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

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A brief overview of dispersive phenomena 89

Then, functional analysis arguments based upon interpolation results (the

so-called TT

∗

result) imply that

u

(R;L

))

≤ C(γ



)

+ γ

−



)

) (5.1.9)

for suitable p and q, which is clearly a decay property. Using scaling arguments

and Sobolev embeddings, it can be proved that

∇u

([0,T ];L

∞

))

≤ C



γ



)

+ γ

−



)



for some s<d/2 depending on the dimension of the space. This kind of

smoothing estimate is used to solve non-linear wave equations locally in time.

The aim of this section is to emphasize basic properties which allow us to

derive dispersion estimates in a much more general framework, as long as waves

propagate in a physical medium. Indeed, the main tool for proving time decay for

solutions of wave equations is multiple integration by parts like in the stationary

phase theorem.

In the next subsection, we shall explain the general way to derive Strichartz

estimates and shall illustrate how this method works on the wave equation in

Subsection 5.1.2.

In Section 5.2, we apply these ideas to the rotating incompressible Navier–

Stokes equations. Dispersion takes place in the direction transverse to the

rotation vector, which inﬂuences the time decay of the associated waves in the L

∞

norm. Finally, the application to the non-linear case of the rotating Navier–Stokes

equations is given in Section 5.3.

5.1.1 Strichartz-type estimates

We now intend to describe mathematically dispersion phenomena in terms of

the time decay in the L

∞

norm in the case of frequency localized functions. It

turns out that dispersion is obtained by integration by parts just like in the

proof of the stationary phase theorem. Let d ≥1 and m ≥1 be two integers,

B a bounded open subset of R

, and Ω an open subset of R

.LetΨ∈D

∞

(B; C)

and a ∈ C

∞

×Ω; R). We deﬁne K : R ×Ω → C by

K(τ,z)=



Ψ(ξ) exp(iτ a(ξ, z)) dξ. (5.1.10)

We also introduce the stationary set

X = {(ξ, z) ∈ B × Ω / ∇

a(ξ,z)=0}.

We assume that the phase a satisﬁes the bounds

∇a ∈ L

∞

(B ×Ω) and ∇

a ∈ L

∞

(B ×Ω). (5.1.11)

We intend to prove the following two theorems.

90 Dispersive cases

Theorem 5.1 Let us consider the non-stationary case X = ∅. Assuming that

inf

B×Ω

|∇

a|≥β>0, (5.1.12)

then K decays in τ at any order: for any k ∈ N, there exists C

> 0 such that

for any τ in R

K(τ,·)

∞

(Ω)

≤ C

(1 + τ )

−k

. (5.1.13)

In the stationary case, we shall prove the following result.

Theorem 5.2 Assume that a function ξ

exists in C

∞

(Ω; B) such that

X = {(ξ

(z),z) ,z∈ Ω} and inf

|x|=1,z∈Ω

a(ξ

(z),z) · x|≥δ>0. (5.1.14)

Then, there exists C>0 such that for all τ ∈ R

∗

K(τ,·)

∞

(Ω)

≤ C|τ|

−d/2

. (5.1.15)

Proof of Theorems 5.1 and 5.2 The main idea in proving the above two

theorems is to perform multiple integrations by parts. As a matter of fact, we

introduce the following operator

L =

1 − iα∂

1+τ|α|

where α(ξ, z)

def

= ∂

a(ξ,z). (5.1.16)

It can be easily checked that Lexp(iτa)=exp(iτ a). On the other hand, the

transposed operator

L is deﬁned by



Lf · gdx=



f ·Lgdx for any (f,g) ∈D(R

; C)

Straightforward computations now yield

L =

1+τα

+ i



∂

α − τα

∂



(1 + τα

)

iα

1+τα

∂

. (5.1.17)

It follows from the assumptions of Theorem 5.1 that for all Ψ ∈D(B),



LΨ(τ,·)



∞

(B×Ω)

≤

1+β



Ψ

∞

(B)

∂

α

∞

(Ω)

+∇Ψ

∞

(B)

α

∞

(Ω)



As a result, it can be deduced by induction that for all k ∈ N

∗

, there exists a

positive C

such that for all τ ≥ 0, we have



(

(τ,·, ·)Ψ



∞

(B×Ω)

≤

(1 + τ )

Ψ

k,∞

(B)

. (5.1.18)

Writing

K(τ,z)=



Ψ(ξ) exp(iτ a(ξ, z)) dξ =



Ψ exp(iτa(ξ, z)) dξ,

A brief overview of dispersive phenomena 91

we deduce from (5.1.18) that

K(τ,·)

∞

(Ω)

≤ Vol(B)

(1 + τ )

Ψ

k,∞

(Ω)

which completes the proof of Theorem 5.1. In order to prove Theorem 5.2, it only

remains to deal with the case when (ξ,z) is localized in a neighborhood B

′

×Ωof

the stationary set X, otherwise Theorem 5.1 would apply and yield the claimed

time decay. For (ξ,z)inB

′

×Ω, we may use the non-degeneracy assumption on

the phase along the stationary set to prove that

|α(ξ,z)|≥|ξ − ξ

(z)|

As a result, we deduce similarly by induction that for k ∈ N and (ξ,z)inB

′

×Ω,

we have



(

Ψ(τ,ξ,z)



≤

(1 + τ |ξ − ξ

(z)|

)

ψ

k,∞

(B)

. (5.1.19)

For a given integer k greater than d/2, we conclude by making the change of

variables ζ =



|τ|(ξ−ξ

(z)) and using the fact that ζ →1/(1+|ζ|

)

is integrable

over R

5.1.2 Illustration of the wave equation

Let us consider again the wave equation (5.1.6). Solutions can be expressed in

terms of

(t, x)=



(ξ) exp(iξ · x ± it|ξ|) dξ. (5.1.20)

The above integral can be rewritten as a function of z = x/t in order to match

the assumptions of Theorem 5.2. It corresponds to the case when

a(ξ,z)=±|ξ| + z · ξ

and to amplitude functions localized in R

\{0}. Observing that

∇

a(ξ,z)=z ±

|ξ|

we deduce that the phase is likely to be stationary when ξ is directed along z. The

idea is then to write the integration over R

as an integration over R ×R

d − 1

for suitable one and d −1 dimensional spaces. Let us assume that the direction

of z is the ﬁrst vector basis e

. Then the phase can be written

a(ξ,z)=ξ

± (|ξ

′

+ |ξ

)

1/2

with ξ =(ξ

,ξ

′

) ∈ R ×R

d−1

and its d − 1 dimensional gradient as

∇

′

a(z,ξ)=±

′

|ξ|

92 Dispersive cases

so that the stationary set over R

d−1

′

is given by X = {(ξ

′

=0,z

)}. The second

derivative in ξ

′

of the phase is expressed as

∇

′

a(ξ

,ξ

′

, 0) =



′

−

′

|ξ|

⊗

′

|ξ|



|ξ|

and hence reduces to I

′

/|ξ

| along the stationary set (we recall that ξ

=0is

forbidden since the amplitude function is supported outside {0}). As a result,

Theorem 5.2 applies and yields the claimed time decay (5.1.8) in t

−(d−1)/2

. The

classical estimate (5.1.9) follows from the so–called TT

∗

argument (see [80]).

There are many applications of the above estimates to ﬂuid mechanics prob-

lems such as the low Mach number limit of compressible ﬂows (otherwise known

as the incompressible limit). In the inviscid case, the Euler equations can be

considered in the isentropic case to simplify to

(ALE

comp

)







∂

ρ + div (ρu)=0

∂

(ρu)+

∇ρ

γε

=0,

where γ>1 denotes the exponent of the pressure law and ε the Mach number of

the ﬂow. As the Mach number vanishes, the density tends to a constant, say 1,

and we ﬁnd div u = 0. Since general initial data do not necessarily have constant

density and incompressible velocity, waves propagate at the very high speed 1/ε.

Introducing the density ﬂuctuation

ψ =

ρ − 1

we get the acoustic wave equations

(WE

)











∂

u +

∇ψ

∂

ψ +

div u

and hence ∂

ψ − ∆ψ/ε

= 0. From the time decay (5.1.8) in L

∞

) deduced

from Theorem 5.2, classical duality arguments allow us to derive bounds such

as (5.1.9) for suitable p and q such that q>2. As a result, rescaling the time

according to the speed of sound 1/ε gives convergence to zero at a rate ε. In the

case of the Euler equations for ﬁnite time or in the viscous case globally in time,

convergence to the incompressible limit system can be proved for general initial

data, namely the local energy of the potential part of the ﬂow, which is carried

out by the acoustic waves, vanishes as the Mach number tends to zero.

Let us emphasize that a number of physically relevant systems can be studied

similarly, as long as waves propagate in an inﬁnite medium. Indeed, a linear-

ization procedure followed by a travelling wave analysis allows us to derive

dispersion relations expressing the pulsation ω as a function of the wavenumber

ξ. Then, the superposition of such waves can be analyzed like the above simple

example.

The particular case of the Rossby operator in R

The next section will illustrate once again dispersion eﬀects in a non-linear

system such as the rotating Navier–Stokes equations, which is the main purpose

of this book.

5.2 The particular case of the Rossby operator in R

In this section, we give another illustration of the preceding dispersion estimates

on a linearized version of the rotating Navier–Stokes equations. In the sequel,

we shall denote e

=(0, 0, 1) the unit vector directed along the x

-coordinate,

ε>0 the Rossby number and ν ≥0 the viscosity of the ﬂuid. The inviscid case

ν = 0 can be treated by using Theorem 5.2 and yields a τ

−1/2

time decay in the

∞

norm. In the Navier–Stokes-like case ν>0, the viscosity provides additional

regularity which can be used for nonlinear applications, so that the proof is

detailed in the sequel. The model equations read as

(VC

)











∂

v − ν∆v +

∧ v

+ ∇p = f

div v =0

|t=0

= v

which yields in Fourier variables ξ ∈ R

(FVC

)







∂

v + ν|ξ|

v +

ξ ∧ v

ε|ξ|



v

|t=0

= v

The matrix Mv

def

ξ ∧v

|ξ|

has three eigenvalues, 0 and ±i

|ξ|

· The associated

eigenvectors are

(ξ)=

(0, 0, 1)

and

(ξ)=

√

2|ξ||ξ



∓ iξ

|ξ|,ξ

± iξ

|ξ|, −|ξ



The precise value of these vectors is not needed for our study; all we need to

know is that the last two are divergence-free, in the sense that ξ · e

(ξ)=0.

Furthermore they are orthogonal, and we leave the proof of the following easy

property to the reader.

Lemma 5.1 Let v ∈H(R

) be given, and deﬁne

def

= F

−1



(v

(ξ) · e

(ξ))e

(ξ)



. (5.2.1)

Then

v

= v



+ v

−



94 Dispersive cases

We are now led to study

ε,±

(τ): g →



g(ξ)e

±iτ

|ξ|

−ντε|ξ|

+ix·ξ

dξ



×R

g(y)e

±iτ

|ξ|

−ντε|ξ|

+i(x−y)·ξ

dξdy,

ﬁrst considering the case when g is supported in C

r,R

for some r<R, deﬁned by

r,R

= {ξ ∈ R

/ |ξ

|≥r and |ξ|≤R}. (5.2.2)

Let us introduce

(t, τ, z)

def



ψ(ξ)e

±iτa(ξ,z)+iz·ξ−νt|ξ|

dξ, (5.2.3)

where

a(ξ,z)

def

|ξ|

and ψ is a function of D(R

\{0}), such that ψ ≡ 1 in a neighborhood of C

r,R

which in addition is radial with respect to the horizontal variable ξ

=(ξ

,ξ

As we shall need anisotropic-type estimates in the next section (when we

apply those estimates to the full non-linear rotating ﬂuid equations), it is

convenient to introduce also

(t, τ, z

,ξ

)

def



ψ(ξ)e

±iτa(ξ,z

)+iz

·ξ

−νt|ξ|

dξ

. (5.2.4)

Note that

(t, τ, z)=



(t, τ, z

,ξ

) dξ

. (5.2.5)

Lemma 5.2 For any (r, R) such that 0 <r<R, a constant C

r,R

exists such

that ∀z ∈ R

, ∀ξ

∈ R,

(t, τ, z)| + |I

(t, τ, z

,ξ

)|≤C

r,R

min{1,τ

−

. (5.2.6)

Proof Due to (5.2.5) and to the fact that ξ

is restricted to r ≤|ξ

|≤R,itis

enough to prove the result on I

and the estimate on K

will follow.

The proof of the estimate on I

follows the lines of Theorem 5.2 in the

previous section, in a very simple way; the only diﬀerence is that we have to take

care of the dependence upon the viscosity. Moreover for the sake of simplicity

we will only consider I

, the term I

−

being dealt with exactly in the same way.

First using the rotation invariance in (ξ

,ξ

), we restrict ourselves to the

case when z

= 0. Next, denoting α(ξ)

def

= − ∂

a(ξ)=ξ

/|ξ|

, we introduce

the following diﬀerential operator:

def

1+τα

(ξ)

(1 + iα(ξ)∂

) ,

The particular case of the Rossby operator in R

which acts on the ξ

variable, and satisﬁes L(e

iτa

)=e

iτa

. Integrating by parts,

we obtain

(t, τ, z

,ξ



iτa(ξ,z

)+iz



L(ψ(ξ)e

−νt|ξ|

)



dξ

Easy computations yield



ψ(ξ)e

−νt|ξ|





1+τα

− i(∂

α)

1 − τα

(1 + τα

)



ψ(ξ)e

−νt|ξ|

−

iα

1+τα

∂



−νt|ξ|

ψ(ξ)



As ξ belongs to the set C

r,R

deﬁned by (5.2.2), we have clearly

|ξ

≤|α(ξ)|≤

hence

1+τα

|1 − τα

1+τα

|α|

1+τα

≤

r,R

1+τξ

An easy computation also shows that |∂

α(ξ)|≤C

r,R

. Finally, since ψ ∈D(R

we have



∂



−νt|ξ|

ψ(ξ)





≤|∂

ψ(ξ)|e

−νtr

+ νt|ξ

||ψ(ξ)|e

−νtr

≤ C

r,R

−

Putting all those estimates together we infer that





ψ(ξ)e

−νt|ξ|





≤

r,R

1+τξ

−

so since r ≤|ξ

|≤R and |e

iτa(ξ,z

)+iz

| = 1, we obtain, for all z

∈ R

and

all ξ

∈ R,

(t, τ, z

,ξ

)|≤C

r,R

−



dξ

1+τξ

which proves Lemma 5.2.

Lemma 5.2 yields the following theorem.

Theorem 5.3 For any positive constants r and R such that r<R, let C

r,R

the frequency domain deﬁned in (5.2.2). Then a constant C

r,R

exists such that

if v

∈ L

) and f ∈ L

; L

)) are two vector ﬁelds such that

Supp v

∪



t≥0

Supp



f(t, ·) ⊂C

r,R

96 Dispersive cases

and if v is the solution of the linear equation (VC

) with forcing term f and

initial data v

, then for all p in [1, +∞],

v

∞

))

≤ C

r,R



v



)

+ f

))



(5.2.7)

and

v

∞,2

)

≤ C

r,R



v



)

+ f

))



, (5.2.8)

where we have noted, for (α, β) ∈ [1, +∞]

f

α,β

def



f(x

, ·)

)



)

Proof Let us start by proving estimate (5.2.7). Its anisotropic counterpart

(estimate (5.2.8)) will be obtained in a similar though slightly more complicated

way.

Before proving the result (5.2.7), we note that Duhamel’s formula enables

us to restrict our attention to the case f = 0. Indeed if f is non-zero then we

write v = v

+ v

−

with

(t)

def

= G

ε,±







ε,±



t − t

′



′

) dt

′

where we have used notation (5.2.1) above. Then by Lemma 5.1 it is enough to

prove the result for v

remembering that

v



)

≤v

)

and f



))

≤f

))

Note that the eigenvalue 0 does not appear in this formula since the correspond-

ing eigenvector is not divergence-free.

If the result (5.2.7) holds when f

= 0, then of course

G

ε,+







∞

))

≤ C

r,R

v



)

Then we write, if p =1,

ε,+

(f)

def





ε,+



t − t

′



′

) dt

′



∞

)

≤





ε,+



t − t

′



′

)



∞

′

≤



∞

′



ε,+



t − t

′



′

)



∞

dtdt

′

≤ C

r,R



f(t

′

)

′

The particular case of the Rossby operator in R

which yields the result if p = 1. Similarly if p =+∞,

ε,+

(f) ≤ sup

t≥0





ε,+



t − t

′



′

)



∞

′

≤ C

r,R

sup

t≥0



f(t

′

)

′

and estimate (5.2.7), for all p ∈ [1, ∞], follows by interpolation. So from now on

we shall suppose that f =0.

As a consequence of Lemma 5.2, considering g such that g is supported in C

r,R

and ψ ∈D(R

\{0}) radial with respect to ξ

, such that ψ ≡ 1 in a neighborhood

of C

r,R

, one has

ε,+

(τ)g(x)=



×R

ψ(ξ)g(y)e

iτ

|ξ|

−ντε|ξ|

+i(x−y)·ξ

dξdy



(ετ,τ,x− y) g(y) dy,

where K

is deﬁned by formula (5.2.3). Moreover, the following estimate holds:



ε,+







∞

≤ C

r,R

−ct

g

Now we shall use a duality argument, otherwise known as the TT

∗

argument.

Once we have observed that

b

∞

))

= sup

ϕ∈B



×R

b(t, x)ϕ(t, x) dxdt,

with

B =

ϕ ∈D(R

×R

), ϕ

∞

))

≤ 1

(

we can write



ε,+







∞

)

= sup

ϕ∈B



×R



,x− y



g(y)ϕ(t, x) dtdxdy

which can be written



ε,+







∞

)

= sup

ϕ∈B



×R

g(y)







,x− y



ϕ(t, x)dx



dtdy.

98 Dispersive cases

A Cauchy–Schwarz inequality then yields, denoting

b(x)=b(−x),



ε,+







∞

)

≤g

with

def

= sup

ϕ∈B







, ·



∗ ϕ(t, ·) dt



By the Fourier–Plancherel theorem, we have

(2π)







, ·



ϕ(t, ·) dt



≤ C



×R





, −ξ



ϕ(t, ξ)





, −ξ



ϕ(s, ξ) dξdtds.

By deﬁnition (5.2.3) of K,wehave



(t, τ, −ξ)=ψ(−ξ)e

iτa(−ξ,z)

−τt|ξ|

Thus the following identity holds





, −ξ







, −ξ







t + s,

t − s

, −ξ



ψ(−ξ).

It follows that

≤ C



)

×R

ψ(ξ)F



t + s,

t − s

, ·



∗ ϕ(t, ·)



ϕ(s, ξ) dξdtds.

We now use the Fourier–Plancherel theorem again to get

≤ C



)

×R

ψ(ξ)



t + s,

t − s

, ·



∗ ϕ(t, ·)



(x)ϕ(s, −x) dxdtds,

≤ C



)





t + s,

t − s

, ·



∗ ϕ(t, ·)



∞

ϕ(s, ·)

dtds,

≤ C



)





t + s,

t − s

, ·





∞

)

ϕ(t, ·)

ϕ(s, ·)

)

dtds.

The dispersion estimate (5.2.6) on K

yields

≤ C

r,R



)

(t − s)

−νr

(t+s)

ϕ(t, ·)

)

ϕ(s, ·)

)

dtds.

Now we conclude simply by writing

≤ C

r,R

ϕ

∞

))



)

(t − s)

dsdt.