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

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

Sylvie Monniaux

To cite this version:

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz

domains. Discrete and Continuous Dynamical Systems - Series S, American Institute of Mathematical

Sciences, 2013, 6 (5), pp.1355-1369. �hal-00651390�

AIMS’ Journals

VolumeX, Number0X, XX200X pp.X–XX

VARIOUS BOUNDARY CONDITIONS FOR NAVIER-STOKES EQUATIONS IN BOUNDED LIPSCHITZ DOMAINS

Sylvie Monniaux

LATP - UMR 6632, Facult´e des Sciences de Saint-J´erˆome Case Cour A, Universit´e Paul C´ezanne

13397 Marseille Cedex 20, France

Abstract. We present here different boundary conditions for the Navier- Stokes equations in bounded Lipschitz domains inR³, such as Dirichlet, Neu- mann 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

 



 

∂

u − ∆u + ∇ π + (u · ∇ )u = 0 in (0, T ) × Ω, div u = 0 in (0, T ) × Ω,

u(0) = u

in Ω,

(1) in a bounded Lipschitz domain Ω ⊂ R

, and a time interval (0, T ) (T ≤ ∞ ), for initial data u

in a critical space, with one of the following boundary conditions on

∂Ω:

• Dirichlet boundary conditions:

u = 0, (2)

• Neumann boundary conditions:

[λ( ∇ u) + ( ∇ u)

]ν − πν = 0, λ ∈ ( − 1, 1], (3)

• Hodge boundary conditions:

ν · u = 0, and ν × curl u = 0, (4) 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 fun- damental 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

-Stokes scale. In Section 3, we adapt the previous proofs in the case of Neumann boundary con- ditions, i.e., for the system {

(1), (3) }

. In Section 4, we study (a slightly mod- ified version of) the system {

(1), (4) }

for initial conditions in the critical space

2000Mathematics Subject Classification. Primary: 35Q30, 35Q35, 35J25.

Key words and phrases. Stokes operator, boundary conditions, Dirichlet, Neumann, Hodge, Lipschitz domains, Navier-Stokes equations.

1

{ u ∈ L

(Ω; R

); 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

vector fields on Ω.

Remark 1. For Ω ⊂ R

a bounded Lipschitz domain, let u ∈ L

(Ω; R

) such that div u ∈ L

(Ω; R ). Then we can define ν · u on ∂Ω in the following weak sense in H

(∂Ω; R ): for ϕ ∈ H

(Ω; R ),

⟨ u, ∇ ϕ ⟩

+ ⟨ div u, ϕ ⟩

= ⟨ ν · u, φ ⟩

(5) where φ = Tr

ϕ, the right hand-side of (5) depends only on φ on ∂Ω and not on the choice of ϕ, its extension to Ω. The notation ⟨· , ·⟩

is for the L

-scalar product on E.

The space L

(Ω; R

) is equal to the orthogonal direct sum H

⊕

G where H

= {

u ∈ L

(Ω; R

); div u = 0 in Ω, ν · u = 0 on ∂Ω }

(6) and G = ∇ H

(Ω; R ). This follows from the following theorem.

Theorem 2.1 (de Rham). Let T be a distribution in C

(Ω; R

)

such that ⟨ T, ϕ ⟩ = 0 for all ϕ ∈ C

(Ω; R

) with div ϕ = 0 in Ω. Then there exists a distribution S ∈ C

(Ω; R )

such that T = ∇ S. Conversely, if T = ∇ S with S ∈ C

(Ω; R )

, then ⟨ T, ϕ ⟩ = 0 for all ϕ ∈ C

(Ω; R

) with div ϕ = 0 in Ω.

Remark 2. In the case of a bounded Lipschitz domain Ω ⊂ R

, the space H

coincides with the closure in L

(Ω; R

) of the space of vector fields u ∈ C

(Ω; R

) with div u = 0 in Ω.

We denote by J : H

↩ → L

(Ω; R

) the canonical embedding and P : L

(Ω; R

) → H

the orthogonal projection. It is clear that P J = Id

. We define now the space V

= H

(Ω; R

) ∩ H

. The embedding J restricted to V

maps V

to H

(Ω; R

): we denote it by J

: V

↩ → H

(Ω; R

). Its adjoint J

= P

: H

(Ω; R

) → V

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

a : V

× V

→ R , a(u, v) =

∑

⟨ ∂

J

u, ∂

J

v ⟩ .

Proposition 1. The Dirichlet-Stokes operator A

is the part in H

of the bounded operator A

: V

→ V

defined by A

u : V

→ R , (A

u)(v) = a(u, v), and satisfies

D(A

) = {

u ∈ V

; P

( − ∆

)J

u ∈ H

} ,

A

u = P

( − ∆

)J

u u ∈ D(A

),

where ∆

denotes the weak vector-valued Dirichlet-Laplacian in L

(Ω; R

). The operator A

is self-adjoint, invertible, − A

generates an analytic semigroup of con- tractions on H

, D(A

1 2

d

) = V

and for all u ∈ D(A

), there exists π ∈ L

(Ω; R ) such that

J A

u = − ∆J

u + ∇ π (7)

and D(A

) admits the following description D(A

) = {

u ∈ V

; ∃ π ∈ L

(Ω; R ) : − ∆J

u + ∇ π ∈ H

} .

Proof. By definition, for u ∈ D(A

), we have, for all v ∈ V

,

⟨ A

u, v ⟩ = a(u, v) =

∑

⟨ ∂

J

u, ∂

J

v ⟩

= −

∑

H⁻¹

⟨ ∂

J

u, J

v ⟩

=

⟨ ( − ∆)J

u, J

v ⟩

=

d

⟨P

( − ∆)J

u, v ⟩

.

The third equality comes from the definition of weak derivatives in L

, the fourth equality comes from the fact that ∑

j=1

∂

= ∆. The last equality is due to the fact that J

= P

. Therefore, A

u and P

( − ∆)J

u are two linear forms which coincide on V

, they are then equal. So we proved here that A

= P

( − ∆)J

: V

→ V

. Moreover, the fact that u ∈ D(A

) implies that A

u is a linear form on H

, so that the linear form P

( − ∆)J

u, originally defined on V

, extends to a linear form on H

(since V

is dense in H

by de Rham’s theorem). The fact that A

is self-adjoint and − A

generates an analytic semigroup of contractions comes from the properties of the form a: a is bilinear, symmetric, sectorial of angle 0, coercive on V

× V

. The property that D(A

1 2

d

) = V

is due to the fact that A

is self-adjoint, applying a result by J.L. Lions [8, Th´ eor` eme 5.3].

To prove the last assertions of this proposition, let u ∈ D(A

). Then A

u ∈ H

and P

J (A

u) = P J (A

u) = u. Moreover, if u ∈ D(A

), u belongs, in particular, to V

. Therefore, J

u ∈ H

(Ω; R

) and ( − ∆)J

u ∈ H

(Ω; R

). We have then, the equalities taking place in V

,

P

( J (A

u) − ( − ∆)J

u )

= P

J (A

u) − P

( − ∆)J

u = A

u − A

u = 0.

By de Rham’s theorem, this implies that there exists p ∈ C

(Ω; R )

such that J (A

u) − ( − ∆) ˜ J u = ∇ p: ∇ p ∈ H

(Ω; R

), which implies that p ∈ L

(Ω; R ).

The relations between the spaces and the operators are summarized in the fol- lowing commutative diagram:

V

A_0,d

 _

d



// H

 _

(−∆^Ω_D)

H

 _

d



// L

P=J^′

oo  _

V

H

P1=J₀^′

oo

In the case of a bounded Lipschitz domain Ω ⊂ R

, we also have the following property of D(A

3 4

d

); see [11, Corollary 5.5].

Proposition 2. The domain of A

is continuously embedded into W

(Ω; R

).

2.2. The nonlinear Dirichlet-Navier-Stokes equations. The system {

(1), (2) } is invariant under the scaling u

(t, x) = λ u(λ

t, λx), (λ

t, λx) ∈ (0, T ) × Ω (λ > 0):

if u is a solution of {

(1), (2) }

in (0, T ) × Ω for the initial value u

, then u

is a solution of {

(1), (2) } in (

0,

) ×

Ω for the initial value x 7→ λu

(λx).

We are interested in finding “mild” solutions of the system {

(1), (2) }

for initial values u

in a critical space, in the same spirit as in [4].

Lemma 2.3. The space D(A

1 4

d

) is a critical space for the Navier-Stokes equations.

Proof. We have to prove that D(A

) is invariant under the scaling u

(x) = λu

(λx) for x ∈

Ω, λ > 0. It suffices to check that ∥ u

∥

= λ

∥ u ∥

and ∥∇ u

∥

= λ

∥∇ u ∥

and apply the fact that D(A

1 4

d

) is the interpolation space (with coefficient

1

2

) between H

and V

= D(A

1 2

d

).

For T > 0, define the space E

by E

=

{

u ∈ C

([0, T ]; D(A

)); u(t) ∈ D(A

), u

(t) ∈ D(A

) for all t ∈ (0, T ] and sup

t∈(0,T)

∥ t

A

u(t) ∥

+ sup

t∈(0,T)

∥ tA

u

(t) ∥

< ∞ } endowed with the norm

∥ u ∥

= sup

t∈(0,T)

∥ A

u(t) ∥

+ sup

t∈(0,T)

∥ t

A

u(t) ∥

+ sup

t∈(0,T)

∥ tA

u

(t) ∥

. The fact that E

is a Banach space is straightforward. Assume now that u ∈ E

, and that (J

u, p) (with p ∈ L

(Ω; R )) satisfy {

(1), (2) }

in H

(Ω; R

): indeed, every term ∇ p, ∂

J

u, − ∆J

u and (J

u ·∇ )J

u independently belong to H

(Ω; R

). We can then apply P

to the equations and obtain

u

(t) + A

u(t) = −P

( (J

u · ∇ )J

u )

since P

∇ p = 0 and P

( − ∆)J

u = A

u. We have then reduced the problem { (1), (2) }

into the abstract Cauchy problem u

(t) + A

u(t) = −P

( (J

u · ∇ )J

u )

u(0) = u

, u ∈ E

, (8)

for which a mild solution is given by the Duhamel formula: u = α + ϕ(u, u), where α(t) = e

u

and

ϕ(u, v)(t) =

∫

e

( −

P

( (J

u(s) · ∇ )J

v(s) + (J

v(s) · ∇ )J

u(s) )) ds.

The strategy to find u ∈ E

satisfying u = α + ϕ(u, u) is to apply a fixed point

theorem. We have then to make sure that E

is a “good” space for the problem, i.e.,

α ∈ E

and ϕ(u, u) ∈ E

. The fact that α ∈ E

follows directly from the properties

of the Stokes operator A

and the semigroup (e

)

.

Proposition 3. The application ϕ : E

× E

→ E

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

, let

f (t) = −

P

( (J

u(t) · ∇ )J

v(t) + (J

v(t) · ∇ )J

u(t) )

, t ∈ (0, T ). (9) By the definition of E

and Sobolev embeddings, it is easy to see that

(J

u(t) · ∇ )J

v(t) + (J

v(t) · ∇ )J

u(t) ∈ L

(Ω; R

) and (J

u(t) · ∇ )J

v(t) + (J

v(t) · ∇ )J

u(t)

2

≤ C t

∥ u ∥

∥ v ∥

where C is a constant independent from t. Indeed, by Proposition 2, if u, v ∈ E

, then ∇ u, ∇ v ∈ L

(Ω, R

) with the estimates

∥∇ u(t) ∥

≤ t

∥ u ∥

and ∥∇ v(t) ∥

≤ t

∥ v ∥

for all t > 0.

Moreover, since D(A

) ↩ → L

(Ω; R

), we also have

∥ u(t) ∥

≤ t

∥ u ∥

and ∥ v(t) ∥

≤ t

∥ v ∥

for all t > 0.

This, combined with the fact that L

· L

↩ → L

, gives the following estimate f(t)

≤ C t

∥ u ∥

∥ v ∥

for all t > 0. (10) Therefore, we have

∥ A

1 4

d

ϕ(u, v)(t) ∥

≤

∫

∥ A

1 4

d

e

∥

C s

∥ u ∥

∥ v ∥

ds

≤ C (∫

0

(t − s)

s

ds

) ∥ u ∥

∥ v ∥

, and since ∫

0

(t − s)

s

ds = ∫

0

(1 − s)

s

ds, we finally obtain the estimate

∥ A

1 4

d

ϕ(u, v)(t) ∥

≤ C ∥ u ∥

∥ v ∥

. (11) The proof of the continuity of t 7→ A

1 4

d

ϕ(u, v)(t) on H

is straightforward once we have the estimate (11). The proof of the fact that

∥ √ tA

3 4

d

ϕ(u, v)(t) ∥

≤ C ∥ u ∥

∥ v ∥

(12) is proved the same way, replacing A

1 4

d

by A

3 4

d

and using the fact that

∥ A

e

∥

≤ C (t − s)

and ∫

0

(t − s)

s

ds = t

∫

(1 − s)

s

ds.

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:

f (s) = −

P

∇ · (

J

u(s) ⊗ J

v(s) + J

v(s) ⊗ J

u(s) )

where u ⊗ v denotes the matrix (u

v

)

and ∇· acts on matrices M = (m

)

the following way:

∇ · M = ( ∑

i=1

∂

m

)

1≤j≤3

.

For u, v ∈ E

and s ∈ (0, T ), we have f

(s) = −

P

∇ · (

J u

(s) ⊗ J

v(s) + J

u(s) ⊗ J v

(s) + J v

(s) ⊗ J

u(s) + J

v(s) ⊗ J u

(s)

)

For all s ∈ (0, T ) we have

s

∥ J u

(s) ⊗ J

v(s) ∥

≤ ∥ sJ u

(s) ∥

∥ s

J

v(s) ∥

≤ ∥ sA

u

(s) ∥

∥ s

A

v(s) ∥

≤ ∥ u ∥

∥ v ∥

,

where the first inequality comes from the fact that L

· L

↩ → L

, the second comes from the Sobolev embeddings D(A

) ↩ → L

(Ω; R

) and D(A

) ↩ → L

(Ω; R

) and the third inequality follows directly from the definition of the space E

. Of course the same occurs for the other three terms J

u(s) ⊗ J v

(s), J v

(s) ⊗ J

u(s) and J

v(s) ⊗ J u

(s). Therefore, since A

1 2

d

maps V

to H

, we obtain sup

0<s<T

∥ s

A

f

(s) ∥

≤ c ∥ u ∥

∥ v ∥

. (13) We have

ϕ(u, v)(t) =

∫

0

e

f (t − s)ds +

∫

0

e

f (s)ds t ∈ (0, T ), and therefore

ϕ(u, v)

(t) = e

f (

) +

∫

0

A

1 2

d

e

A

1 2

0,d

f

(t − s)ds +

∫

0

− A

e

f(s)ds, which yields

∥ A

ϕ(u, v)

(t) ∥

≤ c t

f (

)

2

+ c (∫

0

1 s

1 (t − s)

ds

) ∥ u ∥

∥ v ∥

+ c (∫

0

1 (t − s)

1 s

ds

) ∥ u ∥

∥ v ∥

≤ c t

( 1 +

∫

0

dσ (1 − σ)

σ

) ∥ u ∥

∥ v ∥

,

where we used the estimates (10), (13), and the fact that − A

generates a bounded analytic semigroup, so that ∥ A

e

∥

≤ C t

. This last inequality together with (11) and (12) ensure that ϕ(u, v) ∈ E

whenever u, v ∈ E

.

We conclude this section by applying Picard’s fixed point theorem (see e.g. [17, Theorem A.1]) to obtain the following existence result for the system {

(1), (2) } . Theorem 2.4. Let Ω ⊂ R

be a bounded Lipschitz domain and let u

∈ D(A

1 4

d

).

Let α and ϕ be defined as above.

(i) If ∥ A

1 4

d

u

∥

is small enough, then there exists a unique u ∈ E

solution of

u = α + ϕ(u, u).

(ii) For all u

∈ D(A

), there exists T > 0 and a unique u ∈ E

solution of u = α + ϕ(u, u).

Uniqueness in the larger space C

([0, T ); D(A

)) can be obtained, applying [15, Theorem 1.1].

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 so- lutions being similar to what we done in Section 2.

3.1. The linear Neumann-Stokes operator. The boundary conditions (3) are indexed by λ ∈ ( − 1, 1]: if λ = 1, (3) becomes

T(u, π)ν = 0 on (0, T ) × ∂Ω, (14) where T (u, π) = ∇ u + ( ∇ u)

− πId denotes the stress tensor; if λ = 0, (3) becomes

∂

u − πν = 0 on (0, T ) × ∂Ω. (15) Before defining the Neumann-Stokes operator, we need the following integration by parts formula.

Lemma 3.1. Let λ ∈ R , u, w : Ω → R

, π, ρ : Ω → R sufficiently nice functions defined on the Lipschitz domain Ω ⊂ R

. Let L

u = ∆u + λ ∇ (div u) and define the conormal derivative

∂

(u, π) = (

∇ u + ( ∇ u)

)

ν − πν on ∂Ω. (16)

Then the following integration by parts formula hold

∫

Ω

(L

u − ∇ π) · w dx = −

∫

Ω

[ I

( ∇ u, ∇ w) − π div w ] dx

+

∫

∂Ω

∂

(u, π) · w dσ (17)

=

∫

Ω

(L

w − ∇ ρ) · u dx +

∫

Ω

[ π div w − ρ div u ] dx

+

∫

∂Ω

[ ∂

(u, π) · w − ∂

(w, ρ) · u ]

dσ, (18)

where I

(ξ, ζ) =

∑

(ξ

ζ

+ λξ

ζ

), for ξ = (ξ

)

and ζ = (ζ

)

. Recall that ∇ u = (∂

u

)

.

The space L

(Ω; R

) admits the following orthogonal decomposition: H

⊕

G

, where G

= {

∇ π; π ∈ H

(Ω; R ) } and H

= {

u ∈ L

(Ω; R

); div u = 0 }

. (19)

Following the steps of the previous section, we define V

= H

(Ω; R

) ∩ H

and

J

: H

↩ → L

(Ω; R

) the canonical embedding, P

= J

: L

(Ω; R

) → H

the

orthogonal projection, ˜ J

: V

↩ → H

(Ω; R

) the restriction of J

on V

and ˜ J

=

P ˜

: (H

(Ω; R

))

→ V

, extension of P

to (H

(Ω; R

))

. We can now define the

Neumann-Stokes operator.

Definition 3.2. Let λ ∈ R . The Neumann-Stokes operator is defined as being the associated operator of the bilinear form

a

: V

× V

→ R , a

(u, v) =

∫

Ω

I

( ∇ J ˜

u, ∇ J ˜

v) dx

In the case where λ ∈ ( − 1, 1], the bilinear form a

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

.

The following proposition is a consequence of the integration by parts formula (17), [14, Theorem 6.8] and [8, Th´ eor` eme 5.3].

Proposition 4. Let λ ∈ ( − 1, 1]. The Neumann-Stokes operator A

is the part in H

of the bounded operator A

: V

→ V

defined by (A

u)(v) = a

(u, v).

The operator A

is self-adjoint, invertible, − A

generates an analytic semigroup of contractions on H

, D(A

1 2

λ

) = V

and for all u ∈ D(A

), there exists π ∈ L

(Ω; R ) such that

J

A

u = − ∆ ˜ J

u + ∇ π (20) and D(A

) admits the following description

D(A

) = {

u ∈ V

; ∃ π ∈ L

(Ω; R ) : f = − ∆ ˜ J

u + ∇ π ∈ H

and ∂

(u, π)

= 0 } , where ∂

(u, π)

is defined in a weak sense for all f ∈ (H

(Ω; R

))

by

⟨ ∂

(u, π)

, ψ ⟩

=

⟨ f, Ψ ⟩

+

∫

Ω

I

( ∇ J ˜

u, ∇ Ψ) dx −

⟨ π, div Ψ ⟩

for Ψ ∈ H

(Ω) and ψ = Tr

Ψ.

Remark 3. If f ∈ (H

(Ω; R

))

, the quantity ∂

(u, π)

exists in the Besov space B

−¹2

(∂Ω; R

) = H

(∂Ω, R

) according to [14, Proposition 3.6].

Thanks to [14, Sections 9 & 10], we have a good description of the domain of fractional powers of the Neumann-Stokes operator A

. In particular, in [14, Corollary 10.6] it was established that

D(A

3 4

λ

) is continuously embedded into W

(Ω; R

). (21) 3.2. The nonlinear Neumann-Navier-Stokes equations. The results in 3.1 allow us to prove a result similar to Theorem 2.4 for the system {

(1), (3) } . As in the previous section, it is not difficult to see that D(A

) ↩ → L

(Ω; R

) is a critical space for the system. For T ∈ (0, ∞ ], following the definition of E

in Section 2, we define

F

= {

u ∈ C

([0, T ]; D(A

1 4

λ

)); u(t) ∈ D(A

3 4

λ

), u

(t) ∈ D(A

1 4

λ

) for all t ∈ (0, T ] and sup

t∈(0,T)

∥ t

A

u(t) ∥

+ sup

t∈(0,T)

∥ tA

u

(t) ∥

< ∞ } endowed with the norm

∥ u ∥

= sup

t∈(0,T)

∥ A

1 4

λ

u(t) ∥

+ sup

t∈(0,T)

∥ t

A

3 4

λ

u(t) ∥

+ sup

t∈(0,T)

∥ tA

1 4

λ

u

(t) ∥

.

The same tools as in 2.2 apply, so we can prove the following result (see [14, Theo-

rem 11.3]).

Theorem 3.3. Let Ω ⊂ R

be a bounded Lipschitz domain and let u

∈ D(A

).

Let β and ψ be defined by

β(t) = e

u

, t ≥ 0, and for u, v ∈ F

and t ∈ (0, T ),

ψ(u, v)(t) =

∫

e

( −

P

) (

(J

u(s) · ∇ ) ˜ J

v(s) + J

v(s) · ∇ ) ˜ J

u(s) ) ds.

(i) If ∥ A

u

∥

is small enough, then there exists a unique u ∈ F

solution of u = β + ψ(u, u).

(ii) For all u

∈ D(A

1 4

λ

), there exists T > 0 and a unique u ∈ F

solution of u = β + ψ(u, u).

A comment here may be necessary to link the solution u obtained in Theorem 3.3 and a solution of the system {

(1), (3) }

. If u ∈ F

, then u

∈ H

and (J

u ·∇ ) ˜ J

u ∈ L

(Ω; R

). Moreover, if u satisfies the equation u = β + ψ(u, u), then u is a mild solution of

A

u = − u

− P

( (J

u · ∇ ) ˜ J

u )

∈ H

. Going further, we may write

J

P

( (J

u · ∇ ) ˜ J

u )

= (J

u · ∇ ) ˜ J

u − ∇ q where q ∈ H

(Ω; R ) satisfies

∆q = div (J

u · ∇ ) ˜ J

u) ∈ H

(Ω; R

).

Therefore, we have by definition of A

, there exists π ∈ L

(Ω, R ) such that

− ∆ ˜ J

u + ∇ π = J

(A

u) = − J

u

− (J

u · ∇ ) ˜ J

u + ∇ q

and at the boundary, (u, π) satisfies (3) in the weak sense as in Proposition 4. Since q ∈ H

(Ω; R ), (u, π − q) satisfies also (3). This proves that (u, π − q) is a solution of the system {

(1), (3) } .

The uniqueness is true in a larger space than F

: for each u

∈ D(A

), there is at most one u ∈ C

([0, T ); D(A

)), mild solution of the system {

(1), (3) } . For a more precise statement, see [14, Theorem 11.8].

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

-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

.

4.1. The Hodge-Laplacian. Let H = L

(Ω; R

) and V = {

u ∈ H; curl u ∈ H, div u ∈ L

(Ω; R ) and ν · u = 0 on ∂Ω } . We start by defining on V × V the following form

b : V × V → R , b(u, v) = ⟨ curl u, curl v ⟩ + ⟨ div u, div v ⟩ ,

where ⟨· , ·⟩ denotes either the scalar or the vector-valued L

-pairing.

Remark 4. Contrary to the case of smooth bounded domains (with a C

bound- ary), the space V is not contained in H

(Ω; R

). The Sobolev embedding associated to the space V is as follows: V ↩ → H

(Ω; R

) with the estimate

∥ u ∥

≤ C [

∥ u ∥

+ ∥ curl u ∥

+ ∥ div u ∥

] , u ∈ V ; (22) see for instance [10].

Proposition 5. The Hodge-Laplacian operator B, defined as the associated opera- tor in H of the form b, satisfies

D(B) = {

u ∈ V ; ∇ div u ∈ H, curl curl u ∈ H and ν × curl u = 0 on ∂Ω } Bu = − ∆u, u ∈ D(B).

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

) = 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 H

(∂Ω; R

): for ϕ ∈ H

(Ω; R

),

⟨ curl w, ϕ ⟩

− ⟨ w, curlϕ ⟩

= ⟨ ν × w, ϕ ⟩

(23) where φ = Tr

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

-spaces, we prove that its resolvent admits L

− L

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

u = (Id + t

B)

(χ

f ), we have

∥ χ

u ∥

+ t ∥ χ

div u ∥

+ t ∥ χ

curl u ∥

≤ Ce

∥ χ

f ∥

. (24) Proof. We start by choosing a smooth cut-off function ξ : R

→ R satisfying ξ = 1 on E, ξ = 0on F and ∥∇ ξ ∥

≤

. We then define η = e

where α > 0 is to be chosen later. Next, we take the scalar product of the equation

u − t

∆u = χ

f, u ∈ D(B)

with the function v = η

u. Since η = 1 on F and ∥ u ∥

≤ ∥ χ

f ∥

, it is easy to check then that

∥ ηu ∥

+ t

∥ ηdiv u ∥

+ t

∥ ηcurl u ∥

≤ ∥ χ

f ∥

+ 2α ∥∇ ξ ∥

t

∥ ηu ∥

( ∥ ηdiv u ∥

+ ∥ ηcurl u ∥

)

and therefore, using the estimate on ∥∇ ξ ∥

and choosing α =

, we obtain

∥ ηu ∥

+ t

∥ ηdiv u ∥

+ t

∥ ηcurl u ∥

≤ 2 ∥ χ

f ∥

. Using now the fact that η = e

on E, we finally get

∥ χ

u ∥

+ t ∥ χ

div u ∥

+ t ∥ χ

curl u ∥

≤ √

2e

∥ χ

f ∥

, which gives (24) with C = √

2 and c =

.

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 ∈ Σ

= {

ω ∈ C \ { 0 } ; | arg z | < π − θ }

, f ∈ H and u = (zId + B)

(χ

f ),

we have

| z |∥ χ

u ∥

+ | z |

∥ χ

div u ∥

+ | z |

∥ χ

curl u ∥

≤ C e

1

2

∥ χ

f ∥

. (25) With that in hand and the Sobolev embedding (22), together with the rescaled Sobolev inequality

R

∥ χ

u ∥

≤ C (

∥ χ

u ∥

+ R ∥ χ

div u ∥

+ R ∥ χ

curl u ∥

) (26) where R = diam E, we can prove that, choosing E = Ω ∩ B(x, | z |

) and F

= Ω ∩ (

B(x, 2

| z |

) \ B(x, 2

| z |

) )

for x ∈ Ω and j ∈ N :

| z |∥ χ

u ∥

≤ C | z |

e

∥ f

∥

(27) where f

= χ

f .

Proposition 7. There exists a constant C > 0 such that for all f ∈ L

(Ω; R

) ∩ L

(Ω; R

), z ∈ Σ

, the following estimate holds:

| z |∥ (zId + B)

f ∥

≤ C ∥ f ∥

. (28)

Proof. For x ∈ Ω and r > 0, denote by B

(x, r) the ball centered in x with radius r intersected with Ω. Let u = (zId + B)

f . For x ∈ Ω, let f

= χ

f for F

= B

(x, 2

| z |

) \ B

(x, 2

| z |

) and u

= (zId + B)

f

. From (27) and Fubini’s theorem, keeping in mind that a Lipschitz domain in R

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

), we have

| z |∥ u ∥

≤ C | z | [∫

Ω

( | B

(x, t) |

∫

BΩ(x,t)

| u(y) |

dy )

dx ]

≤ C | z | [∫

Ω

[( | B

(x, t) |

∫

B_Ω(x,t)

| u(y) |

dy )

]

dx ]

≤ C | z | [∫

Ω

[ ∑

j=0

( | B

(x, t) |

∫

B_Ω(x,t)

| u

(y) |

dy )

]

dx ]

≤ C [∫

Ω

( ∑

j=0

Ce

2

( | B

(x, 2

t) |

∫

BΩ(x,2^jt)

| f (y) |

dy )

)

dx ]

≤ C ( ∑

j=0

e

2

) ∥ M ( | f |

) ∥

L³2(Ω;R)

≤ C ∥ f ∥

where we used the notation t = | z |

and M denotes the Hardy-Littlewood maximal

operator (which is bounded on L

for all p ∈ (1, ∞ )).