Chemin J.-Y., Desjardins B., Gallagher I., Grenier E. Mathematical geophysics

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Proof of the regularity of the pressure 29

In order to prove Proposition 1.2, it remains to prove that the range of

∇



→V

′

p →∇p

is closed. The regularity of the boundary ∂Ω turns out to be essential at this

point of the proof.

Let us ﬁrst state the following lemma.

Lemma 1.3 Let Ω be a C

bounded domain of R

and p ∈ H

−1

(Ω). Then p

belongs to L

(Ω) if and only if ∇p belongs to V

′

(Ω). Moreover, there exists a

positive constant C such that

p

≤ C



p

−1

+ ∇p

′



. (1.4.1)

Postponing the proof of Lemma 1.3, we deduce the following corollary, which

implies that ∇L

(Ω) is closed and this ends the proof of Proposition 1.2.

Corollary 1.3 Under the assumptions of Lemma 1.3, there exists a positive

constant C such that for all functions p in

(Ω)

def



p ∈ L

(Ω) /



Ω

p(x) dx =0



one has

p

(Ω)

≤ C∇p

′

(Ω)

Proof Let us assume that there exists a sequence (p

)

n∈N

in L

(Ω) such that

p



(Ω)

= 1 and ∇p



′

(Ω)

converges to 0.

The embedding of L

(Ω) into H

−1

(Ω) is compact due to Theorem 1.3. Thus,

up to an extraction of a subsequence, we may assume that a function p in L

(Ω)

exists such that

→ p weakly in L

(Ω) and lim

n→∞

p

− p

−1

(Ω)

=0.

As the domain Ω is bounded, the constant functions belong to L

(Ω). Thus

the function p is in L

(Ω). Moreover, the assertion above implies in particular

that (p

)

n∈N

converges to p in the distribution sense, and so does (∇p

)

n∈N

to ∇p.As∇p



′

(Ω)

converges to 0, this implies that p = 0, hence p



−1

(Ω)

converges to 0. Passing to the limit in the inequality of Lemma 1.3 gives a

contradiction.

Proof of Lemma 1.3 Since Ω is a C

bounded domain of R

, there exists a

ﬁnite family (U

)

1≤j≤N

of open subsets of R

such that

∂Ω ⊂



1≤j≤N

(1.4.2)

30 Some elements of functional analysis

and, for all j, there exists a C

diﬀeomorphism χ

from U

to W

such that

∩

Ω) = W

∩ R

with R



x =(x

′

) ∈ R

d−1

×R



. (1.4.3)

Finally, there exists an open subset U

of R

such that

⊂ Ω and

Ω ⊂



0≤j≤N

. (1.4.4)

Now let us consider a partition of unity (ϕ

)

0≤j≤N

associated with the

family (U

)

0≤j≤N

and p ∈ H

−1

(Ω). Then, we write

p =



j=0

with p

= ϕ

By deﬁnition of ·

−1

(Ω)

,wehave,ifB denotes the unit ball of H

(Ω),

ap

−1

(Ω)

= sup

u∈B

ap, u

= sup

u∈B

p, au

≤p

−1

(Ω)

sup

u∈B

au

(Ω)

The Leibnitz formula (and Poincar´e’s inequality) implies that

∇(au)

≤ C

u

(Ω)

This gives that ap

−1

(Ω)

≤ C

p

−1

(Ω)

and in particular that

p



−1

(Ω)

≤ Cp

−1

(Ω)

Moreover, if ∇p belongs to V

′

(Ω), then using the fact that ∇p

= p∇ϕ

+ ϕ

∇p,

we deduce that, for all j ∈{0,...,N},

∇p



′

≤ Cp

−1

(Ω)

+ C∇p

′

Then it suﬃces to prove that estimate (1.4.1) holds for each p

Since p

is compactly supported in Ω, estimate (1.4.1) is obtained easily

for p

. As a matter of fact, p

and its derivatives belong to H

−1

) and the

Fourier transform can be used: let us write that, for any function p in L

|p(ξ)|≤|1

B(0,1)

p(ξ)| + |1

(0,1)

p(ξ)|

≤ (1 + |ξ|

)



B(0,1)

(1 + |ξ|

)

−

|p(ξ)|





k=1

(1 + |ξ|

)

(0,1)

iξ

|ξ|

(1 + |ξ|

)

−

F(∂

p)(ξ)



≤

√

2(1 + |ξ|

)

−

|p(ξ)| + C



k=1

(1 + |ξ|

)

−

|F(∂

p)(ξ)|.

Proof of the regularity of the pressure 31

The Fourier–Plancherel theorem implies that

p

≤ Cp

−1

)

+ ∇p

−1

)

, (1.4.5)

which proves the result for p

. For each j ≥ 1, χ

being a C

diﬀeomorphism,

one has

−1



∩ R

) →H

∩ Ω)

v →v ◦χ

−1

A change of variables allows us to deduce that

∗



−1

∩ Ω) → H

−1

∩ R

)

f →



v →f, Jχ

◦ χ

−1

× v ◦χ

−1





As a result, the proof reduces to the case when Ω = R

. Let us introduce the

mappings Q and R from H

)toH

) given for u ∈ H

)by

Qu(x)=



0ifx

< 0,

u(x)+3u(x

′

, −x

) − 4u(x

′

, −2x

) otherwise,

and

Ru(x)=



0ifx

< 0,

u(x) − 3u(x

′

, −x

)+2u(x

′

, −2x

) otherwise.

It can easily be checked that Q and R map continuously H

)toH

)

and L

)toL

). Moreover, one has

∂

∂x

◦ Q = Q ◦

∂

∂x

if j = d and

∂

∂x

◦ R = Q ◦

∂

∂x

· (1.4.6)

Next, we consider the transposed maps on H

−1

, namely



−1

) →H

−1

)

f →

Qf/

Qf, v = f,Qv

and similarly for R. Taking the transpose of (1.4.6) leads to

∂

∂x

◦

Q =

Q ◦

∂

∂x

if j = d and

∂

∂x

◦

Q =

R ◦

∂

∂x

· (1.4.7)

Applying (1.4.7) to p

= χ

∗

∈ H

−1

) yields

Qp

∈ H

−1

) and

∂

∂x

(

Qp

) ∈ H

−1

Using (1.4.5), we deduce that

Qp

∈ L

) and



Qp



)

≤ C





Qp



−1

)

+ ∇

Qp



′

)



32 Some elements of functional analysis

Since the restriction of Q on compactly supported functions in R

is the identity

map, the restriction of

Qp

to R

is p

. Therefore, we have

p



)

≤ C



p



−1

)

+ ∇p



′

)



so that p

belongs to L

(Ω) and

p



(Ω)

≤ C



p

−1

)

+ ∇p

′

)



which proves Lemma 1.3, hence Proposition 1.2.

Remark For a general open subset Ω, it can be proved that there exists p

in L

loc

(Ω) such that if f ∈V

◦

(Ω), then f = ∇p.

Weak solutions of the Navier–Stokes equations

The mathematical analysis of the incompressible Stokes and Navier–Stokes

equations in a possibly unbounded domain Ω of R

(d = 2 or 3) is the purpose

of this chapter. Notice that no regularity assumptions will be required on the

domain Ω.

2.1 Spectral properties of the Stokes operator

Because of the compactness result stated in Theorem 1.3, page 27, the case

of bounded domains will be diﬀerent (in fact slightly simpler) than the case

of general domains.

2.1.1 The case of bounded domains

The study of the spectral properties of the Stokes operator previously deﬁned

relies on the study of its inverse, which is in fact much easier. We shall restrict

ourselves here to the case of the homogeneous Stokes operator which is adapted

to the case of a bounded domain.

Deﬁnition 2.1 Let us denote by B the following operator



H→V

⊂H

f → u/− ∆u − f ∈V

◦

This operator has the following properties.

Proposition 2.1 The operator B is continuous, self-adjoint, positive, one-to-

one and thus the range of B is dense in H.

Proof Let us show that B is symmetric: for all vector ﬁelds f and g in H,

denoting u = Bf and v = Bg, one has

(Bf|g)

= −∆v, u =(∇u|∇v)

= −∆u, v =(f|Bg)

The fact that B is bounded is due to the following computation. Let f be a vector

ﬁeld in H(Ω) and denote u = Bf . Then, using the Poincar´e inequality, we have

(Bf|f)

= −∆u, u≥cu

= cBf

34 Weak solutions of the Navier–Stokes equations

This proves that B is continuous and positive. Finally to show that B is one-to-

one, we observe that the kernel of B is equal to V

◦

∩H, which is exactly V

⊥

By deﬁnition of H as the closure of V

in L

, this space reduces to {0} and the

proposition follows. As the closure in H of the range of B (denoted by R(B)) is

equal to the orthogonal space of ker B, we infer that R(B) is dense in H.

The basic theorem is the following.

Theorem 2.1 Let Ω be a bounded domain of R

. A Hilbertian basis (e

)

k∈N

of H and a non-decreasing sequence of positive eigenvalues converging to

inﬁnity (µ

)

k∈N

exist such that

−∆e

+ ∇π

= µ

, (2.1.1)

where (π

)

k∈N

is a sequence of L

loc

(Ω) functions, the gradient of which belongs

to V

◦

. Moreover, (µ

−1

)

k∈N

is an orthonormal basis of V

endowed with the H

scalar product.

Moreover for any f ∈V

′

we have

f

′



j∈N

−2

f,e



Remark As claimed by Proposition 1.2, the functions π

are in L

(Ω) when

the boundary of Ω is C

Proof of Theorem 2.1 Let us consider the operator B deﬁned in

Deﬁnition 2.1. Proposition 2.1 claims that the operator B is positive, self-adjoint

and one-to-one. As the range of B is included in V

which is included in H

, the

operator is compact as inferred by Theorem 1.3. Applying the spectral theorem

for self-adjoint compact operators in a Hilbert space, we get the existence of a

Hilbertian basis (e

)

k∈N

of H and a non-decreasing sequence of positive eigen-

values converging to inﬁnity (µ

)

k∈N

such that Be

= µ

−2

. By deﬁnition of B,

this implies that there exists π

∈ L

loc

(Ω) such that

= −∆Be

+ ∇π

= −µ

−2

∆e

+ ∇π

Moreover, denoting π

= µ

π

, it is obvious that (e

)

k∈N

satisﬁes (2.1.1)

and that

′

)

= −∆e

′



= −∇π

+ µ

′



= µ

′

)

= µ

k,k

′

Spectral properties of the Stokes operator 35

This proves (µ

−1

)

k∈N

is an orthonormal family of V

endowed with the H

scalar product. Let us prove that the vector space generated by (e

)

k∈N

is dense

in V

. Let us consider any u in V

such that

∀k ∈ N, (u|e

)

=0.

We have, by deﬁnition of the H

scalar product,

(u|e

)

=(∇u|∇e

)

= −∆e

,u.

Using (2.1.1), we infer that

(u|e

)

= µ

+ ∇π

,u

= µ

e

,u

= µ

|u)

As the vector space generated in H by the sequence (e

)

k∈N

is dense in H,we

deduce that u = 0 and thus that (e

)

k∈N

is an orthonormal basis of V

Now let us consider a vector ﬁeld f ∈V

′

and let us compute its norm in V

′

By deﬁnition of f

′

we have

f

′

= sup

v∈V

v≤1

f,v

= sup

(α

)

ℓ

≤1





j≤k

−1



Using the characterization of ℓ

(N), we infer that

f

′

= sup

(α

)

ℓ

≤1

sup







j≤k

−1

f,e









(µ

−1

f,e

)



ℓ

This concludes the proof of Theorem 2.1.

Deﬁnition 2.2 Let us deﬁne







′

→V

f →



j≤k

f,e

e

36 Weak solutions of the Navier–Stokes equations

Remark

• The operators P

are the spectral projectors of the Stokes problem (see the

forthcoming Theorem 2.2 on page 38 for the general deﬁnition of spectral

projectors).

• As (e

)

k∈N

is a Hilbert basis on H, we have, for any vector ﬁeld in L

(Ω)

Pv =



(v|e



v, e

e

where P is the Leray projector on L

divergence-free vector ﬁelds deﬁned

in Deﬁnition 1.3.

Corollary 2.1 The operators P

satisfy

P

f

′

≤f

′

and ∀f ∈V

′

, lim

k→∞

P

f − f

′

=0.

Proof The ﬁrst point of the corollary is clear. As to the second one, according

to Theorem 2.1 we have

P

f − f

′



j≥k

−2

f,e



This goes to zero as k goes to inﬁnity, as the remainder of a convergent series.

2.1.2 The general case

Let us ﬁrst deﬁne the inverse of the inhomogeneous Stokes operator.

Deﬁnition 2.3 Let us denote by B the following operator



H→V

⊂H

f →u/u− ∆u − f ∈V

◦

This operator has the following properties.

Proposition 2.2 The operator B is continuous, self-adjoint and one-to-one.

Moreover it satisﬁes B

L(H)

≤ 1.

Proof The fact that B is bounded is due to the following computation. Let f

be a vector ﬁeld in H(Ω) and denote u = Bf. Then

(Bf|f)

= u|u − ∆u≥u

= Bf

This proves that B is continuous and that B

L(H)

≤ 1.

Now let us show that B is symmetric. For all vector ﬁelds f and g in H(Ω),

denoting u = Bf and v = Bg, one has

(Bf|g)

= u, v −∆v =(u|v)

+(∇u|∇v)

= u − ∆u, v =(f|Bg)

Spectral properties of the Stokes operator 37

Finally to show that B is one-to-one, we observe that the kernel of B is equal

to V

◦

∩H, which is exactly V

⊥

. By the deﬁnition of H as the closure of V

in L

this space reduces to {0} and the proposition follows.

As the closure in H of the range of B (denoted by R(B)) is equal to the

orthogonal space of ker B, we infer from Proposition 2.2 that R(B)isdense

in H. This allows us to deﬁne the inverse of B as an unbounded operator A with

a dense domain of deﬁnition.

Deﬁnition 2.4 Let us denote by A the following operator



R(B) →H

u →f/Bf= u.

This is exactly the operator A deﬁned in (1.2.1) on page 19.

Lemma 2.1 The operator A is self-adjoint with domain R(B).

This lemma is a classical result of operator theory. For the reader’s

convenience, we give a proof of it.

Proof of Lemma 2.1 Of course A is symmetric, so the only point to check

is that the domain D(A

∗

)ofA

∗

is R(B). By deﬁnition,

D(A

∗

)={v ∈H|∃C>0, ∀u ∈ R(B), (Au|v)

≤ Cu

Since A is symmetric, we have of course R(B) ⊂D(A

∗

). Now let us prove

that D(A

∗

) ⊂ R(B). Let us deﬁne the graph norm

v

∗

def

= v

+ A

∗

v

The fact that (D(A

∗

), ·

∗

) is a Hilbert space is left as an exercise to the

reader. The equality D(A

∗

)=R(B) will result from the fact that R(B) is closed

and dense for the ·

∗

norm. For any f ∈H, we have A

∗

Bf = ABf = f hence

Bf

∗

= Bf

+ f

which immediately gives that f

≤Bf

∗

, hence that the space R(B)is

closed in (D(A

∗

), ·

∗

). Now let v be in the orthogonal space of R(B)inthe

sense of the (·|·)

∗

scalar product. By deﬁnition, we have, for any f in H,

(Bf|v)

+(A

∗

Bf|A

∗

=0.

As B is self-adjoint and A

∗

B = Id, we get for all f in H,

(f|Bv)

+(f|A

∗

=0.

38 Weak solutions of the Navier–Stokes equations

This implies that Bv + A

∗

v =0. In particular A

∗

v belongs to R(B)=D(A)

and by application of the operator A we get v + AA

∗

v =0. By deﬁnition of A

∗

we have

(AA

∗

v|v)

= A

∗

v

thus v

∗

= 0 and the lemma is proved.

As the Stokes operator A = A − Id is also self-adjoint, we are now ready to

apply the spectral theorem to the operator A (see, for instance, [108], Chapter

VII for a proof).

Theorem 2.2 There exists a family of orthogonal projections on H, denoted

by (P

)

λ∈R

, which commutes with A and satisﬁes the following properties.

The family (P

)

λ∈R

is increasing, in the following sense:

′

= P

inf(λ,λ

′

)

, for any (λ, λ

′

) ∈ R

. (2.1.2)

For λ<0, P

=0and for any u ∈H,

lim

λ→∞

P

u − u

=0. (2.1.3)

The family (P

)

λ∈R

is continuous on the right, which means that

∀u ∈H, lim

′

→λ, λ

′

>λ

P

′

u − P

u

=0. (2.1.4)

For any u ∈H, the function λ → (P

u|u)

= P

u

is increasing, and

u



d(P

u|u) and ∇u

=(Au|u)



λd(P

u|u). (2.1.5)

Let us state a corollary which shows that the operators P

should be

understood as smoothing operators; they are an extension of frequency cut-oﬀ

operators.

Corollary 2.2 For any u ∈H, the vector ﬁeld P

u is an element of V

and

∇P

u

≤ λ

u

For any u ∈V

, we have

(Id −P

)u

≤

u

Proof Using (2.1.5) we get

∇P

u



′

d(P

′

u|u).