L’INSTITUTFOURIER DE ANNALES

N A L E S E D L ’IN IT ST T U

F O U R IE

ANNALES

DE

L’INSTITUT FOURIER

Yong ZHOU

Local well-posedness for the incompressible Euler equations in the critical Besov spaces

Tome 54, n^o3 (2004), p. 773-786.

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LOCAL WELL-POSEDNESS FOR THE INCOMPRESSIBLE EULER

EQUATIONS IN THE CRITICAL BESOV SPACES

by Yong

^ZHOU

1.

Introduction.

In this _paper, we consider the

incompressible

^Euler

equations

ⁱⁿ

N &#x3E; 3,

where

v(x, t)

^E

^JRN

stands for the

velocity field, P(x, t)

^{is the pressure,}

while

f (x, t)

^{is the}

^force,

which will be assumed ^{as zero}

just

^for

simplicity.

Our main results can be _gone

through

^for any

f

^E

L~(0, T; B’P, 1

For the local

well-posedness

^{of the}

sysytem (1.1),

^we mention the

following

results. Given vo E ^m &#x3E;

N/2 + ^1,

^Kato

[9] proved

^local

existence and

uniqueness

^{for a solution}

belonging

^to

C([O, T]; Hm(JRN))

with T ⁼ Later on, many various function _spaces

(see [3], [4], [5], [10], [13])

^{are used to} establish the local existence and

uniqueness

^{for the}

incompressible

^Euler

equations.

^For

example,

^{with s} &#x3E;

N /p + 1,

1 p ^oo is used in

[10]

^and

Fp,q

^{for s} &#x3E;

N/p

is used in

[3].

^In

particular,

^Vishik

[17]

showed the

(global) well-posedness

Keywords: Well-posedness - ^Euler equations - Besov spaces.

Math. classification: 76D03 - 35Q35 - ^46E35.

for 2-D

incompressible

^Euler

equations

ⁱⁿ the critical

(borderline)

^Besov

spaces

Bp~p~1 (I~2). ^Later,

^Vishik

^{[18] proved}

the existence

(N

⁼

2)

^and

uniqueness (TV ~ 2)

result for

(1.1)

with initial

vorticity belonging

^{to a}

space of Besov

type.

^The purpose of this paper is to establish local well-

posedness

^of

( 1.1 )

^{in for} any 3.

THEOREM 1.1. ^- Let 1 p ^oo. Given _any i

there exists a T ⁼ and a

unique

^solution

(v, VP)

^to

(1. 1)

p1‘

such that

The maximum local existence

time,

say

T*,

^is called the

lifespan

^of

the solution. If T* is

finite,

^we say that the solution blows up at time T*.

In our case it is

limsuPt~T* ||v (.,t)

^oo.

^Beal,

^{Kato and}

Majda [1]

p,l

established the

following blow-up

criterion for the smooth solution

v (x, t)

JU

where w - curl v is the

vorticity

^{field. Later} on, ^some refined results were

proved

ⁱⁿ

~11~, [12].

^The

^blow-up

criterion for our case reads

THEOREM 1.2. ^- The local solution constructed in Theorem l.l blows up at time T* if and

only

^if

2.

Littlewood-Paley decomposition

^{and Besov} spaces.

We start

by recalling

^the

Littlewood-Paley decomposition

^of

temper-

ate distributions. Let S be the class of Schwartz class of

rapidly decreasing

functions. Given

f

^E

S,

the Fourier transform is defined as

One can extend and

.~-1

^to S’ ⁱⁿ the usual _way,

where S’

denotes the set of all

tempered

distributions.

Let 0

^{E S}

satisfying

for ~ ^I

any

f

^E

S’,

^{we define}

Then the

homogeneous

Besov semi-norm and Triebel-Lizorkin semi-norm are defined

by

The _space

Bp,q

^and

Fp,q

^are

quasi-normed

spaces with the above

quasi-norm given by

Definition 2.1. For s &#x3E;

0, (p, q) E (1,00)

^{x we}

define the

inhomogeneous

^Besov space ^norm and

inhomogeneous

Triebel-Lizorkin _space

norm

_p,q ^of

f

^E

^S’

^as

The

inhomogeneous

Besov and Triebel-Lizorkin _spaces ^are Banach _spaces

equipped

^{with the}

respectively.

Let us now state some classical results.

LEMMA 2.2

[14], [16] (Bernstein’s ^Lemma).

^- Assume that k E

Z~,

and

Supp

^. then there exists a

constant

C(k)

such that the

follouTing inequality

^holds.

For _any k E

Z+,

there exists a constant

C(k)

such that the

following

inequality is

^true.

LEMMA 2.3

[14], [16] (Embeddings).

^-

^{Let p E} (1, oo),

^then

PROPOSITION 2.4

(Product).

belong

^{to . , and}

In

particular, for ,

^{, there holds}

This

proposition

will be showed in the

appendix.

with

3.

Proof

of Theorem 1.1.

In this

section,

^{C denotes a absolute}

constant,

^which

maybe

^different

from line to line.

Consider the

following

^linear

system

We have the

following

local existence theorem for

(3.1),

which will be

proved

in the

appendix.

PROPOSITION 3.1. ^- Assume that

div w = 0,

for some T &#x3E; 0. Then for _any

unique

^solution

r-

to

(3.1).

^And

consequently,

^{B7 P can}

be determined

uniquely.

In order to prove the existence

part

^{of the main}

theorem,

^{we consider}

the

following approximate

^{linear iteration}

system

^for

(1.1)

where

v° =

0. In

[3],

^{Chae used a similar}

(not same)

^iterative

system

^to

construct the local solution. But

unfortunately,

the linear

system (3.32)- (3.33)

^{on page 671 of}

[3]

^{is not}

solvable,

^{since the}

system

itself lacks consistence.

If we have the uniform estimate for the _sequence v"

by induction,

which satisfies the conditions in

Proposition 3.1,

^{then the}

system (3.2)

^can

be solved with solution Uniform estimates.

First

multiply (3.2)

coordinate

by

coordinate with

where is the 1-th coordinate of the ^vector field

vn+l,

^and

integrate

over

JRN. Taking

^the

divergence

^free

property

^{of Vn into}

account,

^{we have}

therefore,

Note that for p ⁼

2, multiplying (3.2) by

^and

integrating by parts,

we have

Now

takingAj

^on

(3.2),

^we

get

Multiplying (3.4)

coordinate

by

coordinate with and

integrating

^{over we have}

Then

apply

^{’. on}

(3.5)

^{and take}

summation,

Now we turn our attention to the estimates for

B7 pn+l. Taking divergence

on the both sides of

(3.2),

^it follows that

thus

Thanks to the

divergence

^free

property

^of

vn,

^{we obtain}

For 1 p m, it was

proved [14], [16]

^{that -}

^{and Ri}

^{is bounded}

from LP into itself

[8], [15].

^{Due to} Bernstein’s

lemma,

^{we have}

It follows from

(3.7)

^that

where we used that

Ri

is bounded from

Bp,q

^{into itself}

^[8].

Combining (3.3), (3.6), (3.8)

^and

(3.9),

Note that

although

the above constants C

maybe depend

^{on N and} p, it is

nothing

^{to do with} n, therefore we can obtain uniform estimates

by

induction.

In

fact,

suppose that the initial datum vo satisfies then the

following inequality

^holds

for all n &#x3E;

0, provided

^that

Tl (independent

^of

n)

^is

sufficiently

^small.

(3.11)

^{can be showed}

easily by

mathematical induction.

First,

^{it is}

true for n = 0.

Suppose (3.11)

^{holds for n, we} want to prove it is true for

n + 1. It follows from

(3.10)

^that

Hence, (3.11) ^holds,

^{if we choose}

^Tl

^{so small that}

Moreover, Tl

^is

independent

^{of n.}

’

Convergence.

To _prove the convergence, it is sufficient to estimate the difference of the iteration. Take the difference between the

equation (3.2)

^{for the}

(n

⁺

l)-th ^step

and the n-th

step,

^{and set}

then we

get

^the

equation

^{as follows}

I - -11 1 ~ - ~ I 1 ~

Just as what done for

vn+l, multiplying (3.12)

coordinate

by

^coordinate

with Thanks to Holder’s

inequality,

^{we have}

Taking A.

^on

(3.12),

^we

^get

Multiplying (3.14)

coordinate

by

coordinate with

integrating

^over

R ,

^{we have}

Then

apply 2 jnlp

^on

(3.15)

^{and take}

summation,

Combining (3.13)

^and

(3.16),

^{we have}

We ^can estimate

B7IIn+l

^as follows. From the

equation (3.2),

^{it follows}

Thanks to the

divergence

^{free of}

vn,

^{we have}

11 T

Therefore,

^{we have}

where we used the Holder

inequality

^and

embedding

Lemma 2.3. And

similarly,

where we used

(2.6).

Then

integrate (3.17)

^{on the time interval}

(0,T) by taking (3.18)

^and

(3.19)

^into

account,

So if we choose

T2 Tl sufficiently

small such that

where

C1

^{is the constant obtained} for the uniform

estimate,

^{then it follows}

from

(3.20)

^that

Hence due to

(3.21),

^{it is clear that}

as n tends ^to

infinity.

Therefore,

the solution to the

system (1.1)

^{is obtained}

by taking

^the

limit for the

approximate

sequence

vn+’. Moreover,

^{from the}

equation,

^we

have v E

C(o, T; BP 1p+1 ) .

^This

^completes

^the

^proof

of local existence

part.

Uniqueness.

Suppose (vi, PI)

^and

(v2 , P2 )

^{are two solutions to}

( 1.1 )

^{with the same}

initial datum. If we set v = _{vl - v2} and P =

Pl - P2,

^{then we}

get

^{a similar}

system

^as

(3.12)

Just as what done for the convergence

part

^{for the} sequences, ^we

obtain,

from

(3.22),

If we choose

T x min{T1, T2}

^{such that}

where

C1

^{is the constant obtained}

by

the existence

part

^{such that}

then, (3.23)

^{tells us, on}

(0, T),

this

implies

^the

uniqueness.

4.

Proof

of Theorem 1.2.

The

proof

is easy.

Indeed, just

^as the uniform estimate which ^was done in section

3,

^{we have the}

following

estimate for the solution to

(1.1).

On the other

hand,

^the pressure ^can be estimated as

Therefore,

^{it follows from}

(4.1)

^and

(4.2)

^that

/ rot t

Then use the known fact that

where P is a

singular integral operator homogeneous

^of

degree -N

^{and A is}

a constant matrix.

By

the boundedness of the

singular integral operator [8],

we have

So Theorem 1.2 follows from

(4.3)

^and

(4.4).

5.

Appendix.

Proof of

Proposition

^{2.4. - We use}

Bony’s decomposition [2], [5]

^to

present

^the

product

^as

where

By compactness

^{of the}

supports

of the series of Fourier

transform,

^for any

it follows that

Similarly,

It follows from

Bony’s

formula that

Therefore, by

^Minkowski

inequality,

^{we have}

Then

(2.5)

follows from

(5.1), (5.2)

^and

(5.3).

Remark 5.1. ^-

Actually,

^{we can} prove the

following

^Moser

type inequality

provided

^that

Proof of

Proposition

^{3.1. - The idea is to}

approximate (3.1) ^by

linear

transport equations.

^{First it is} easy to check that

(3.1)

^is

equivalent

to the

following system.

So we

approximate (5.4) by

^linear

transport equations

The existence theorem for

(5.5)

is well-known for each n. Just as the

proof

of Theorem

1.1,

^{we should}

give

^a uniform estimates for the _sequence

vn+1

and the convergence of the

corresponding

sequence. In order to do _so, ^we

only

^{need to do a}

priori

^{estimates for the}

equivalent system (5.4).

The estimate for the _pressure is _{easy now,}

Therefore,

^{it follows from}

(5.6)

^that

Apply

^Gronwall

inequality

^on

(5.7),

^then

Since we have the a

priori

^estimate

(5.8),

the existence and

uniqueness

^of

solutions for the

system (5.4)

^can be obtained

by

^the

approximate

sequence solutions to

(5.5).

This finishes the

proof

^of

Proposition

^3.1.

6.

Acknowledgment.

The author would like to express sincere

gratitude

^{to his}

supervisor

Professor

Zhouping

Xin for enthusiastic

guidance

^and constant encourage- ment. And thanks also to Professor

Shing-Tung

^{Yau and Professor}

Zhoup- ing

^{Xin for}

providing

^{an excellent}

study

and research environment in The Institute of Mathematical Sciences. This work is

partially supported by Hong Kong

^RGC Earmarked Grants CUHK-4219-99P and CUHK-4279- OOP.

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Manuscrit reCu le 20 juin 2003, accepté ^{le 18 novembre 2003.}

Yong ZHOU,

Chinese University ^of Hong Kong

Institute of Mathematical Sciences Department ^of Mathematics

Shatin, N.T.,

(Hong Kong).

&#x26;

Xiamen University

Department ^of Mathematics

Xiamen, Fujian

(Chine).

yzhou~math.cuhk.edu.hk