖 ∂ ⁡ t - Δ ⁢ 𝒖 + ∇ ⁡ π = 𝒇 , div ⁡ 𝒖 = 0 in ⁢ Ω × ( 0 , T ) , 𝒖 ⋅ 𝒏 = 0 , [ 𝔻 ⁢ ( 𝒖 ) ⁢ 𝒏 ] 𝝉 = 𝟎 on ⁢ Γ × ( 0 , T ) , 𝒖 ⁢ ( 0 ) = 𝒖 0 in ⁢ Ω ,

where 𝒖 0 ∈ 𝑳 σ , τ p ( Ω ) ) , 𝒇 ∈ C 1 ( 0 , T ; 𝑳 p ( Ω ) and 1 < p , q < ∞ . Notice that a function

𝒖 ∈ C ( ] 0 , + ∞ [ , 𝐃 ( A p ) ) ∩ C 1 ( ] 0 , + ∞ [ , 𝑳 σ , τ p ( Ω ) )

solves (4.1) if and only if there exists a function π ∈ C ( ] 0 , ∞ [ ; W 1 , p ( Ω ) / ℝ ) such that ( 𝒖 , π ) solves (4.2). Indeed, let 𝒖 be a solution to (4.1). Thus

A p ⁢ 𝒖 = - P ⁢ Δ ⁢ 𝒖 = 𝒇 - ∂ ⁡ 𝒖 ∂ ⁡ t ,

where P is the Helmholtz projection defined by (2.4) and (2.5). Since ( 𝒖 , 𝒇 - ∂ ⁡ 𝒖 ∂ ⁡ t ) ∈ 𝐃 ⁢ ( A p ) × 𝑳 σ , τ p ⁢ ( Ω ) , due to [7, Theorem 4.1] there exists π ∈ W 1 , p ⁢ ( Ω ) / ℝ such that

A p ⁢ 𝒖 = - Δ ⁢ 𝒖 + ∇ ⁡ π = - 𝒇 - ∂ ⁡ 𝒖 ∂ ⁡ t .

Moreover, we have the estimate

∥ 𝒖 ∥ 𝑾 2 , p ⁢ ( Ω ) / 𝓣 p ⁢ ( Ω ) + ∥ π ∥ W 1 , p ⁢ ( Ω ) / ℝ ≤ C ⁢ ( Ω , p ) ⁢ ( ∥ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) + ∥ ∂ ⁡ 𝒖 ∂ ⁡ t ∥ 𝑳 p ⁢ ( Ω ) ) ,

where 𝓣 p ⁢ ( Ω ) is the kernel of the Stokes operator with Navier-slip boundary condition described above. This means the mapping 𝒇 - ∂ ⁡ 𝒖 ∂ ⁡ t ↦ π is continuous from 𝑳 σ , τ p ⁢ ( Ω ) to W 1 , p ⁢ ( Ω ) . As a result, π ∈ C ( ] 0 , ∞ [ ; W 1 , p ( Ω ) / ℝ ) 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 𝒖 0 ∈ 𝑳 σ , τ p ⁢ ( Ω ) and when 𝒇 = 𝟎 , problem (1.1) has a unique solution ( 𝒖 , π ) satisfying

𝒖 ∈ C ( [ 0 , + ∞ [ , 𝑳 σ , τ p ( Ω ) ) ∩ C ( ] 0 , + ∞ [ , 𝐃 ( A p ) ) ∩ C 1 ( ] 0 , + ∞ [ , 𝑳 σ , τ p ( Ω ) ) ,

𝒖 ∈ C k ( ] 0 , + ∞ [ , 𝐃 ( A p ℓ ) ) for all k ∈ ℕ , ℓ ∈ ℕ ∗ ,

π ∈ C ( ] 0 , ∞ [ ; W 1 , p ( Ω ) / ℝ ) .

Moreover, the following estimates hold:

(4.3) ∥ 𝒖 ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) ≤ C ⁢ ( Ω , p ) ⁢ ∥ 𝒖 0 ∥ 𝑳 p ⁢ ( Ω ) ,

(4.4) ∥ ∂ ⁡ 𝒖 ⁢ ( t ) ∂ ⁡ t ∥ 𝑳 p ⁢ ( Ω ) ≤ C ⁢ ( Ω , p ) t ⁢ ∥ 𝒖 0 ∥ 𝑳 p ⁢ ( Ω ) ,

(4.5) ∥ 𝔻 ⁢ ( 𝒖 ) ∥ 𝑳 p ⁢ ( Ω ) ≤ C ⁢ ( Ω , p ) t ⁢ ∥ 𝒖 0 ∥ 𝑳 p ⁢ ( Ω ) .

Remark 4.1.

In the case where the domain Ω is not obtained by rotation around a vector b ∈ ℝ 3 , the Stokes semigroup decays exponentially and we can extend estimates (4.3)–(4.5) to the following L p - L q estimates. More precisely, for every p and q such that 1 < p ≤ q < ∞ , for every 𝒖 0 ∈ 𝑳 σ , τ p ⁢ ( Ω ) and 𝒇 = 𝟎 , there exists a constant δ > 0 such that the unique solution 𝒖 ⁢ ( t ) to problem (1.1) belongs to 𝑳 σ , τ q ⁢ ( Ω ) and satisfies

(4.6) ∥ 𝒖 ⁢ ( t ) ∥ 𝑳 q ⁢ ( Ω ) ≤ C ⁢ e - δ ⁢ t ⁢ t - 3 / 2 ⁢ ( 1 / p - 1 / q ) ⁢ ∥ 𝒖 0 ∥ 𝑳 p ⁢ ( Ω ) ,

(4.7) ∥ 𝔻 ⁢ ( 𝒖 ⁢ ( t ) ) ∥ 𝑳 q ⁢ ( Ω ) ≤ C ⁢ e - δ ⁢ t ⁢ t - 1 / 2 ⁢ t - 3 / 2 ⁢ ( 1 / p - 1 / q ) ⁢ ∥ 𝒖 0 ∥ 𝑳 p ⁢ ( Ω ) ,

(4.8) ∥ ∂ m ∂ ⁡ t m ⁢ A p n ⁢ 𝒖 ⁢ ( t ) ∥ 𝑳 q ⁢ ( Ω ) ≤ C ⁢ e - δ ⁢ t ⁢ t - ( m + n ) ⁢ t - 3 / 2 ⁢ ( 1 / p - 1 / q ) ⁢ ∥ 𝒖 0 ∥ 𝑳 p ⁢ ( Ω ) for all ⁢ m , n ∈ ℕ .

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 ∈ ℝ we have

∥ A α ⁢ 𝒖 ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) = ∥ A α ⁢ e - t ⁢ A p ⁢ 𝒖 0 ∥ 𝑳 p ⁢ ( Ω ) ≤ C ⁢ e - δ ⁢ t ⁢ t - α ⁢ ∥ 𝒖 0 ∥ 𝑳 p ⁢ ( Ω ) .

Consider now the non-homogeneous case, where 𝒖 0 = 𝟎 and 𝒇 ∈ L q ⁢ ( 0 , T ; 𝑳 σ , τ p ⁢ ( 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 𝒖 ∈ C ⁢ ( 0 , T ; 𝑳 σ , τ p ⁢ ( 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 L p - L q regularity:

𝒖 ∈ L q ⁢ ( 0 , T ; 𝐃 ⁢ ( A p ) ) , ∂ ⁡ 𝒖 ∂ ⁡ t ∈ L q ⁢ ( 0 , T ; L σ , τ p ⁢ ( Ω ) ) , π ∈ L q ⁢ ( 0 , T ; W 1 , p ⁢ ( Ω ) / ℝ ) .

In what follows we prove maximal L p - L q 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 < ∞ , 𝐟 ∈ 𝐋 q ⁢ ( 0 , T ; 𝐋 σ , τ p ⁢ ( Ω ) ) and 𝐮 0 = 0 . Problem (1.1) has a unique solution ( 𝐮 , π ) such that

(4.9) 𝒖 ∈ L q ⁢ ( 0 , T 0 ; 𝑾 2 , p ⁢ ( Ω ) ) , T 0 ≤ T ⁢ if ⁢ T < ∞ , T 0 < T ⁢ if ⁢ T = ∞ ,

π ∈ L q ⁢ ( 0 , T ; W 1 , p ⁢ ( Ω ) / ℝ ) , ∂ ⁡ 𝒖 ∂ ⁡ t ∈ L q ⁢ ( 0 , T ; 𝑳 p ⁢ ( Ω ) ) ,

(4.10) ∫ 0 T ∥ ∂ ⁡ 𝒖 ∂ ⁡ t ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t + ∫ 0 T ∥ A p ⁢ 𝒖 ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t + ∫ 0 T ∥ π ⁢ ( t ) ∥ W 1 , p ⁢ ( Ω ) / ℝ q ⁢ d t ≤ C ⁢ ( p , q , Ω ) ⁢ ∫ 0 T ∥ 𝒇 ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t .

Proof.

As stated above, problem (1.1) has a unique solution 𝒖 ∈ C ⁢ ( 0 , T ; 𝑳 σ , τ p ⁢ ( Ω ) ) . Let us prove that this solution satisfies the maximal L p - L q regularity (4.9). Indeed, let μ > 0 and set 𝒖 μ ⁢ ( t ) = e - 1 / μ ⁢ t ⁢ 𝒖 ⁢ ( t ) . The function 𝒖 μ ⁢ ( t ) is a solution to the following problem:

(4.11) { ∂ ⁡ 𝒖 μ ∂ ⁡ t + ( 1 μ ⁢ I + A p ) ⁢ 𝒖 μ ⁢ ( t ) = e - 1 μ ⁢ t ⁢ 𝒇 , div ⁡ 𝒖 μ ⁢ ( t ) = 0 in ⁢ Ω × ( 0 , T ) , 𝒖 μ ⁢ ( t ) ⋅ 𝒏 = 0 , [ 𝔻 ⁢ ( 𝒖 μ ⁢ ( t ) ) ⁢ 𝒏 ] τ = 𝟎 on ⁢ Γ × ( 0 , T ) , 𝒖 μ ⁢ ( 0 ) = 𝒖 ⁢ ( 0 ) = 𝟎 in ⁢ Ω .

Since the pure imaginary powers of the operator ( 1 μ ⁢ I + A p ) are bounded in 𝑳 σ , τ p ⁢ ( Ω ) (see Theorem 3.7) and since for all 1 < p < ∞ , 𝑳 σ , τ p ⁢ ( Ω ) is ζ-convex, we can apply the result of [16, Theorem 2.1] to the operator ( 1 μ ⁢ I + A p ) . Thus, the solution 𝒖 μ ⁢ ( t ) to problem (4.11) satisfies the following maximal L p - L q regularity:

𝒖 μ ∈ L q ⁢ ( 0 , T 0 ; 𝐃 ⁢ ( A p ) ) ∩ W 1 , q ⁢ ( 0 , T ; 𝑳 σ , τ p ⁢ ( Ω ) ) ,

with T 0 ≤ T if T < ∞ and T 0 < T if T = ∞ . Furthermore, 𝒖 μ ⁢ ( t ) satisfies the following estimate:

∫ 0 T ∥ ∂ ⁡ 𝒖 μ ∂ ⁡ t ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t + ∫ 0 T ∥ ( 1 μ ⁢ I + A p ) ⁢ 𝒖 μ ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t ≤ C ⁢ ( p , q , Ω ) ⁢ ∫ 0 T ∥ e - 1 μ ⁢ t ⁢ 𝒇 ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t

(4.12) ≤ C ⁢ ( p , q , Ω ) ⁢ ∫ 0 T ∥ 𝒇 ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t ,

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

𝒖 μ → 𝒖 in ⁢ L q ⁢ ( 0 , T 0 ; 𝐃 ⁢ ( A p ) ) ∩ W 1 , q ⁢ ( 0 , T ; 𝑳 σ , τ p ⁢ ( Ω ) ) as ⁢ μ → ∞ .

Thus the solution 𝒖 to problem (1.1) satisfies

(4.13) 𝒖 ∈ L q ⁢ ( 0 , T 0 ; 𝐃 ⁢ ( A p ) ) ∩ W 1 , q ⁢ ( 0 , T ; 𝑳 σ , τ p ⁢ ( Ω ) ) ,

with T 0 ≤ T if T < ∞ and T 0 < T if T = ∞ .

Using now the fact that A p ⁢ 𝒖 = - Δ ⁢ 𝒖 + ∇ ⁡ π = 𝒇 - ∂ ⁡ 𝒖 ∂ ⁡ t , thanks to [7, Theorem 4.1] we have

(4.14) ∥ 𝒖 ⁢ ( t ) ∥ 𝑾 2 , p ⁢ ( Ω ) / 𝓣 p ⁢ ( Ω ) + ∥ π ∥ W 1 , p ⁢ ( Ω ) / ℝ ≤ C ⁢ ( ∥ ∂ ⁡ 𝒖 ∂ ⁡ t ∥ 𝑳 p ⁢ ( Ω ) + ∥ 𝒇 ∥ 𝑳 p ⁢ ( Ω ) ) .

As a result we deduce that π ∈ L q ⁢ ( 0 , T ; W 1 , p ⁢ ( Ω ) / ℝ ) .

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

∥ 𝒗 ∥ 𝑾 2 , p ⁢ ( Ω ) ≃ ∥ 𝒗 ∥ 𝐃 ⁢ ( A p ) ≃ ∥ ( 1 μ ⁢ I + A p ) ⁢ 𝒗 ∥ 𝑳 p ⁢ ( Ω ) for all ⁢ 𝒗 ∈ 𝐃 ⁢ ( A p ) ,

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

(4.15) ∫ 0 T ∥ ∂ ⁡ 𝒖 μ ∂ ⁡ t ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t + ∫ 0 T ∥ 𝒖 μ ⁢ ( t ) ∥ 𝐃 ⁢ ( A p ) q ⁢ d t ≤ C ⁢ ( p , q , Ω ) ⁢ ∫ 0 T ∥ 𝒇 ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t ,

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

∫ 0 T ∥ ∂ ⁡ 𝒖 ∂ ⁡ t ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t + ∫ 0 T ∥ 𝒖 ⁢ ( t ) ∥ 𝐃 ⁢ ( A p ) q ⁢ d t ≤ C ⁢ ( p , q , Ω ) ⁢ ∫ 0 T ∥ 𝒇 ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) q ⁢ d t .

Finally, using (4.14) and the fact that ∥ 𝒖 ∥ 𝐃 ⁢ ( A p ) is equivalent to ∥ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) + ∥ A p ⁢ 𝒖 ∥ 𝑳 p ⁢ ( Ω ) , estimate (4.10) follows directly. We also note that the regularity (4.13) allows us to deduce that the operator ∂ ∂ ⁡ t + A p is an isomorphism from L q ⁢ ( 0 , T 0 ; 𝐃 ⁢ ( A p ) ) ∩ W 1 , q ⁢ ( 0 , T ; 𝑳 σ , τ p ⁢ ( Ω ) ) to L q ⁢ ( 0 , T ; 𝑳 σ , τ p ⁢ ( Ω ) ) . Thus estimate (4.10) follows directly. ∎

The boundedness of the pure imaginary powers of the operators λ ⁢ I + B p and λ ⁢ I + C p , with λ > 0 on the spaces [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ and [ 𝑻 p ′ ⁢ ( 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 [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ and [ 𝑻 p ′ ⁢ ( 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 ≤ ∞ , 𝐮 0 = 0 and let 𝐟 ∈ L q ⁢ ( 0 , T ; [ 𝐇 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ ) . The evolutionary Stokes problem (1.1) has a unique solution ( 𝐮 , π ) satisfying

𝒖 ∈ L q ⁢ ( 0 , T 0 ; 𝑾 1 , p ⁢ ( Ω ) ) , T 0 ≤ T ⁢ if ⁢ T < ∞ , T 0 < T ⁢ if ⁢ T = ∞ ,

π ∈ L q ( 0 , T ; L p ( Ω ) / ℝ ) , ∂ ⁡ 𝒖 ∂ ⁡ t ∈ L q ( 0 , T ; ∈ [ 𝑯 0 p ′ ( div Ω ) ] σ , τ ′ ) ,

(4.16) ∫ 0 T ∥ ∂ ⁡ 𝒖 ∂ ⁡ t ∥ [ 𝑯 0 p ′ ⁢ ( div ⁡ Ω ) ] ′ q ⁢ d t + ∫ 0 T ∥ B p ⁢ 𝒖 ⁢ ( t ) ∥ [ 𝑯 0 p ′ ⁢ ( div ⁡ Ω ) ] ′ q ⁢ d t + ∫ 0 T ∥ π ⁢ ( t ) ∥ L p ⁢ ( Ω ) / ℝ q ⁢ d t ≤ C ⁢ ( p , q , Ω ) ⁢ ∫ 0 T ∥ 𝒇 ⁢ ( t ) ∥ [ 𝑯 0 p ′ ⁢ ( div ⁡ Ω ) ] ′ q ⁢ d t .

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 1 μ ⁢ I + B p , μ > 0 , on [ 𝑯 0 p ′ ⁢ ( div , Ω ) ] σ , τ ′ and the change of variable 𝒖 μ ⁢ ( t ) = e - 1 / μ ⁢ t ⁢ 𝒖 ⁢ ( t ) , we obtain that problem (1.1) has a unique solution satisfying

𝒖 ∈ L q ⁢ ( 0 , T 0 ; 𝑾 1 , p ⁢ ( Ω ) ) ∩ W 1 , q ⁢ ( 0 , T ; [ 𝑯 0 p ′ ⁢ ( div ⁡ Ω ) ] σ , τ ′ ) ,

with T 0 ≤ T if T < ∞ and T 0 < T if T = ∞ . Next, using that B p ⁢ 𝒖 = - Δ ⁢ 𝒖 + ∇ ⁡ π = 𝒇 - ∂ ⁡ 𝒖 ∂ ⁡ t , thanks to [7, Theorems 3.7, 3.9] we have

∥ 𝒖 ⁢ ( t ) ∥ 𝑾 1 , p ⁢ ( Ω ) / 𝓣 p ⁢ ( Ω ) + ∥ π ∥ L p ⁢ ( Ω ) / ℝ ≤ C ⁢ ( ∥ ∂ ⁡ 𝒖 ∂ ⁡ t ∥ [ 𝑯 0 p ′ ⁢ ( div ⁡ Ω ) ] ′ + ∥ 𝒇 ∥ [ 𝑯 0 p ′ ⁢ ( div ⁡ Ω ) ] ′ ) .

As a result we deduce that π ∈ L q ⁢ ( 0 , T ; L p ⁢ ( Ω ) / ℝ ) and estimate (4.16) follows. ∎

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

Let 1 < p , q < ∞ , 0 < T ≤ ∞ , 𝐮 0 = 0 and let 𝐟 ∈ L q ⁢ ( 0 , T ; [ 𝐓 p ′ ⁢ ( Ω ) ] σ , τ ′ ) . The evolutionary Stokes problem (1.1) has a unique solution ( 𝐮 , π ) satisfying

𝒖 ∈ L q ⁢ ( 0 , T 0 ; 𝑳 p ⁢ ( Ω ) ) , T 0 ≤ T ⁢ if ⁢ T < ∞ , T 0 < T ⁢ if ⁢ T = ∞ ,

π ∈ L q ( 0 , T ; W - 1 , p ( Ω ) / ℝ ) , ∂ ⁡ 𝒖 ∂ ⁡ t ∈ L q ( 0 , T ; ∈ [ 𝑻 p ′ ( Ω ) ] σ , τ ′ ) ,

∫ 0 T ∥ ∂ ⁡ 𝒖 ∂ ⁡ t ∥ [ 𝑻 p ′ ⁢ ( Ω ) ] ′ q ⁢ d t + ∫ 0 T ∥ C p ⁢ 𝒖 ⁢ ( t ) ∥ [ 𝑻 p ′ ⁢ ( Ω ) ] ′ q ⁢ d t + ∫ 0 T ∥ π ⁢ ( t ) ∥ W - 1 , p ⁢ ( Ω ) / ℝ q ⁢ d t ≤ C ⁢ ( p , q , Ω ) ⁢ ∫ 0 T ∥ 𝒇 ⁢ ( t ) ∥ [ 𝑻 p ′ ⁢ ( Ω ) ] ′ q ⁢ d t .

Proof.

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

𝒖 ∈ L q ⁢ ( 0 , T 0 ; 𝑳 p ⁢ ( Ω ) ) ∩ W 1 , q ⁢ ( 0 , T ; [ 𝑻 p ′ ⁢ ( Ω ) ] σ , τ ′ ) ,

with T 0 ≤ T if T < ∞ and T 0 < T if T = ∞ . Next, using the fact C p ⁢ 𝒖 = - Δ ⁢ 𝒖 + ∇ ⁡ π = 𝒇 - ∂ ⁡ 𝒖 ∂ ⁡ t , thanks to [7, Theorem 5.5] we have

∥ 𝒖 ⁢ ( t ) ∥ 𝑳 p ⁢ ( Ω ) / 𝓣 p ⁢ ( Ω ) + ∥ π ∥ W - 1 , p ⁢ ( Ω ) / ℝ ≤ C ⁢ ( ∥ ∂ ⁡ 𝒖 ∂ ⁡ t ∥ [ 𝑻 p ′ ⁢ ( Ω ) ] ′ + ∥ 𝒇 ∥ [ 𝑻 p ′ ⁢ ( Ω ) ] ′ ) .

As a result we deduce that π ∈ L q ⁢ ( 0 , T ; W - 1 , p ⁢ ( Ω ) / ℝ ) and estimate (4.16) follows. ∎

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

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 ℋ ⁢ ( X 1 , X 0 ) the set of all A ∈ ℒ ⁢ ( X 1 , X 0 ) such that - A considered as a linear operator in X 0 with domain X 1 is the infinitesimal generator of a strongly continuous analytic semigroup ( e - t ⁢ A ) t ≥ 0 on X 0 .

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

X 1 ↪ 𝑑 X 0 .

The couple ( X 0 , X 1 ) is called a densely injected Banach couple and 𝔸 ∈ ℋ ⁢ ( X 1 , X 0 ) . Observe that the Banach space X 0 is the completion of X 1 in the norm u ↦ ∥ 𝔸 1 - 1 ⁢ u ∥ 1 = ∥ u ∥ 0 , with 𝔸 1 the X 1 -realization of 𝔸 . Notice that if 𝔸 is not invertible, it suffices to replace 𝔸 by λ ⁢ I + 𝔸 with λ > 0 .

The above description allows to introduce a superspace X - 1 := ( X - 1 , ∥ ⋅ ∥ - 1 ) of X 0 by choosing for X - 1 a completion of X 0 in the norm u ↦ ∥ u ∥ - 1 := ∥ 𝔸 - 1 ⁢ u ∥ 0 . Then ( X - 1 , X 0 ) is a densely injected Banach space as well and it is not difficult to show that 𝔸 0 := 𝔸 extends continuously to an operator 𝔸 - 1 ∈ ℋ ⁢ ( X 0 , X - 1 ) .

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

X 1 ↪ 𝑑 X α ↪ 𝑑 X β ↪ 𝑑 X - 1 , - 1 < β < α < 1 .

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

𝔸 α - 1 ∈ ℋ ⁢ ( X α , X α - 1 ) , 0 ≤ α ≤ 1 .

These extensions are natural in the sense that e - t ⁢ 𝔸 α - 1 is the restriction to X α - 1 of e - t ⁢ 𝔸 β - 1 for 0 ≤ β ≤ α ≤ 1 .

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

( 𝔸 α ) z ⁢ x = 𝔸 z ⁢ x for all ⁢ x ∈ X 1 ∩ X α + 1 .

Moreover, if 𝔸 belongs to the class ℬ ⁢ ℐ ⁢ 𝒫 ⁢ ( X 0 ) of operators with bounded imaginary powers on X 0 , then the interpolation-extrapolation scale ( X 0 , 𝔸 ) α ∈ ℝ is equivalent to the fraction power scale generated by ( X 0 , 𝔸 ) (see [4, Chapter V, Theorem 1.5.4]). If in addition 𝔸 belongs to the class ℬ ⁢ ℐ ⁢ 𝒫 ⁢ ( X 0 , M , ϑ ) of operators with uniformly bounded imaginary powers on X 0 with constant M and angle ϑ, then 𝔸 α ∈ ℬ ⁢ ℐ ⁢ 𝒫 ⁢ ( X 0 , 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:

(A.1) 𝒖 ⋅ 𝒏 = 0 , 𝑷 Γ ⁢ ( 𝔻 ⁢ ( 𝒖 ) ⁢ 𝒏 ) + α ⁢ 𝒖 = 𝟎 on ⁢ Γ × ( 0 , T ) ,

where

𝑷 Γ = I - 𝒏 ⊗ 𝒏

and α is the coefficient of friction. Their work is based on the theory of weighted L p -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:

𝒖 ⋅ 𝒏 = 0 , 𝑷 Γ ⁢ ( ℝ ⁢ ( 𝒖 ) ⁢ 𝒏 ) = 𝟎 on ⁢ Γ × ( 0 , T ) ,

with

ℝ ⁢ ( 𝒖 ) = ∇ ⁡ 𝒖 - ∇ T ⁡ 𝒖 .

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 A N ⁢ T be the Stokes operator with the Navier-type boundary conditions

𝒖 ⋅ 𝒏 = 0 , 𝐜𝐮𝐫𝐥 ⁡ 𝒖 × 𝒏 = 𝟎 on ⁢ Γ × ( 0 , T )

in 𝑳 σ , τ p ⁢ ( Ω ) . Using [2, Proposition 6.4], we know that

( λ ⁢ I + A N ⁢ T ) ∈ ℬ ⁢ ℐ ⁢ 𝒫 ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) , M , θ N ⁢ T ) , ∥ ( λ ⁢ I + A N ⁢ T ) i ⁢ s ∥ ℒ ⁢ ( 𝑳 σ , τ p ⁢ ( Ω ) ) ≤ M ⁢ e | s | ⁢ θ N ⁢ T

for λ > 0 with a constant M being independent of λ and 0 < θ N ⁢ T < π 2 .

Let us employ the Amann interpolation-extrapolation theory to the pair ( X 0 , A 0 ) where

A 0 = A N ⁢ T , X 0 = 𝑳 σ , τ p ⁢ ( Ω ) , X 1 = 𝐃 ⁢ ( A N ⁢ T ) .

This yields the scale ( X α , A α ) α ∈ ℝ with

A α = A 0 α = A N ⁢ T α , X α = 𝐃 ⁢ ( A 0 α ) , α > 0

and

X - α = ( X α ♯ ) ′ , α > 0 .

For a Banach space E we denote by E ♯ the dual space of E, i.e. E ♯ := E ′ . For α = - 1 2 , we obtain the isomorphism ( λ ⁢ I + A N ⁢ T ) - 1 / 2 : X 1 / 2 → X - 1 / 2 , where

X 1 / 2 = 𝑽 σ , τ p ⁢ ( Ω ) and X - 1 / 2 = [ 𝑽 σ , τ p ′ ⁢ ( Ω ) ] ∗ .

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

( λ ⁢ I + A N ⁢ T ) - 1 / 2 ∈ ℬ ⁢ ℐ ⁢ 𝒫 ⁢ ( X - 1 / 2 , M , θ - 1 / 2 )

and

∥ [ ( λ ⁢ I + A N ⁢ T ) - 1 / 2 ] i ⁢ s ∥ ℒ ⁢ ( [ 𝑽 σ , τ p ′ ⁢ ( Ω ) ] ∗ ) ≤ M ⁢ e | s | ⁢ θ - 1 / 2 ,

with a constant M being independent of λ and θ - 1 / 2 = θ N ⁢ T ∈ ( 0 , π 2 ) . In addition, the Stokes operator with the Navier-type boundary conditions (1.2) has the following natural weak formulation: For all 𝒖 ∈ 𝑽 σ , τ p ⁢ ( Ω ) and all 𝒗 ∈ 𝑽 σ , τ p ′ ⁢ ( Ω ) ,

〈 ( λ ⁢ I + A N ⁢ T ) W ⁢ 𝒖 , 𝒗 〉 Ω = λ ⁢ ∫ Ω 𝒖 ⋅ 𝒗 ⁢ d x + ∫ Ω 𝐜𝐮𝐫𝐥 ⁡ 𝒖 ⋅ 𝐜𝐮𝐫𝐥 ⁡ 𝒗 ⁢ d ⁢ x ,

with 〈 ⋅ , ⋅ 〉 Ω = 〈 ⋅ , ⋅ 〉 ( 𝑽 σ , τ p ′ ⁢ ( Ω ) ) ′ × 𝑽 σ , τ p ′ ⁢ ( Ω ) and A N ⁢ T , W the weak Stokes operator in [ 𝑽 σ , τ p ′ ⁢ ( Ω ) ] ∗ (the dual of 𝑽 σ , τ p ′ ⁢ ( Ω ) ). It can be shown that

A N ⁢ T , W := A - 1 / 2 and ( λ ⁢ I + A N ⁢ T ) W := ( λ ⁢ I + A N ⁢ T ) - 1 / 2 .

In addition ( λ ⁢ I + A N ⁢ T ) - 1 / 2 ∈ ℋ ⁢ ( 𝑽 σ , τ p ⁢ ( Ω ) , [ 𝑽 σ , τ p ′ ⁢ ( Ω ) ] ∗ ) . Observe that

X 1 W = X 1 / 2 = 𝑽 σ , τ p ⁢ ( Ω ) = 𝐃 ⁢ ( A N ⁢ T , W ) and X 0 W = X - 1 / 2 = [ 𝑽 σ , τ p ′ ⁢ ( Ω ) ] ∗ .

Next let us denote by A N the Stokes operator with the Navier-slip boundary conditions (1.1b) and by A N , W the weak Stokes operator with the same boundary conditions (1.1b). By using Corollary 2.1, an easy computation shows that for all 𝒖 ∈ 𝑽 σ , τ p ⁢ ( Ω ) and all 𝒗 ∈ 𝑽 σ , τ p ′ ⁢ ( Ω ) ,

〈 A N , W ⁢ 𝒖 , 𝒗 〉 Ω = ∫ Ω 𝐜𝐮𝐫𝐥 ⁡ 𝒖 ⋅ 𝐜𝐮𝐫𝐥 ⁡ 𝒗 ⁢ d ⁢ x + 〈 2 ⁢ 𝚲 ⁢ 𝒖 , 𝒗 〉 Γ = 〈 A N ⁢ T , W ⁢ 𝒖 , 𝒗 〉 Ω + 〈 2 ⁢ 𝚲 ⁢ 𝒖 , 𝒗 〉 Γ ,

with 〈 ⋅ , ⋅ 〉 Ω = 〈 ⋅ , ⋅ 〉 ( 𝑽 σ , τ p ′ ⁢ ( Ω ) ) ′ × 𝑽 σ , τ p ′ ⁢ ( Ω ) and 〈 ⋅ , ⋅ 〉 Γ = 〈 ⋅ , ⋅ 〉 𝑾 - 1 / p , p ⁢ ( Γ ) × 𝑾 1 / p , p ′ ⁢ ( Γ ) .

Similarly, for any λ > 0 , for any 𝒖 ∈ 𝑽 σ , τ p ⁢ ( Ω ) and 𝒗 ∈ 𝑽 σ , τ p ′ ⁢ ( Ω ) we have

〈 ( λ ⁢ I + A N , W ) ⁢ 𝒖 , 𝒗 〉 Ω = 〈 ( λ ⁢ I + A N ⁢ T , W ) ⁢ 𝒖 , 𝒗 〉 Ω + 〈 2 ⁢ 𝚲 ⁢ 𝒖 , 𝒗 〉 Γ .

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

( λ ⁢ I + A N , W ) ∈ ℬ ⁢ ℐ ⁢ 𝒫 ⁢ ( X 0 W , M , θ N ⁢ W )

and

∥ ( λ ⁢ I + A N , W ) i ⁢ s ∥ ℒ ⁢ ( [ 𝑽 σ , τ p ′ ⁢ ( Ω ) ] ∗ ) ≤ M ⁢ e | s | ⁢ θ N ⁢ W ,

where M is independent of λ and 0 < θ N ⁢ W < π 2 .

Consider now the interpolation-extrapolation scale ( X α W , 𝒜 α , W ) α ∈ ℝ generated by ( X 0 W , 𝒜 0 , W ) where

X 0 W = X - 1 / 2 = [ 𝑽 σ , τ p ′ ⁢ ( Ω ) ] ∗ , 𝒜 0 , W = A N , W , X 1 W = X 1 / 2 = 𝑽 σ , τ p ⁢ ( Ω ) = 𝐃 ⁢ ( A N ⁢ T , W ) .

It can be shown that

X 1 / 2 W = 𝑳 σ , τ p ⁢ ( Ω ) , 𝒜 1 / 2 , W = A N , 𝐃 ⁢ ( 𝒜 1 / 2 , W ) = X 3 / 2 W = 𝐃 ⁢ ( A N ) .

Since 𝒜 1 / 2 , W is the restriction of 𝒜 0 , W to X 3 / 2 W , it follows that for all λ > 0 , ( λ ⁢ I + A N ) ∈ ℬ ⁢ ℐ ⁢ 𝒫 ⁢ ( 𝑳 σ , τ p ⁢ ( 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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Received: 2017-01-22

Revised: 2017-04-04

Accepted: 2017-07-23

Published Online: 2017-08-15

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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

Abstract

1 Introduction

2 Preliminaries

2.1 Stokes operator

Corollary 2.1.

3 Fractional powers of the Stokes operator

Proposition 3.1.

Proof.

Corollary 3.2.

Proof.

Remark 3.3.

Proposition 3.4.

Proof.

Remark 3.5.

Corollary 3.6.

Proof.

Theorem 3.7.

Theorem 3.8.

Proof.

Theorem 3.9.

Theorem 3.10.

Proof.

Remark 3.11.

Proposition 3.12.

Proof.

4 Applications to the Stokes problem

Remark 4.1.

Theorem 4.2 (Strong solution to the Stokes problem).

Proof.

Theorem 4.3 (Weak solution to the Stokes problem).

Proof.

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

Proof.

A Appendix

A.1 Amann interpolation-extrapolation scales

A.2 Proof of Theorem 3.7

Acknowledgements

References

Journal and Issue

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Maximal L^p -L^q regularity to the Stokes problem with Navier boundary conditions