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

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The Leray theorem 49

We have, after integration by parts,

P

∆u

(t), Ψ(t) = −ν



Ω

∇u

(t, x):∇P

Ψ(t, x) dx,

P

Q(u

(t),u

(t)), Ψ(t) =



Ω

(t, x) ⊗ u

(t, x):∇P

Ψ(t, x) dx

and

u

(t),

Ψ(t) =



Ω

(t, x) · ∂

Ψ(t, x) dx.

By time integration between 0 and t, we infer that



Ω

(t, x) ·Ψ(t, x) dx



Ω

(ν∇u

: ∇P

Ψ − u

⊗ u

: ∇P

Ψ − u

·∂

Ψ) (t

′

,x) dxdt

′



Ω

(0,x) · Ψ(0,x) dx +



f

′

), Ψ(t

′

)dt

′

We now have to pass to the limit.

Let us start by proving the following preliminary lemma.

Lemma 2.4 Let H be a Hilbert space, and let (A

)

n∈N

be a bounded sequence

of linear operators on H such that

∀h ∈ H, lim

n→∞

A

h − h

=0.

Then if ψ ∈ C([0,T]; H) we have

lim

n→∞

sup

t∈[0,T ]

A

ψ(t) − ψ(t)

=0.

Proof The function ψ is continuous in time with values in H, so for all pos-

itive ε, the compact ψ([0,T]) can be covered by a ﬁnite number of balls of

radius

2(A +1)

with A

def

= sup

A



L(H)

and center (ψ(t

ℓ

))

0≤ℓ≤N

. Then we have, for all t in [0,T] and ℓ in {0,...,N},

A

ψ(t) − ψ(t)

≤A

ψ(t) − A

ψ(t

ℓ

)

+ A

ψ(t

ℓ

) − ψ(t

ℓ

)

+ ψ(t

ℓ

) − ψ(t)

The assumption on A

implies that for any ℓ, the sequence (A

ψ(t

ℓ

))

n∈N

tends

to ψ(t

ℓ

). Thus, an integer n

exists such that, if n ≥ n

∀ℓ ∈{0,...,N}, A

ψ(t

ℓ

) − ψ(t

ℓ

)

50 Weak solutions of the Navier–Stokes equations

We infer that, if n ≥ n

, for all t ∈ [0,T] and all ℓ ∈{0,...,N},

A

ψ(t) − ψ(t)

≤A

ψ(t) − A

ψ(t

ℓ

)

+ ψ(t

ℓ

) − ψ(t)

For any t,letuschooseℓ such that

ψ(t) − ψ(t

ℓ

)

≤

2(A +1)

The lemma is proved.

We can apply the lemma to H = V, and to Ψ ∈ C([0,T]; V

). We ﬁnd that

lim

k→∞

sup

t∈[0,T ]

P

Ψ(t) − Ψ(t)

=0.

This implies that

lim

k→∞



Ω

∇u

: ∇P

Ψ dxdt

′

= lim

k→∞



Ω

∇u

: ∇Ψ dxdt

′

and the weak convergence of (u

)

k∈N

to u in L

loc

; V

) ensures that

lim

k→∞



Ω

∇u

: ∇Ψ dxdt

′

= ν



Ω

∇u : ∇Ψ dxdt

′

By (2.2.7) we have, for any non-negative t,

lim

k→∞



Ω

(t, x) · Ψ(t, x) dx = lim

k→∞

u

(t), Ψ(t) = u(t), Ψ(t)

and by (2.2.6)

lim

k→∞

sup

t∈[0,T ]





Ω

(t, x) · ∂

Ψ(t, x) dx −u(t),∂

Ψ(t)



=0.

The two terms associated with the initial data and the bulk forces are convergent

by construction of the sequence (f

)

k∈N

and because of the fact that, thanks to

Theorem 2.2, (P

)

k∈N

tends to u

in L

Now let us pass to the limit in the non-linear term. As above we have in fact

lim

k→∞



Ω

⊗ u

: ∇P

Ψ)(t

′

,x) dxdt

′

= lim

k→∞



Ω

⊗ u

: ∇Ψ)(t

′

,x) dxdt

′

so it is enough to prove that

lim

k→∞



Ω

⊗ u

: ∇Ψ)(t

′

,x) dxdt

′



Ω

(u ⊗ u : ∇Ψ)(t

′

,x) dxdt

′

The Leray theorem 51

Let (K

)

n∈N

be an exhaustive sequence of compact subsets of Ω. Applying

Lemma 2.4 with H = (L

)

and A

def

= 1

f,wehave

lim

n∞

1

∇Ψ −∇Ψ

([0,T ];L

)

=0.

Using the fact that the sequence (u

⊗ u

)

k∈N

is bounded in L

([0,T]; L

(Ω)),

it is enough to prove that, for any compact subset K of Ω, we have

lim

k→∞

u

⊗ u

− u ⊗ u

([0,T ];L

)

= 0 (2.2.12)

which will be implied by

lim

k→∞

u

− u

([0,T ];L

)

=0. (2.2.13)

Recall that L

denotes the space of L

functions supported in K. Using

Corollary 1.2, we have, for any vector ﬁeld a ∈V,

a(t)

≤ Ca(t)

1−

∇a(t)

H¨older’s inequality implies that

a

([0,T ];L

)

≤ Ca

1−

([0,T ]×K)

∇a

([0,T ]×Ω)

We therefore have

u

− u

([0,T ];L

)

≤ Cu

− u

1−

([0,T ]×K)

∇(u

− u)

([0,T ]×Ω)

Proposition 2.7 allows us to conclude the proof of the fact that u is a solution

of (NS

) in the sense of Deﬁnition 2.5.

It remains to prove the energy inequality (2.2.1). Assertion (2.2.7) of

Proposition 2.7 implies in particular that for any time t ≥ 0 and any v ∈V

lim inf

k→∞

(t)|v)

=(u(t)|v)

As V

is dense in H, we get that for any t ≥ 0, the sequence (u

(t))

k∈N

converges

weakly towards u(t) in the Hilbert space H. Hence

u(t)

≤ lim inf

k→∞

u

(t)

for all t ≥ 0.

On the other hand, (u

)

k∈N

converges weakly to u in L

loc

; V), so that for

all non-negative t,wehave



∇u(t

′

)

′

≤ lim inf

k→∞



∇u

′

)

′

Taking the lim inf

k→∞

in the energy equality for approximate solutions (2.2.3)

yields the energy inequality (2.2.1).

To conclude the proof of Theorem 2.3 we just need to prove the time

continuity of u with values in V

′

. That result is obtained by using the fact

that u satisﬁes (S

) in particular with a function Ψ independent of time.

52 Weak solutions of the Navier–Stokes equations

Choosing such a function yields that for any Ψ ∈V

u(t), Ψ−u(t

′

), Ψ =



′

f(t

′′

), Ψ dt

′′



′



Ω

(ν∇u(t

′′

):∇Ψ − u ⊗ u(t

′′

):∇Ψ) dxdt

′′

Using the inequality

u(t)

≤u(t)

1−

∇u(t)

we infer, by energy inequality, that u belongs to L

([0,T]; L

(Ω)). Then we

deduce that

|u(t), Ψ−u(t

′

), Ψ| ≤ |t − t

′

1−

u

([0,T ];L

)

Ψ

+ |t − t

′

(f

([0,T ];V

′

)

+ ν∇u

([0,T ];L

)

)Ψ

which concludes the proof of Theorem 2.3.

Stability of Navier–Stokes equations

In this chapter we intend to investigate the stability of the Leray solutions con-

structed in the previous chapter. It is useful to start by analyzing the linearized

version of the Navier–Stokes equations, so the ﬁrst section of the chapter is

devoted to the proof of the well-posedness of the time-dependent Stokes system.

The study will be applied in Section 3.2 to the two-dimensional Navier–Stokes

equations, and the more delicate case of three space dimensions will be dealt

with in Sections 3.3–3.5.

3.1 The time-dependent Stokes problem

Given a positive viscosity ν, the time-dependent Stokes problem reads as follows:

(ES

)











∂

u − ν∆u = f −∇p

div u =0

|∂Ω

|t=0

= u

∈H.

Let us deﬁne what a solution of this problem is.

Deﬁnition 3.1 Let u

be in H and f in L

loc

; V

′

). We shall say that u

is a solution of (ES

) with initial data u

and external force f if and only

if u belongs to the space

C(R

; V

′

) ∩ L

∞

loc

; H) ∩ L

loc

; V

)

and satisﬁes, for any Ψ in C

; V

u(t), Ψ(t) +



[0,t]×Ω

(ν∇u : ∇Ψ − u · ∂

Ψ) (t

′

,x) dxdt

′



Ω

(x) · Ψ(0,x) dx +



f(t

′

), Ψ(t

′

)dt

′

The following theorem holds.

Theorem 3.1 The problem (ES

) has a unique solution in the sense of the

above deﬁnition. Moreover this solution belongs to C(R

; H) and satisﬁes

u(t)

+ ν



∇u(t

′

)

′

u





f(t

′

),u(t

′

)dt

′

54 Stability of Navier–Stokes equations

Proof In order to prove uniqueness, let us consider some function u

in C(R

; V

′

) ∩ L

loc

; V

) such that, for all Ψ in C

; V

u(t), Ψ(t) +



Ω

(ν∇u : ∇Ψ − u · ∂

Ψ) (t

′

,x) dxdt

′

=0.

This is valid in particular for a time-independent function P

Ψ where Ψ is any

given vector ﬁeld in V

. We may write

P

u(t), Ψ = u(t), P

Ψ

= −ν



Ω

∇u(t

′

,x):∇P

Ψ(x) dxdt

′

= ν



u(t

′

), ∆P

Ψdt

′

We have, thanks to the spectral Theorem 2.2,

P

u(t), Ψ = ν



P

u(t

′

), ∆P

Ψdt

′

≤ ν∆P

Ψ



P

u(t

′

)

′

≤ νkP

Ψ



P

u(t

′

)

′

By the deﬁnition of H, the space V

is dense in H, and we have

P

u(t)

= sup

Ψ

Ψ∈V

P

(t)u, Ψ

≤ νk



P

u(t

′

)

′

The Gronwall lemma ensures that P

u(t) = 0 for any t and k. This implies

uniqueness.

In order to prove existence, let us consider a sequence (f

)

k∈N

associated

with f by Lemma 2.2, page 44, and then the approximated problem

(ES

ν,k

)



∂

− νP

∆u

= f

|t=0

= P

Again thanks to the spectral Theorem 2.2, page 38, it is a linear ordinary diﬀer-

ential equation on H

which has a global solution u

which is C

; H

). By

the energy estimate in (ES

ν,k

) we get that

u

(t)

+ ν∇u

(t)

= f

(t),u

(t).

The time-dependent Stokes problem 55

A time integration gives

u

(t)

+ ν



∇u

′

)

′

P

u(0)



f

′

),u

′

)dt

′

(3.1.1)

In order to pass to the limit, we write the energy estimate for u

− u

k+ℓ

, which

gives

k,ℓ

(t)

def

(u

− u

k+ℓ

)(t)

+ ν



∇(u

− u

k+ℓ

)(t

′

)

′

(P

− P

k+ℓ

)u(0)



(f

− f

k+ℓ

)(t

′

),u

′

)dt

′

≤

(P

− P

k+ℓ

)u(0)



(f

− f

k+ℓ

)(t

′

)

′



∇(u

− u

k+ℓ

)(t

′

)

′



(u

− u

k+ℓ

)(t

′

)

′

This implies that

−νt

(u

− u

k+ℓ

)(t)

+ νe

−νt



∇(u

− u

k+ℓ

)(t

′

)

′

≤



(P

− P

k+ℓ

)u(0)



(f

− f

k+ℓ

)(t

′

)

′



This implies immediately that the sequence (u

)

k∈N

is a Cauchy sequence in the

space C(R

; H) ∩ L

loc

; V

). Let us denote by u the limit and prove that u

is a solution in the sense of Deﬁnition 3.1. As u

is a C

solution of the ordinary

diﬀerential equation (ES

ν,k

),wehave,foraΨinC

; V

u

(t), Ψ(t) = ν∆u

(t), Ψ(t) + f

(t), Ψ(t) + u

(t),∂

Ψ(t).

By time integration, we get

u

(t), Ψ(t) = −ν



Ω

∇u

′

,x):∇Ψ(t

′

,x) dxdt

′



f

′

), Ψ(t

′

)dt

′

+ P

u(0), Ψ(0) +



u

′

),∂

Ψ(t

′

)dt

′

Passing to the limit in the above equality and in (3.1.1) gives the theorem.

Remark This proof works independently of the nature of the domain Ω. In

the case when the domain Ω is bounded, the solution is given by the explicit

56 Stability of Navier–Stokes equations

formula

u(t)=



j∈N

(t)e

with

(t)

def

= e

−νµ

)



−νµ

(t−t

′

)

f(t

′

),e

dt

′

. (3.1.2)

In the case of the whole space R

, we have the following analogous formula

u(t, x)=(2π)

−d



ix·ξ

u(t, ξ) dξ with

u(t, ξ)

def

= e

−ν|ξ|

u

(ξ)+



−ν|ξ|

(t−t

′

)



f(t

′

,ξ) dt

′

. (3.1.3)

3.2 Stability in two dimensions

In a two-dimensional domain, the Leray weak solutions are unique and even

stable. More precisely, we have the following theorem.

Theorem 3.2 For any data u

in H and f in L

loc

; V

′

), the Leray weak

solution is unique. Moreover, it belongs to C(R

; H) and satisﬁes, for any (s, t)

such that 0 ≤ s ≤ t,

u(t)

+ ν



∇u(t

′

)

′

u(s)



f(t

′

),u(t

′

)dt

′

. (3.2.1)

Furthermore, the Leray solutions are stable in the following sense. Let u (resp. v)

be the Leray solution associated with u

(resp. v

)inH and f (resp. g)inthe

space L

loc

; V

′

). Then

−νt

(u − v)(t)

+ νe

−νt



∇(u − v)(t

′

)

′

≤



u

− v





(f − g)(t

′

)

′



exp



(t)



with

E(t)

def

= e

νt

min



u





f(t

′

)

′

, v





g(t

′

)

′



Remark When the domain Ω satisﬁes the Poincar´e inequality, the estimate

becomes

(u − v)(t)

+ ν



∇(u − v)(t

′

)

′

≤



u

− v





(f − g)(t

′

)

′



exp



(t)



with E

(t)

def

= e

−νt

E(t).

Stability in two dimensions 57

Proof of Theorem 3.2 As u belongs to L

∞

loc

; H) ∩L

loc

; V

), thanks

to Lemma 2.3, page 44, the non-linear term Q(u, u) belongs to L

loc

; V

′

Thus u is the solution of (ES

) with initial data u

and external force f + Q(u, u).

Theorem 3.1 immediately implies that u belongs to C(R

; H) and satisﬁes, for

any (s, t) such that 0 ≤ s ≤ t,

u(t)

+ ν



∇u(t

′

)

′

u(s)



f(t

′

),u(t

′

)dt

′



Q(u(t

′

),u(t

′

)),u(t

′

)dt

′

Using Lemma 2.3, we get the energy equality (3.2.1).

To prove the stability, let us observe that the diﬀerence w

def

= u − v is the

solution of (ES

) with data u

− v

and external force

f − g + Q(u, u) − Q(v, v).

Theorem 3.1 implies that

w(t)

+2ν



∇w(t

′

)

′

= w(0)



(f − g)(t

′

),w(t

′

)dt

′



(Q(u, u) − Q(v, v))(t

′

),w(t

′

)dt

′

The non-linear term is estimated thanks to the following lemma.

Lemma 3.1 In two-dimensional domains, if a and b belong to V

, we have

|(Q(a, a) − Q(b, b)),a− b| ≤ C∇(a − b)

a − b

∇a

a

Proof It is a nice exercise in elementary algebra to deduce from Lemma 2.3 that

Q(a, a) − Q(b, b),a− b = Q(a − b, a),a− b. (3.2.2)

Using Lemma 2.3 again, we get the result.

Let us go back to the proof of Theorem 3.2. Using the well-known fact

that 2ab ≤ a

+ b

,weget

w(t)



∇w(t

′

)

′

≤w(0)



(f − g)(t

′

)

′

+ C



∇w(t

′

)

w(t

′

)

∇u(t

′

)

u(t

′

)

′

+ Cν



w(t

′

)

′

Note that if the domain satisﬁes the Poincar´e inequality, then the last term on

the right-hand side can be omitted.

58 Stability of Navier–Stokes equations

Using (with θ =1/4) the convexity inequality (1.3.5), page 26 we infer that

w(t)

+ ν



∇w(t

′

)

′

≤w(0)



(f − g)(t

′

)

′



w(t

′

)



∇u(t

′

)

u(t

′

)

+ ν



′

Gronwall’s lemma implies that

w(t)

+ ν



∇w(t

′

)

′

≤



w(0)



(f − g)(t

′

)

′



× exp



νt +

sup

τ∈[0,t]

u(τ)



∇u(t

′

)

′



The energy estimate tells us that

sup

τ∈[0,t]

u(τ)



∇u(t

′

)

′

≤



u





f(t

′

)

′



2νt

As u and v play the same role, the theorem is proved.

3.3 Stability in three dimensions

In order to obtain stability, we need to enforce the time regularity of the Leray

solution. The precise stability theorem is the following.

Theorem 3.3 Let u be a Leray solution associated with initial velocity u

in H and bulk force f in L

([0,T]; V

′

). We assume that u belongs to the

space L

([0,T]; V

) for some positive T . Then u is unique, belongs to C([0,T]; H)

and satisﬁes, for any (s, t) such that 0 ≤ s ≤ t ≤ T ,

u(t)

+ ν



∇u(t

′

)

′

u(s)



f(t

′

),u(t

′

)dt

′

. (3.3.1)

Let v be any solution associated with v

in H and g in L

loc

([0,T]; V

′

). Then, for

all t in [0,T],

−νt

(u − v)(t)

+ νe

−νt



∇(u − v)(t

′

)

′

≤



u

− v





(f − g)(t

′

)

′



exp





∇u(t

′

)

′

