This paper studies maximal - regularity for the Stokes problem with Navier-slip boundary conditions
where is the stress tensor and Ω is a bounded domain of of class . A unit normal vector to the boundary can be defined almost everywhere; it is denoted by . Here and π denote respectively unknowns velocity field and the pressure of a fluid occupying the domain Ω, while and 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  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  who considered the stationary Stokes problem with the boundary conditions (1.1b) in bounded or unbounded domains of and proved the existence and regularity of solutions to this problem.
The author together with Amrouche and Rejaiba  proved the analyticity of the Stokes semigroup with the boundary conditions (1.1b) in -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 ) and proved the existence of strong, weak and very weak solutions to this problem. In this paper we shall prove maximal - regularity for the non-homogeneous case (i.e. problem (1.1) with ). 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 to certain elliptic and parabolic problem with initial data of low regularity was introduced by Lions and Magenes  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 regularity for the Stokes problem we can cite  by Solonnikov among the first works on this problem. He constructed a solution to the initial value Stokes problem with Dirichlet boundary conditions ( on ). His proof was based on methods in the theory of potentials. However, when Ω is not bounded, the result in  was not global in time. Later on, Giga and Sohr  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 in space and time. To derive such global maximal - 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 .
Saal considered in  the Stokes problem in spatial regions with moving boundary and proved maximal - 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 , Saal proved maximal - regularity for the Stokes problem with homogeneous Robin boundary conditions in the half space . To this end, he proved that the associated Stokes operator is sectorial and admits a bounded -calculus on . Nevertheless, as shown by Shimada in  the same approach can not be applied to the Stokes problem with non-homogeneous Robin boundary condition. For this reason Shimada derived the maximal - 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.  considered the realization of the Hodge Laplacian operator defined by
in a domain with a suitably smooth boundary. Geissert et al.  proved that for all , the realization of the operator admits a bounded -calculus. They also showed that in the case where Ω is simply connected their result is true for . 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 - regularity to magneto-hydrodynamic equation with perfectly conducting wall condition.
Following the results in , the author together with Amrouche and Escobedo proved in  maximal - 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  the Stokes problem (1.1a) with the following boundary conditions:
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 ). 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 . 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  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 spaces. For more information on the “Amann interpolation-extrapolation scales” we refer to . We note that Wilke and Prüss  studied the critical spaces for the Navier–Stokes equations with Navier boundary conditions. Their work is based on the theory of weighted -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  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 - 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.
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 .
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 given by
The operator P in (2.3) is the Helmholtz projection defined by
where is the unique solution of the following weak Neumann problem (cf. ):
An easy computation shows that
where is the unique solution up to an additive constant of the problem
Consider the space
Observe that the Stokes operator can be defined by the following weak formulation: for all and all we have
We also note (see [7, Theorem 3.9]) that the operator is sectorial and generates a bounded analytic semigroup on for all . We denote by the analytic semigroup associated to the operator in .
Next we consider the space
equipped with the graph norm. For every , the space is dense in (cf. [9, Section 2] and [6, Proposition 2.3]). In addition, for any function in the normal trace exists and belongs to and the closure of in is equal to
We have denoted by the set of infinitely differentiable functions with compact support in Ω and by the restriction to Ω of infinitely differentiable functions with compact support in . The dual space of can be characterized as follows (cf. [26, Proposition 1.0.4]): A distribution belongs to if and only if there exist and such that and
We consider the following space:
We note (cf. ) that for a function in the dual space such that , the normal trace value exists and belongs to the space .
The Stokes operator can be extended to the space (cf. [7, Section 3.2]). This extension is a closed linear densely defined operator
where is the unique solution up to an additive constant of the problem
The operator generates a bounded analytic semigroup on for all (see [7, Theorem 3.10]). We note that the trace value for a function in (2.7) is justified by the fact that for a function such that , the trace value exists and belongs to (see [7, Lemma 2.4]).
Consider also the following space:
Thanks to [8, Lemmas 4.11, 4.12] we know that is dense in and a distribution if and only if there exist and such that , with
We consider the subspace
We recall from  that for a function in the dual space such that the normal trace exists and belongs to .
The Stokes operator with Navier-slip boundary condition can also be extended to the space (see [7, Section 3.3]). This extension is a densely defined closed linear operator
and in Ω for all , with the unique solution up to an additive constant of the problem
The operator generates a bounded analytic semigroup on for all (see [7, Theorem 3.12]). We recall from [7, Lemma 5.4] that for a function such that , the trace value exists and belongs to . This give a meaning to the trace of a function 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 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 along each family of curves, respectively, are a possible system of coordinates in W. We denote by the unit tangent vectors to the boundary. With this notation, we have
where and . As a result for any the following formulas hold (see ):
In the particular case on Γ, the following equality holds:
It can be seen from (2.9) that the Navier-slip boundary (1.1b) differs from (1.2) only by the term 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).
For any vector such that and on Γ, we have
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 on . Since the Stokes operator in 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 , , are well, densely defined and closed linear operators on with domain . Furthermore,
We denote by the set of complex number, , and by the set of divergence free infinitely differentiable functions with compact support in Ω
Nevertheless, as described above, since the Stokes operator 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 . We start by the following proposition.
There exists an angle such that the resolvent set of the operator contains the sector
Moreover, the following estimate holds:
with a constant κ independent of λ.
Since the operator generates a bounded analytic semigroup on , the operator is an isomorphism from in . We recall that is given by (2.2). Let such that . It is clear that the operator is an isomorphism from to . Using [7, Theorem 3.8], one has, since ,
where the constant is independent of λ. This means that the resolvent set of the operator contains the set where the estimate (3.2) is satisfied. Using the result of [33, Chapter VIII, Theorem 1], we deduce that there exists an angle , such that the resolvent set of contains the sector . In addition for every such that estimate (3.1) holds. ∎
Let be as in Proposition 3.1.
holds for all with a constant C independent of λ.
Let be fixed and let such that and . Then one has
with a constant κ independent of λ.
Let and such that . Observe that satisfies
Using [7, Theorem 3.8], we have
Next let be fixed. Then for all such that and we have
Observe that estimate (3.4) holds for all . ∎
One may say that it is superfluous to prove an estimate of type (3.4) for the operator since . 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 .
Next we state and prove our results on the complex and pure imaginary powers of the operator . We start by the following proposition.
Let be as in Proposition 3.1. For all with we have
with a constant depending on .
Let such that . Thanks to Proposition 3.1 we know that the operator 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. ):
This means that
In addition, we know that , where is the principal argument of . An easy computation shows that
As a result we have
Next, we write in the form
In other words
Thanks to our assumption on z we can verify that the improper integrals and are convergent and satisfy
with a constant depending on .
Finally, substituting in (3.7), we have for all with ,
This completes the proof of the proposition. ∎
(i) We recall from [18, Propositions 4.7, 4.10] that for all the operator is analytic in z for .
(ii) Observe that if we replace in (3.6) by
with some constant , we obtain for all with ,
Let , taking the limit as tends to 0, we obtain that for all ,
Using then the density of in , we obtain estimate (3.9) for all .
Let and . The operator is bounded from to . Furthermore, there exists an angle and a constant such that
Let , and let such that in Ω. Using Proposition 3.4, we deduce that there exists an angle and a constant such that
Next observe that
Therefore one has estimate (3.10). This means that for all , the operator is bounded from to and satisfies estimate (3.10). We recall that the operator has a bounded inverse and thus for all with , the operator is an isomorphism from to (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 in , since the second term on the right-hand side of (3.8) blows up as 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 . 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 . 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  and it will be done in Appendix A.
There exist an angle and a constant such that for all and all ,
where M is independent of λ.
There exist and a constant such that for all and all ,
Using that , we deduce that
In the case where the domain Ω is not obtained by rotation around a vector , 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.
Suppose that the domain Ω is not obtained by rotation around a vector . There exist and a constant such that for all ,
Next we study the domains of fractional powers of the operator on . 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 and their norms. It follows from  that for , the domain does not depend on and coincides with for , that is,
We also know from [31, Theorem 1.15.3] that the boundedness of the pure imaginary powers of the operator allows us to determine the domain of definition of , and then of for any complex number α satisfying using complex interpolation theory. In addition for all , the map is a norm on . This is due to the fact (cf. [31, Theorem 1.15.2, part (e)]) that the operator has a bounded inverse, and thus for all with , the operator is an isomorphism from to .
The following theorem characterizes the domain of .
For all , (given by (2.6)) with equivalent norms.
Consider a function (see (2.2) for the definition of ) and set and . It is clear that if , then and . In addition, if , then and . Now, let , then
As a result, , , in Ω and on Γ. Thanks to , we know that for all , the norm is equivalent to . As a result and
Next we prove the second inclusion. Since has a bounded inverse, the operator is an isomorphism from to for all . This means that for all there exists a unique solution of
Let and observe that
We recall that is the unique solution of problem (3.14) and that the adjoint operator of is equal to the operator . We also recall that the dual of is equal to . Using the density of in , we obtain estimate (3.15) for all and then
In the case where the domain Ω is not obtained by rotation around a vector , the Stokes operator is invertible with bounded inverse and the following equivalence holds:
The following proposition gives us an embeddings of Sobolev type for the domains of fractional powers of the Stokes operator .
For all and all such that the following Sobolev embedding holds:
Moreover, for all the following estimate holds:
In the particular case where the domain Ω is not obtained by rotation around a vector , the following estimate holds:
Consider first the case where and recall that
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 , we proceed as in the proof of [2, Corollary 6.11]. This result is similar to the result of Borchers and Miyakawa  who proved the result for the Stokes operator with Dirichlet boundary conditions in exterior domains for .
In the particular case where the domain Ω is not obtained by rotation around a vector , the operator is an isomorphism from to . Thus one has estimate (3.18). ∎
4 Applications to the Stokes problem
Consider first the two problems
where , and . Notice that a function
Moreover, we have the estimate
where is the kernel of the Stokes operator with Navier-slip boundary condition described above. This means the mapping is continuous from to . As a result, and solves (4.2). Conversely, let be a solution of (4.2). Applying the Helmholtz projection P to the first equation of problem (4.2), one gets directly that solves (4.1).
For the homogeneous problem (i.e. ), the analyticity of the semigroup gives us a unique solution satisfying all the regularity desired. As stated in , when the initial data and when , problem (1.1) has a unique solution satisfying
Moreover, the following estimates hold:
In the case where the domain Ω is not obtained by rotation around a vector , the Stokes semigroup decays exponentially and we can extend estimates (4.3)–(4.5) to the following - estimates. More precisely, for every p and q such that , for every and , there exists a constant such that the unique solution to problem (1.1) belongs to and satisfies
Consider now the non-homogeneous case, where and , with and . It is well known (cf. ) that for such , problem (1.1) has a unique solution . It is also known that for such the analyticity of the Stokes semigroup is not enough to obtain a unique solution satisfying the following maximal - regularity:
Theorem 4.2 (Strong solution to the Stokes problem).
Let , , and . Problem (1.1) has a unique solution such that
As stated above, problem (1.1) has a unique solution . Let us prove that this solution satisfies the maximal - regularity (4.9). Indeed, let and set . The function is a solution to the following problem:
Since the pure imaginary powers of the operator are bounded in (see Theorem 3.7) and since for all , is ζ-convex, we can apply the result of [16, Theorem 2.1] to the operator . Thus, the solution to problem (4.11) satisfies the following maximal - regularity:
with if and if . Furthermore, satisfies the following estimate:
where the constant is independent of μ. In addition,
Thus the solution to problem (1.1) satisfies
with if and if .
Using now the fact that , thanks to [7, Theorem 4.1] we have
As a result we deduce that .
It remains to prove estimate (4.10). We recall first the following equivalence of norms:
independently of . Then, substituting in (4.12), we have
where the constant 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
Finally, using (4.14) and the fact that is equivalent to , estimate (4.10) follows directly. We also note that the regularity (4.13) allows us to deduce that the operator is an isomorphism from to . Thus estimate (4.10) follows directly. ∎
The boundedness of the pure imaginary powers of the operators and , with on the spaces and , 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 and 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 , , and let . The evolutionary Stokes problem (1.1) has a unique solution satisfying
Proceeding in the same way as in the proof of Theorem 4.2, using the boundedness of the pure imaginary powers of the operators , , on and the change of variable , we obtain that problem (1.1) has a unique solution satisfying
with if and if . Next, using that , thanks to [7, Theorems 3.7, 3.9] we have
As a result we deduce that and estimate (4.16) follows. ∎
Theorem 4.4 (Very weak solution to the Stokes problem).
Let , , and let . The evolutionary Stokes problem (1.1) has a unique solution satisfying
The proof is similar to the proof of Theorem 4.2. First we use the boundedness of the pure imaginary powers of , with on the space and the change of variable . We obtain then a unique solution to problem (1.1) satisfying
with if and if . Next, using the fact , thanks to [7, Theorem 5.5] we have
As a result we deduce that and estimate (4.16) follows. ∎
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 the set of all such that considered as a linear operator in with domain is the infinitesimal generator of a strongly continuous analytic semigroup on .
Let us fix a negative generator of a strongly continuous analytic semigroup on an arbitrary Banach space and assume for simplicity that has a bounded inverse on . Thus is a closed densely defined operator in having a non-empty resolvent set. If we denote by the domain of , then defines a norm on which is equivalent to the graph norm. Thus is a Banach space such that
The couple is called a densely injected Banach couple and . Observe that the Banach space is the completion of in the norm , with the -realization of . Notice that if is not invertible, it suffices to replace by with .
The above description allows to introduce a superspace of by choosing for a completion of in the norm . Then is a densely injected Banach space as well and it is not difficult to show that extends continuously to an operator .
Next, given , we choose an interpolation functor of exponent θ with the property that is dense in whenever is a densely injected Banach couple, an “admissible interpolation functor”. Then we put and . This defines a scale of Banach spaces
Furthermore, denoting by with the -realization of the , it follows that
These extensions are natural in the sense that is the restriction to of for .
This interpolation-extrapolation technique is rather flexible, has many applications and is crucial for our work. Indeed, let be as above and let be the interpolation-extrapolation scale generated by and , . It is well known that the complex power can be defined for every . Similarly we can define and
Moreover, if belongs to the class of operators with bounded imaginary powers on , then the interpolation-extrapolation scale is equivalent to the fraction power scale generated by (see [4, Chapter V, Theorem 1.5.4]). If in addition belongs to the class of operators with uniformly bounded imaginary powers on with constant M and angle ϑ, then (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  in another context of functional framework in the study of the critical spaces for the Navier–Stokes equations subject to the following boundary conditions:
and α is the coefficient of friction. Their work is based on the theory of weighted -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:
Let be the Stokes operator with the Navier-type boundary conditions
in . Using [2, Proposition 6.4], we know that
for with a constant M being independent of λ and .
Let us employ the Amann interpolation-extrapolation theory to the pair where
This yields the scale with
For a Banach space E we denote by the dual space of E, i.e. . For , we obtain the isomorphism , where
with a constant M being independent of λ and . In addition, the Stokes operator with the Navier-type boundary conditions (1.2) has the following natural weak formulation: For all and all ,
with and the weak Stokes operator in (the dual of ). It can be shown that
In addition . Observe that
Next let us denote by the Stokes operator with the Navier-slip boundary conditions (1.1b) and by the weak Stokes operator with the same boundary conditions (1.1b). By using Corollary 2.1, an easy computation shows that for all and all ,
with and .
Similarly, for any , for any and we have
where M is independent of λ and .
Consider now the interpolation-extrapolation scale generated by where
It can be shown that
Since is the restriction of to , it follows that for all , and that it satisfies (3.11).
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
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.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0