Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains

We present here different boundary conditions for the Navier-Stokes equations in 
bounded Lipschitz domains in $\mathbb{R}^3$, such as Dirichlet, Neumann or Hodge 
boundary conditions. 
We first study the linear Stokes operator associated to the boundary conditions. Then 
we show how the properties of the operator lead to local solutions or global solutions for 
small initial data.

1. Introduction. The aim of this paper is to describe how to find solutions of the Navier-Stokes equations in a bounded Lipschitz domain Ω ⊂ R 3 , and a time interval (0, T ) (T ≤ ∞), for initial data u 0 in a critical space, with one of the following boundary conditions on ∂Ω: • Dirichlet boundary conditions: • Neumann boundary conditions: [λ(∇u) + (∇u) ⊤ ]ν − πν = 0, λ ∈ (−1, 1], • Hodge boundary conditions: ν · u = 0, and ν × curl u = 0, where ν(x) denotes the unit exterior normal vector on a point x ∈ ∂Ω (defined almost everywhere when ∂Ω is a Lipschitz boundary). The strategy is to find a functional setting in which the Fujita-Kato scheme applies, such as in their fundamental paper [4]. The paper is organized as follows. In Section 2, we define the Dirichlet-Stokes operator and then show the existence of a local solution of the system { (1), (2) } for initial values in a critical space in the L 2 -Stokes scale. In Section 3, we adapt the previous proofs in the case of Neumann boundary conditions, i.e., for the system { (1), (3) } . In Section 4, we study (a slightly modified version of) the system { (1), (4) } for initial conditions in the critical space 2 SYLVIE MONNIAUX { u ∈ L 3 (Ω; R 3 ); div u = 0 in Ω, ν · u = 0 on ∂Ω } . Finally, in Section 5, we discuss some open problems related to this subject.
2. Dirichlet boundary conditions. For a more complete exposition of the results in this section, as well as an extension to more general domains, the reader can refer to [16] and [11]. The case where Ω is smooth was solved by Fujita and Kato in [4]. In [2], the case of bounded Lipschitz domains Ω was studied for initial data not in a critical space.
2.1. The linear Dirichlet-Stokes operator. We start with a remark about L 2 vector fields on Ω.
The space L 2 (Ω; R 3 ) is equal to the orthogonal direct sum H d and G = ∇H 1 (Ω; R). This follows from the following theorem.

Remark 2.
In the case of a bounded Lipschitz domain Ω ⊂ R 3 , the space H d coincides with the closure in L 2 (Ω; R 3 ) of the space of vector fields u ∈ C ∞ c (Ω; R 3 ) with div u = 0 in Ω.
We denote by J : H d ↩→ L 2 (Ω; R 3 ) the canonical embedding and P : L 2 (Ω; R 3 ) → H d the orthogonal projection. It is clear that PJ = Id H d . We define now the space is then an extension of the orthogonal projection P. We are now in the situation to define the Dirichlet-Stokes operator.
Definition 2.2. The Dirichlet-Stokes operator is defined as being the associated operator of the bilinear form

Proposition 1. The Dirichlet-Stokes operator
The third equality comes from the definition of weak derivatives in L 2 , the fourth equality comes from the fact that ∑ n j=1 ∂ 2 j = ∆. The last equality is due to the fact that J ′ 0 = P 1 . Therefore, A d u and P 1 (−∆)J 0 u are two linear forms which coincide on V d , they are then equal. So we proved here that . We have then, the equalities taking place in V ′ d , By de Rham's theorem, this implies that there exists p ∈ C ∞ c (Ω; R) ′ such that J(A d u) − (−∆)Ju = ∇p: ∇p ∈ H −1 (Ω; R 3 ), which implies that p ∈ L 2 (Ω; R).
The relations between the spaces and the operators are summarized in the following commutative diagram: In the case of a bounded Lipschitz domain Ω ⊂ R 3 , we also have the following property of D(A 3 4 d ); see [11,Corollary 5.5]. 3 4 d is continuously embedded into W 1,3 0 (Ω; R 3 ).

Proposition 2. The domain of A
2.2. The nonlinear Dirichlet-Navier-Stokes equations. The system λ Ω for the initial value x → λu 0 (λx). We are interested in finding "mild" solutions of the system { (1), (2) } for initial values u 0 in a critical space, in the same spirit as in [4].
λ Ω, λ > 0. It suffices to check that ∥u λ ∥ 2 = λ − 1 2 ∥u∥ 2 and ∥∇u λ ∥ 2 = λ 1 2 ∥∇u∥ 2 and apply the fact that D(A 1 4 d ) is the interpolation space (with coefficient and sup The fact that E T is a Banach space is straightforward. Assume now that u ∈ E T , and that (J 0 u, p) (with p ∈ L 2 (Ω; R)) satisfy . We can then apply P 1 to the equations and obtain for which a mild solution is given by the Duhamel formula: The strategy to find u ∈ E T satisfying u = α + ϕ(u, u) is to apply a fixed point theorem. We have then to make sure that E T is a "good" space for the problem, i.e., α ∈ E T and ϕ(u, u) ∈ E T . The fact that α ∈ E T follows directly from the properties of the Stokes operator A d and the semigroup (e −tA d ) t≥0 .

Proposition 3. The application ϕ : E T × E T → E T is bilinear, continuous and symmetric.
Proof. The fact that ϕ is bilinear and symmetric is immediate, once we have proved that it is well-defined. For u, v ∈ E T , let By the definition of E T and Sobolev embeddings, it is easy to see that and Therefore, we have The proof of the continuity of t → A 1 4 d ϕ(u, v)(t) on H d is straightforward once we have the estimate (11). The proof of the fact that is proved the same way, replacing A 1 4 d by A 3 4 d and using the fact that It remains to prove the estimate on the derivative with respect to t of ϕ(u, v)(t).
Let us rewrite f as defined in (9) as follows: where u⊗v denotes the matrix (u i v j ) 1≤i,j≤3 and ∇· acts on matrices M = (m i,j ) 1≤i,j≤3 the following way: hal-00651390, version 1 -13 Dec 2011 For all s ∈ (0, T ) we have where the first inequality comes from the fact that L 3 · L 6 ↩→ L 2 , the second comes from the Sobolev embeddings D(A We have and therefore where we used the estimates (10), (13), and the fact that −A d generates a bounded analytic semigroup, so that This last inequality together with (11) and (12)   3. Neumann boundary conditions. In this section, we study the system { (1), (3) } . We will only survey the results proved in [14], the method to prove existence of solutions being similar to what we done in Section 2.
In the case where λ ∈ (−1, 1], the bilinear form a λ is continuous, symmetric, coercive and sectorial. So its associated operator is self-adjoint, invertible and the negative generator of an analytic semigroup of contractions on H n . The following proposition is a consequence of the integration by parts formula (17) for Ψ ∈ H 1 (Ω) and ψ = Tr ∂Ω Ψ.
and sup The same tools as in 2.2 apply, so we can prove the following result (see [14, Theorem 11.3]).
and for u, v ∈ F T and t ∈ (0, T ), ds.
Going further, we may write 4. Hodge boundary conditions. Most of the results presented here are proved thoroughly in [12] for the linear theory and [13] for the nonlinear system. We start with the study of the linear Hodge-Laplacian on L p -spaces and then move to the Hodge-Stokes operator before applying the properties of this operator to prove the existence of mild solutions of the Hodge-Navier-Stokes system in L 3 .

The Hodge-Laplacian. Let
We start by defining on V × V the following form where ⟨·, ·⟩ denotes either the scalar or the vector-valued L 2 -pairing.
Since the form b is continuous, bilinear, symmetric, coercive and sectorial, the operator −B generates an analytic semigroup of contractions on H, B is self-adjoint and D(B 1 2 ) = V . Remark 5. As in Remark 1 for a bounded Lipschitz domain Ω and a vector field w ∈ H satisfying curl w ∈ H, we can define ν × w on ∂Ω in the following weak sense in where φ = Tr ∂Ω ϕ, the right hand-side of (23) depends only on φ on ∂Ω and not on the choice of ϕ, its extension to Ω.
To prove that B extends to L p -spaces, we prove that its resolvent admits L 2 − L 2 off-diagonal estimates. This was proved in [12, Section 6] Proposition 6. There exist two constants C, c > 0 such that for any open sets E, F ⊂ Ω such that dist (E, F ) > 0 and for all t > 0, f ∈ H and (24) Proof. We start by choosing a smooth cut-off function ξ : R 3 → R satisfying ξ = 1 on E, ξ = 0on F and ∥∇ξ∥ ∞ ≤ k dist (E,F ) . We then define η = e αξ where α > 0 is to be chosen later. Next, we take the scalar product of the equation with the function v = η 2 u. Since η = 1 on F and ∥u∥ 2 ≤ ∥χ F f ∥ 2 , it is easy to check then that and therefore, using the estimate on ∥∇ξ∥ ∞ and choosing α = dist (E,F ) 4kt , we obtain Using now the fact that η = e α on E, we finally get which gives (24) with C = √ 2 and c = 1 4k .

hal-00651390, version 1 -13 Dec 2011
With a slight modification of the proof, we can show that for all θ ∈ (0, π) there exist two constants C, c > 0 such that for any open sets E, F ⊂ Ω such that dist (E, F ) > 0 and for all z we have With that in hand and the Sobolev embedding (22), together with the rescaled Sobolev inequality where R = diam E, we can prove that, ) for x ∈ Ω and j ∈ N: where f j = χ Fj f .

Proposition 7.
There exists a constant C > 0 such that for all f ∈ L 2 (Ω; R 3 ) ∩ L 3 (Ω; R 3 ), z ∈ Σ π−θ , the following estimate holds: Proof. For x ∈ Ω and r > 0, denote by B Ω (x, r) the ball centered in x with radius r intersected with Ω. (27) and Fubini's theorem, keeping in mind that a Lipschitz domain in R n is a n-set in the terminology of [7] (which means that balls centered in Ω with radius r intersected with Ω have a volume equivalent to r n ), we have where we used the notation t = |z| − 1 2 and M denotes the Hardy-Littlewood maximal operator (which is bounded on L p for all p ∈ (1, ∞)).

Corollary 1. The semigroup (e −tB ) t≥0 extends to a bounded analytic semigroup on
Proof. For p = 3, this comes directly from Proposition 7. We obtain the result for all p ∈ [2,3] by interpolation and for all p ∈ [ 3 2 , 2] by duality (since the operator B is self-adjoint).
We can actually prove that the semigroup (e −tB ) t≥0 extends to a bounded analytic semigroup on L p (Ω; R 3 ) for p in an interval containing [ 6 5 , 6]. In an open interval (p Ω , q Ω ) containing [ 3 2 , 3], the negative generator B p of this semigroup satisfies To obtain estimates in L p for p > 3, the method is in the same spirit as what we have just done, combined with a bootstrap argument and regularity results for B. For a complete proof, the reader may refer to [12,Section 5].

The nonlinear Hodge-Navier-Stokes equations.
Granted that u is a sufficiently smooth vector field, we have the following identification (u · ∇)u = 1 2 ∇|u| 2 + u × curl u. That is, replacing π in (1) by π + 1 2 |u| 2 , the system Before trying to solve this system, we need some facts about the Hodge-Stokes operator. In [3], it was proved that the orthogonal projection P defined in Section 2 on L 2 (Ω; R 3 ) extends to a bounded projection on L p (Ω; R 3 ) for p in an open interval (p Ω , q Ω ) containing ; denote it by P p . In [12,Lemma 3.7], it was proved that P p and B p , the Hodge-Laplacian in L p (Ω; R 3 ) commute on D(B p ). This allows us to define the Hodge-Stokes operator A p on The results we proved for the Hodge-Laplacian naturally extend to the Hodge-Stokes operator as stated in the following theorem.
and ν × curl u = 0 on ∂Ω } We now rewrite the nonlinear Hodge-Navier-Stokes system for initial data in the ciritical space H 3 in the abstract form whenever p ∈ (p Ω , q Ω ), q ∈ (p, q Ω ) with 1 p − α 3 = 1 q for some α ∈ (0, 1). The proof of this results relies on the possibility to find an "inverse of the curl" modulo gradient vectors and uses results proved in [9].
With these properties of the Hodge-Stokes semigroup in hand, the following existence result for (31) is almost immediate. For T ∈ (0, ∞], we define the space G T by where ε > 0 is such that 3(1 + ε) < q Ω .

Theorem 4.2.
Let Ω ⊂ R 3 be a bounded Lipschitz domain and let u 0 ∈ H 3 . Let γ and Φ be defined by γ(t) = e −tAp u 0 , t ≥ 0, and for u, v ∈ G T , and t ∈ (0, T ), For a complete proof of this theorem, we refer to [13, Section 5].

5.
Remarks and open problems.
It is common to identify the Navier's slip boundary conditions (35) with the Hodge boundary conditions (4). This is true only on flat parts of the boundary. In the case of a C 2 domain Ω, it can be proved that (35) and (4) differ only by a zero-order term. For more informations on this subject, the interested reader could refer to [13, Section 2].

Open problems.
In the case of a smooth bounded domain in R n , it was proved by Y. Giga and T. Miyakawa in [6] that the Dirichlet-Navier-Stokes system admits a local mild solution for initial values in L n (critical space for the system in dimension n). Their method relies on the fact that the Dirichlet-Stokes operator, as defined in Section 2, extends to all L p spaces and is the negative generator of an analytic semigroup there, which was proved in [5]. The situation in Lipschitz domains is different. For instance, P. Deuring provided in [1] an example of a domain with one conical singularity such that the Dirichlet-Stokes semigroup does not extend to an analytic semigroup in L p for p large (or p small), away from 2. As already mentioned, E. Fabes, O. Mendez and M. Mitrea proved in [3] that the orthogonal projection P defined in Section 2 on L 2 (Ω; R 3 ) extends to a bounded projection on L p (Ω; R 3 ) for p in an open interval containing (if Ω is C 1 , then this interval is (1, ∞)). This led M. Taylor in [18] to formulate the conjecture that the Dirichlet-Stokes semigroup defined originally on H d extends to an analytic semigroup on L p for p in the same interval as in [3]. Remark 6. This conjecture is actually true when, instead of considering Dirichlet boundary conditions, we consider Hodge boundary conditions, as proved in Section 4.
In the same paper [3], the authors proved that the orthogonal projection P n defined in Section 3 on L 2 (Ω; R 3 ) also extends to a bounded projection on L p (Ω; R 3 ) for p in the same open interval containing ] . This leads to the conjecture similar to Taylor's that the Neumann-Stokes semigroup defined originally on H n extends to an analytic semigroup on L p for p in the same interval.
As for now, no positive result is known in L p for p ̸ = 2 for these two conjectures. To apply the Fujita-Kato scheme as in Sections 2 & 3, proving that the Stokes semigroup extends to an analytic semigroup in L 3 seems to be the first step to obtain mild solutions of the Navier-Stokes system with either Dirichlet or Neumann boundary conditions. hal-00651390, version 1 -13 Dec 2011