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

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The Stokes problem 19

where u is the unique solution of the Dirichlet problem

(Id −∆)u = f.

We recall, indeed, that Id −∆ is a one-to-one isometry from H

to H

−1

When the boundary of the domain is regular, namely the boundary is a C

hypersurface, it can be checked that the map f → f · n is well deﬁned on the

space of L

vector ﬁelds with L

divergence (with values in the space H

−

the boundary). In that case, the space H is exactly the space of L

divergence-free

vector ﬁelds such that f · n

|∂Ω

=0.

Now we are ready to deﬁne the well-known Leray projector.

Deﬁnition 1.3 We denote by P the orthogonal projection of (L

(Ω))

on H.

1.2 The Stokes problem

The Stokes system is deﬁned as follows. Let f ∈V

′

. We shall say that u in V

solves the inhomogeneous Stokes problem Au = f if, for all v ∈V

u − ∆u, v = f,v. (1.2.1)

In other words, u − ∆u − f ∈V

◦

Again assuming that f ∈V

′

, we shall say that u in V

solves the homogeneous

Stokes problem Au = f if, for all v ∈V

−∆u, v = f,v. (1.2.2)

In other words, −∆u − f ∈V

◦

When considering the Navier–Stokes equations (NS

), the homogeneous

version of the Stokes system arises in a very natural way. However, the inhomo-

geneous Stokes problem allows us to control the L

norm of the velocity, which

is particularly convenient in the case of unbounded domains. As a matter of fact,

the following existence and uniqueness result holds.

Theorem 1.1 Given f ∈V

′

, there exists a unique solution u in V

of the

inhomogeneous Stokes problem (1.2.1). When the domain Ω is bounded, there is

a unique solution u in V

of the homogeneous Stokes problem (1.2.2).

Proof The proof is nothing but the Lax–Milgram theorem. For the reader’s

convenience, we recall it. Given f in V

′

, a linear map V

→ R on the Hilbert

space V

(a closed subspace of V endowed with the H

scalar product) can be

deﬁned as v →f,v. Thanks to the Riesz theorem, a unique u exists in V

such that

∀v ∈V

, (u|v)

= f,v.

20 Some elements of functional analysis

By deﬁnition of the H

scalar product, we get

∀v ∈V

, (u|v)

+(∇u|∇v)

= f,v.

As u and v belong to V,weget

∀v ∈V

, u − ∆u, v = f,v,

and the theorem is proved (the case of the homogeneous Stokes problem in a

bounded domain is strictly analogous thanks to Poincar´e’s inequality).

Let us point out that relations (1.2.1) and (1.2.2) are equalities on linear

forms on V

. It is therefore natural to introduce the following deﬁnition.

Deﬁnition 1.4 We denote by V

′

(Ω) the space of continuous linear forms

on the space V

(Ω). The norm in V

′

(Ω) is given by

f

(Ω)

def

= sup

v∈V

v

≤1

f,v.

The following proposition will be very useful in the following.

Proposition 1.1 For any function f ∈V

′

(Ω), a sequence of functions (f

)

n∈N

in H(Ω) exists such that

lim

n→∞

f

− f

′

(Ω)

=0.

Proof By the density of (L

)

in V

′

there exists a sequence (g

)

n∈N

in L

(Ω)

(we drop the exponent d to simplify the notation) such that

lim

n→∞

g

− g

′

(Ω)

=0.

We have in particular, of course,

lim

n→∞

g

− g

′

(Ω)

=0.

Then we just need to write

f − Pg



′

(Ω)

≤g

− Pg



′

(Ω)

+ g

− f

′

(Ω)

The Stokes problem 21

and to notice that

g

− Pg



′

(Ω)

= sup

v∈V

v

≤1

g

− Pg

,v

= sup

v∈V

v

≤1

− Pg

|v)

= sup

v∈V

v

≤1

|v − Pv)

=0.

The proposition is proved, on choosing f

= Pg

Remarks

• The fact that a vector ﬁeld g ∈V

′

belongs to the polar space V

◦

of V

implies in particular that for all ϕ ∈D, one has for all 1 ≤ i, j ≤ d

g, ψ

 =0, where ψ

= ε

∂

ϕ − ε

∂

ϕ ∈V

where ε

∈ R

has zero components except 1 on the i

component, so that

g

, −∂

ϕ + g

,∂

ϕ =0,

hence ∂

− ∂

=0inD

′

, i.e. curl g =0.

• There exist simple examples of smooth domains Ω and vector ﬁelds g such

that curl g = 0 but which do not appear as a gradient. For instance, consider

the two-dimensional ring

Ω={x ∈ R

/ 0 <R

< |x| <R

}

and g =(−∂

log |x|,∂

log |x|). Then g is clearly divergence-free, and irrota-

tional since x → log |x| is harmonic on Ω. Assume that g can be written as

a gradient of a function p. Since g is smooth on Ω, so is p.Letnowx

∈ Ω

such that g(x

) = 0, and let t → γ(t) be the unique solution of

dγ

(t)=g(γ(t)),γ(0) = x

Note that the fact that the level lines of x → log |x| are circles implies that γ

is periodic. Then, we have

p ◦ γ(t)=

dγ

·∇p(γ(t)) = |∇p(γ(t))|

≥ 0,

which yields a contradiction to the periodicity of γ, since the derivative at

time t = 0 does not vanish.

• However, the condition for a vector ﬁeld g ∈V

′

to belong to V

◦

stronger than the assumption that g is irrotational. As a matter of fact,

22 Some elements of functional analysis

when g belongs to V

◦

, one can prove that a locally L

function p exists

such that g = ∇p. Moreover, this function p is L

loc

(

Ω) when Ω has a C

boundary. In order to give the ﬂavor of this kind of result, let us state the

following proposition, which will be proved in Section 1.4.

Proposition 1.2 Let Ω be a C

bounded domain of R

and g ∈V

◦

. Then there

exists p ∈ L

(Ω) such that g = ∇p.

• In all that follows, we shall denote by ∇p any element of V

◦

1.3 A brief overview of Sobolev spaces

The aim of this section is to recall the basic features of Sobolev spaces in the

whole space R

and useful compactness results in the case of bounded domains.

1.3.1 Deﬁnition in the case of the whole space R

In the whole space R

(d = 2 or 3), Sobolev spaces are deﬁned in terms of integ-

rability properties in frequency space, using the Fourier transform. Let us recall

that for all u ∈D

′

, the Fourier transform Fu, also denoted by u,isdeﬁnedby

∀ξ ∈ R

, Fu(ξ)=u(ξ)

def



−ix·ξ

u(x) dx.

The inverse Fourier transform allows us to recover u from u:

u(x)=F

−1

u(x)=(2π)

−d



ix·ξ

u(ξ) dξ.

For all s ∈ R we introduce the inhomogeneous Sobolev spaces

def



u ∈S

′

/u

def



(1 + |ξ|

)

|u(ξ)|

dξ < +∞



and similarly the homogeneous Sobolev spaces

def



u ∈S

′

/u ∈ L

loc

and u

def



|ξ|

|u(ξ)|

dξ < +∞



Let us notice that when s ≥ d/2

is not a Hilbert space. Let us deﬁne the

following sequence. Let C be a ring included in the unit ball B(0, 1) such that

C∩2C = ∅ and let us deﬁne

def

= F

−1



q=1

q(s+

)

−q

(ξ).

It is left as an exercise to the reader to check that the sequence (f

)

n∈N

is a

Cauchy sequence in

which does not converge in

if s ≥ d/2.

A brief overview of Sobolev spaces 23

1.3.2 Sobolev embeddings

The purpose of this section is to recall Sobolev embeddings, which will be

extensively used in this book. In the whole space R

, the statement is the

following.

Theorem 1.2 If s is positive and smaller than d/2 then the space

) is

continuously embedded in L

d−2s

Proof First of all, let us show how a scaling argument allows us to ﬁnd the

critical index p =2d/(d−2s). Let f be a function on R

, and let us denote by f

ℓ

the function f

ℓ

(x)

def

= f(ℓx)(ℓ>0). We have, for all p ∈ [1, ∞[,

f

ℓ



)

= ℓ

−

f

)

and

f

ℓ



)



|ξ|



ℓ

(ξ)|

dξ

= ℓ

−2d



|ξ|



f(ℓ

−1

ξ)|

dξ

= ℓ

−d+2s

f

)

As soon as an inequality of the type f

)

≤ Cf

)

holds for any

smooth function f , it holds also for f

ℓ

for any positive ℓ. This leads to the

relation p =2d/(d − 2s).

Let us now prove the theorem. In order to simplify the computations, we may

assume without loss of generality that f

)

=1.

Let us start by observing that for any p ∈ [1, +∞[, Fubini’s theorem allows

us to write for any measurable function f,

f

)

def



|f(x)|

= p



|f(x)|

p−1

dΛ dx

= p



∞

p−1

meas



x ∈ R

/|f(x)| > Λ



dΛ.

As quite often in this book, let us decompose the function into low and high

frequencies. More precisely, we shall write f = f

1,A

+ f

2,A

, with

1,A

= F

−1

B(0,A)



f) and f

2,A

= F

−1

(0,A)



f), (1.3.1)

where A>0 will be determined later. As the support of the Fourier transform

of f

1,A

is compact, the function f

1,A

is bounded. More precisely we have the

following lemma.

24 Some elements of functional analysis

Lemma 1.1 Let s be in ]−∞,d/2[ and let K be a compact subset of R

.Iff

belongs to

and if Supp



f is included in K, then we have

f

∞

≤ (2π)

−d





dξ

|ξ|



f

Proof The inverse Fourier formula together with the Cauchy–Schwarz inequal-

ity allows us to write

f

∞

)

≤ (2π)

−d





f

)

≤ (2π)

−d



|ξ|

−s

|ξ|



f(ξ)|dξ

≤ (2π)

−d





dξ

|ξ|



f

The lemma is proved.

Applying this lemma, we get

f

1,A



∞

≤ C

−s

. (1.3.2)

The triangle inequality implies that for any positive A we have



x ∈ R

/|f(x)|>Λ



⊂



x ∈ R

/2|f

1,A

(x)|>Λ



∪



x ∈ R

/2|f

2,A

(x)|>Λ



From the above inequality (1.3.2) we infer that

A = A

def





=⇒ meas



x ∈ R

/|f

1,A

(x)| > Λ/2



=0.

From this we deduce that

f

)

≤ p



∞

p−1

meas



x ∈ R

/|f

2,A

(x)| > Λ/2



dΛ.

It is well known (this is the so-called Bienaim´e–Tchebychev inequality) that

meas



x ∈ R

/|f

2,A

(x)| > Λ/2





{

x∈R

/|f

2,A

(x)|>Λ/2

}

≤



{

x∈R

/|f

2,A

(x)|>Λ/2

}

4|f

2,A

(x)|

≤

f

2,A



)

Thus we infer

f

)

≤ 4p



∞

p−3

f

2,A



)

dΛ.

A brief overview of Sobolev spaces 25

But we know that the Fourier transform is (up to a constant) a unitary transform

of L

.Thuswehave

f

)

≤ 4p(2π)

−d



∞

p−3



(|ξ|≥A

)



f(ξ)|

dξdΛ. (1.3.3)

Then by the deﬁnition of A

we have

|ξ|≥A

⇐⇒ Λ ≤ 4C

|ξ|

Then, Fubini’s theorem implies that

f

)

≤ 4p(2π)

−d









|ξ|

p−3

dΛ







f(ξ)|

dξ

≤

p − 2

(2π)

−d

(4C

)

p−2



|ξ|

d(p−2)



f(ξ)|

dξ.

As 2s = d(p − 2)/p the theorem is proved.

The following corollary will be useful in the future.

Corollary 1.1 If p belongs to ]1, 2], then

) ⊂

) with s = −d



−



Proof This corollary is proved by duality. Let us write

a

= sup

ϕ

−s

≤1

a, ϕ.

As −s = d



−



= d



−



1 −



, we have that ϕ

−s

≥ Cϕ

′

and thus

a

≤ C sup

ϕ

′

≤1

a, ϕ

≤ Ca

This concludes the proof.

Other useful estimates are Gagliardo–Nirenberg inequalities.

Corollary 1.2 If p ∈ [2, ∞[ such that 1/p > 1/2 − 1/d, then a constant C

exists such that for any domain Ω in R

, we have for any u ∈ H

(Ω),

u

(Ω)

≤ Cu

1−σ

(Ω)

∇u

(Ω)

where σ =

d(p − 2)

· (1.3.4)

26 Some elements of functional analysis

Proof By density arguments, we may suppose that u is in D(Ω). Then, Sobolev

embeddings yield

u

(Ω)

≤ Cu

d(p−2)

)

The convexity of the Sobolev norms

u

)

≤u

1−σ

)

u

)

for all σ ∈ [0, 1],

allows us to conclude.

The following lemma, known as Bernstein inequalities, will be useful in the

future, especially in Part III.

Lemma 1.2 Let B beaballofR

. A constant C exists so that, for any non-

negative integer k, any couple of real numbers (p, q) such that 1 ≤ p ≤ q ≤ +∞

and any function u of L

, we have

Supp u ⊂ λB ⇒ sup

|α|=k

∂

u

≤ C

k+1

k+d(

−

)

u

Proof Using a dilation of size λ, we can assume throughout the proof that

λ =1.Letφ be a function of D(R

) the value of which is 1 near B.Asu(ξ)=

φ(ξ)u(ξ), we can write, if g denotes the inverse Fourier transform of φ,

∂

u = ∂

g⋆u

where ⋆ denotes the convolution operator

g⋆u(x)=



g(x − y)u(y) dy.

By Young’s inequality, we get

∂

u

≤∂

g

u

with

+1=

ByaH¨older estimate

∂

g

≤∂

g

1−

∞

∂

g

Then using the general convexity inequality

∀(a, b) ∈ R

×R

, ∀θ ∈]0, 1[,ab≤ θa

+(1−θ)b

1−θ

, (1.3.5)

we get

∂

g

≤∂

g

∞

+ ∂

g

≤ 2(1 + |·|

)

∂

g

∞

≤ 2(Id −∆)



(·)

φ)

The lemma is proved.

A brief overview of Sobolev spaces 27

1.3.3 A compactness result

In the next chapter, we shall use the following compactness theorem, known as

Rellich’s theorem.

Theorem 1.3 Let Ω be a bounded domain of R

. The embedding of H

(Ω)

into L

(Ω) is compact and so is that of L

(Ω) into H

−1

(Ω).

Proof For the reader’s convenience, we present here a proof based on the

Fourier transform of periodic functions. Without any loss of generality, we may

assume that

Ω is included in Q

def

=]0, 2π[

. Let us deﬁne, for any function u in

(Ω), the 2π Z-periodic function

u(x)=



j∈Z

u(x − 2πj).

Let us deﬁne its Fourier coeﬃcients for k ∈ Z

u

def



−ik·x

u(x)

(2π)

Using integration by parts and the Fourier–Plancherel theorem, we obtain



k∈Z

(1 + |k|

)|u

≤ C∇u

Let us deﬁne the sequence (T

)

N∈N

of linear maps by











(Ω) →L

(Q)

u →



|k|≤N

u

ik·x

where L

(Q) is identiﬁed with 2π Z

-periodic L

functions. Obviously, the range

of T

is a ﬁnite-dimensional vector space. Moreover, the following property holds

u − (T

|Ω



(Ω)

≤u − T

u

(Q)

≤





|k|>N

u

ik·x



(Q)

≤

(N +1)

∇u

(Ω)

. (1.3.6)

Thus we end up with the inequality

Id −T

|Ω



L(H

(Ω);L

(Ω))

≤

N +1

28 Some elements of functional analysis

The operator Id appears as a limit in the norm of L(H

(Ω); L

(Ω)) of operators

of ﬁnite rank. This proves the compactness of the identity as a map from H

(Ω)

into L

(Ω).

Now let us prove the compactness of the identity as a map from L

(Ω)

into H

−1

(Ω). For any u in L

(Ω), we have

u − (T

|Ω



−1

(Ω)

= sup

ϕ∈H

(Ω)

∇ϕ

u − (T

|Ω

,ϕ

= sup

ϕ∈H

(Ω)

∇ϕ



Ω

(u − T

u)(x)ϕ(x) dx

≤ sup

ϕ∈H

(Q)

ϕ



(u − T

u)(x)ϕ(x) dx.

The Fourier–Plancherel theorem implies, by deﬁnition of T

, that

u − (T

|Ω



−1

(Ω)

≤ (2π)

−d

sup

ϕ∈H

(Q)

ϕ



|k|>N

u

ϕ

≤ N

−1

(2π)

−d

sup

ϕ∈H

(Q)

ϕ



|k|>N

u

(1 + |k|

)

ϕ

The Cauchy–Schwarz inequality implies that

u − (T

|Ω



−1

(Ω)

≤ N

−1

u

The theorem is proved.

1.4 Proof of the regularity of the pressure

The aim of this section is the proof of Proposition 1.2. Let Ω be a C

bounded

domain of R

and ∇L

(Ω) be the set of vector ﬁelds f ∈V

′

(Ω) be such that

there exists a function p ∈ L

(Ω) satisfying f = ∇p.Letv ∈V(Ω) such that v

belongs to (∇L

(Ω))

∗

. Then, for all functions p ∈ L

(Ω),

div v, p = −∇p, v

= −f,v

=0.

It follows that div v = 0, which means that (∇L

(Ω))

∗

⊂V

.ThusV

◦

is a subset

of ((∇L

)

∗

)

◦

. Thanks to the Hahn–Banach theorem, one has

◦

⊂ ((∇L

)

∗

)

◦

∇L

−1

In other words, if f ∈V

◦

, there exists a sequence (p

)

n∈N

of L

(Ω) functions

such that ∇p

converges to f in V

′