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Yong ZHOU

Local well-posedness for the incompressible Euler equations in the criticalBesov spacesTome 54, no 3 (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 thefollowing results. Given vo E m &#x3E; N/2 + 1, Kato [9] proved localexistence 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 theincompressible 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.

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for 2-D incompressible Euler equations in the critical (borderline) Besovspaces Bp~p~1 (I~2). Later, Vishik [18] proved the existence (N = 2) anduniqueness (TV ~ 2) result for (1.1) with initial vorticity belonging to aspace 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 ofthe 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 decreasingfunctions. Given f E S, the Fourier transform is defined as

One can extend and .~-1 to S’ in the usual way, where S’ denotes theset of all tempered distributions. Let 0 E S satisfying

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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 abovequasi-norm given by Definition 2.1. For s &#x3E; 0, (p, q) E (1,00) x we

define the inhomogeneous Besov space norm and inhomogeneousTriebel-Lizorkin space norm of f E S’ as

p,q

The inhomogeneous Besov and Triebel-Lizorkin spaces are Banach spacesequipped 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 followinginequality is true.

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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 differentfrom line to line.

Consider the following linear system

We have the following local existence theorem for (3.1), which will be provedin the appendix.

PROPOSITION 3.1. - Assume that div w = 0,for some T &#x3E; 0. Then for any

unique solutionr-

to (3.1). And consequently, B7 P canbe determined uniquely.

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In order to prove the existence part of the main theorem, we considerthe following approximate linear iteration system for (1.1)

where v° = 0. In [3], Chae used a similar (not same) iterative system toconstruct 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) canbe 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 integrateover 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 withand integrating over we have

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Then apply ’. on (3.5) and take summation,

Now we turn our attention to the estimates for B7 pn+l. Taking divergenceon 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 boundedfrom 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),

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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 byinduction.

In fact, suppose that the initial datum vo satisfiesthen 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 forn + 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 followsI - -11 1 ~ - ~ I 1 ~

Just as what done for vn+l, multiplying (3.12) coordinate by coordinatewith Thanks to Holder’s inequality, we have

Taking A. on (3.12), we get

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Multiplying (3.14) coordinate by coordinate withintegrating 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 have11 T

Therefore, we have

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where we used the Holder inequality and embedding Lemma 2.3. Andsimilarly,

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 followsfrom (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 thelimit for the approximate sequence vn+’. Moreover, from the equation, wehave 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 sameinitial datum. If we set v = vl - v2 and P = Pl - P2, then we get a similar

system as (3.12)

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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 wasdone 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 isa constant matrix. By the boundedness of the singular integral operator [8],we have

So Theorem 1.2 follows from (4.3) and (4.4).

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5. Appendix.

Proof of Proposition 2.4. - We use Bony’s decomposition [2], [5] topresent 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

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Then (2.5) follows from (5.1), (5.2) and (5.3).

Remark 5.1. - Actually, we can prove the following Moser typeinequality

provided that

Proof of Proposition 3.1. - The idea is to approximate (3.1) bylinear transport equations. First it is easy to check that (3.1) is equivalentto 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 proofof Theorem 1.1, we should give a uniform estimates for the sequence vn+1and 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

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Since we have the a priori estimate (5.8), the existence and uniqueness ofsolutions 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 supervisorProfessor 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 TheInstitute of Mathematical Sciences. This work is partially supported byHong 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 KongInstitute of Mathematical Sciences

Department of MathematicsShatin, N.T., (Hong Kong).&#x26;

Xiamen UniversityDepartment of MathematicsXiamen, Fujian (Chine).yzhou~math.cuhk.edu.hk