On the Uniqueness of Weak Solutions for the 3D Navier-Stokes Equations

www.elsevier.com/locate/anihpc

On the uniqueness of weak solutions for the 3D Navier–Stokes equations

Qionglei Chen

, Changxing Miao

, Zhifei Zhang

aInstitute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, PR China bSchool of Mathematical Sciences, Peking University, 100871, PR China

Received 23 August 2008; received in revised form 21 January 2009; accepted 28 January 2009 Available online 23 February 2009

Abstract

In this paper, we improve some known uniqueness results of weak solutions for the 3D Navier–Stokes equations. The proof uses the Fourier localization technique and the losing derivative estimates.

©2009 Elsevier Masson SAS. All rights reserved.

Keywords:Navier–Stokes equations; Uniqueness; Weak solution; Fourier localization; Losing derivative estimates

1. Introduction

We consider the three-dimensional Navier–Stokes equations inR³

⎧⎨

⎩

ut−u+u· ∇u+ ∇p=0, divu=0,

u(0)=u₀(x),

(1.1)

whereu=(u¹(t, x), u²(t, x), u³(t, x))andp=p(t, x)denote the unknown velocity vector and the unknown scalar pressure of the fluid respectively, whileu₀(x)is a given initial velocity vector satisfying divu₀=0.

In a seminal paper [21], J. Leray proved the global existence of weak solution with finite energy, that is, u(t, x)∈LTdef

=L^∞

0, T;L²

∩L²

0, T;H¹

for anyT >0.

It is well known that weak solution is unique and regular in two spatial dimensions. In three dimensions, however, the question of regularity and uniqueness of weak solution is an outstanding open problem in mathematical fluid mechanics. In this paper, we are interested in the classical problem of finding sufficient conditions for weak solutions of (1.1) such that they become regular and unique. Let us firstly recall the definition of weak solution.

* Corresponding author.

E-mail addresses:chen_qionglei@iapcm.ac.cn (Q. Chen), miao_changxing@iapcm.ac.cn (C. Miao), zfzhang@math.pku.edu.cn (Z. Zhang).

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.01.008

Definition 1.1. Let u₀∈L²(R³) with divu₀=0. A measurable function u is called a weak solution of (1.1) on (0, T )×R³if it satisfies the following conditions:

(1) u∈LT ∩C_w([0, T];L²), whereC_w([0, T];L²)consists of all weak continuous functions with respect to time in L²(R³);

(2) divu=0 in the sense of distribution;

(3) For any functionψ∈C₀^∞([s, t] ×R³)with divψ=0, there holds t

s

R³

u·ψ_t− ∇u· ∇ψ+ ∇ψ:(u⊗u) (t, x) dx dt=

R³

u(t, x)·ψ (t, x) dx−

R³

u(s, x)·ψ (s, x) dx.

In addition, ifusatisfies the energy inequality u(t )²

2+2 t 0

∇u(t)²

2dt u₀ ²₂,

it is also called a Leray–Hopf weak solution.

The Leray–Hopf weak solutions are unique and regular in the class P=L^q

0, T;L^r

with2 q +3

r =1, 3r∞ [11,14,15,25,27], or P=L^q

0, T;W^1,r

with2 q +3

r =2, 3

2< r∞ [1], or P=L^q

0, T;W^s,r

with 2 q +3

r =1+s, 3

1+s< r∞, s0 [26].

Recently, there are many researches devoted to refine the above results. First of all, we have the following refined regularity criterion in the framework of Besov spaces: the weak solutions are regular in the class

P=C

[0, T];B_∞⁻¹_,∞

or P=L^q

0, T;B_p,^r _∞ ,

with ²_q+³_p=1+r, ₁₊³_r < p∞,and−1< r1, see [4,8,17,18]. Concerning the refined uniqueness criterion of weak solutions, Kozono and Taniuchi [16] proved the uniqueness of the Leray–Hopf weak solutions in the class

P=L²(0, T;BMO).

Gallagher and Planchon [12] proved the uniqueness in the class P=L^q

0, T; ˙B⁻¹⁺

3 p+²_q p,q

with 3 p+2

q >1.

Lemarié-Rieusset [19] proved the uniqueness in the class P=C

[0, T];X₁⁽⁰⁾

or P=L^1−2r(0, T;Xr) withr∈ [0,1).

Finally, Germain [13] proved the uniqueness in the class P=C

[0, T];X₁⁽⁰⁾

or P=L^1−r2 (0, T;X_r) withr∈ [−1,1).

HereB_p,q^s denotes the Besov space and

X_s:=

⎧⎪

⎨

⎪⎩

MH˙^s, L²

, ifs∈(0,1], Λ^sBMO, ifs∈(−1,0], Lip, ifs= −1,

whereM(H˙^s, L²)is the space of distributions such that their pointwise product with a function inH˙^s belongs toL², Λ^s=(1−)^s2.X⁽⁰⁾_s denotes the closure of the Schwartz class inX_s. We want to point out that

X_s→Λ^sBMO, ifs∈(0,1]. (1.2)

We refer to [13] for more properties aboutXs. The key step of their proofs is to find a path spacePso that the trilinear form

F (u, v, w):=

T 0

R³

u· ∇v·w dx dt

is continuous from(LT)²×PtoR. Germain also pointed out that the path spacePhe found is optimal in some sense (see [13, p. 400] for precise meaning).

The purpose of this paper is to improve the above uniqueness results.

Theorem 1.2.Letu₀, v₀∈L²(R³)withdivu₀=divv0=0. Letuandvbe two Leray–Hopf weak solutions of (1.1) on(0, T )with the initial datau₀andv₀respectively. Assume that

u∈L^q

0, T;B_p,^r _∞ ,

with_q²+_p³=1+r, ₁₊³_r < p∞, r∈(0,1], and(p, r)=(∞,1). Then there holds u(t )−v(t )²

2+ t 0

∇(u−v)(t)²

2dt u₀−v₀ ²₂exp

C t 0

e+u(t)

B_p,^r_∞

q

dt

.

In particular, ifu₀=v₀, thenu=va.e. on(0, T )×R³. Remark 1.3.Due to the embedding relation

B_p,q^s B_p,^s _∞, q <+∞ and Λ^−rBMOB_∞^r _,∞,

Theorem 1.2 is an improvement of the corresponding results given by Gallagher and Planchon [12] and Germain [13].

The proof only uses an important observation that ifu∈L^q(0, T;B_p,^r _∞)with(p, q, r)as in Theorem 1.2, thenucan be decomposed as

u=u^l+u^h withu^l∈L¹(0, T;Lip)andu^h∈L^q˜

0, T;L^p˜ for somep,˜ q˜satisfying_q²_˜ +_p³_˜=1,p >˜ 3, see Lemma 3.1.

In the case where eitherr0 or(p, r)=(∞,1), using Bony’s decomposition and the losing derivative estimates, we prove

Theorem 1.4.Letu0∈L²(R³)withdivu0=0. Letuandv be two weak solutions of (1.1)on(0, T )with the same initial datau0. Assume thatuandvsatisfy one of the following two conditions:

(a) u∈L^q1(0, T;B_p^r1_1,∞)andv∈L^q2(0, T;B_p^r2_2,∞), where 2

q1+ 3

p1 =1+r₁, 2 q2+ 3

p2 =1+r₂, withr₁, r₂∈(−1,0],r₁+r₂>−1, ₁₊³_r

1 < p₁∞, ₁₊³_r

2 < p₂∞.

(b) u, v∈L¹(0, T;B_∞¹_,∞).

Thenu=va.e. on(0, T )×R³.

Remark 1.5.Due to the embedding relation X_sΛ^sBMOB_∞^−s_,∞, s∈(0,1],

the condition imposed on weak solution in Theorem 1.4 is weaker than that of Germain [13] and Lemarié- Rieusset [19]. However, the price to pay is to impose the conditions on both weak solutions.

Remark 1.6.The main novelty of Theorem 1.4 is that weak solutions are unique in the class L¹(0, T;B_∞¹_,∞). In particular, from the inequality

u _B1

∞,∞C

u ₂+ curlu _B0

∞,∞

(see Section 2 for its proof), (1.3)

we can obtain theBeale–Kato–Majdatype uniqueness criterion: if weak solutionsuandvwith the same initial data satisfy

curlu,curlv∈L¹

0, T;B_∞⁰_,∞ ,

thenu=von(0, T )×R³. Secondly, Theorem 1.4 allows us to impose different conditions on both weak solutions.

Thirdly, we do not impose the energy inequality on weak solutions.

Remark 1.7. Chemin and Lemarié-Rieusset [6,20] proved the uniqueness of weak solutions in the class C([0, T];B_∞⁻¹_,∞). While, Theorem 1.4 gives the uniqueness in the class L¹(0, T;B_∞¹_,∞). It is natural to expect that the uniqueness also holds in the classL^1+r2 (0, T;B_∞^r _,∞)forr∈(−1,1)from the viewpoint of interpolation. This problem remains unknown for the case ofr∈(−1,−¹₂].

Remark 1.8. The result (b) in Theorem 1.4 is also valid for the Euler equation. In detail, let u, v ∈ C_ω([0, T );L²(R³))∩L¹([0, T );B_∞¹_,∞(R³)be two weak solutions of the Euler equation with the same initial data, thenu=va.e. on[0, T ).

Notation.Throughout the paper,Cstands for a generic constant. We will use the notationABto denote the relation ACB, and · pdenotes the norm of the Lebesgue spaceL^p.

2. Preliminaries

Let us firstly recall some basic facts on the Littlewood–Paley decomposition, one may check [5] for more details.

Choose two nonnegative radial functions χ, ϕ∈S(R³)supported respectively in B= {ξ ∈R³,|ξ| ⁴₃}andC= {ξ∈R³, ³₄|ξ|⁸₃}such that for anyξ ∈R³,

χ (ξ )+

j0

ϕ 2^−jξ

=1. (2.1)

Leth=F⁻¹ϕandh˜=F⁻¹χ, the frequency localization operator_jandS_j are defined by jf =ϕ

2^−jD f=2^3j

R³

h 2^jy

f (x−y) dy, forj0,

S_jf =χ 2^−jD

f=

−1kj−1

_kf =2^3j

R³

h˜ 2^jy

f (x−y) dy,

and

₋₁f=S₀f, _jf=0 forj−2.

With our choice ofϕ, one can easily verify that

_jkf =0 if|j−k|2 and

_j(S_k−1f _kf )=0 if|j−k|5. (2.2)

For anyf∈S(R³), we have by (2.1) that f =S0(f )+

j0

jf, (2.3)

which is called the Littlewood–Paley decomposition. In the sequel, we will constantly use the Bony’s decomposition from [2] that

uv=T_uv+T_vu+R(u, v), (2.4)

with

Tuv=

j

Sj−1ujv, R(u, v)=

|j−j|1

ju_jv,

and we also denote

T_uv=Tuv+R(u, v).

With the introduction of_j, let us recall the definition of the inhomogenous Besov space from [29]:

Definition 2.1.Lets∈R, 1p, q∞, the inhomogenous Besov spaceB_p,q^s is defined by B_p,q^s =

f ∈S R³

; f _Bp,q^s <∞ , where

f _B^s

p,q :=

⎧⎨

⎩ (_∞

j=−12^{j sq} _jf ^q_p)^1q, forq <∞, sup_j−12^{j s} _jf _p, for q= ∞.

Let us point out thatB_∞^s _,∞is the usual Hölder spaceC^s fors∈R\Zand the following inclusion relations hold LipB_∞¹_,∞, Λ^−sBMOB_∞^s _,∞ fors∈R.

We refer to [13,29] for more properties.

The following Bernstein’s inequalities will be frequently used throughout the paper.

Lemma 2.2.(See [5].) Let1pq∞. Assume thatf ∈L^p, then there hold suppfˆ⊂

|ξ|C2^j ⇒ ∂^αf

qC2^{j|α|+3j (p1−q1)} f _p, suppfˆ⊂

1

C2^j|ξ|C2^j

⇒ f _pC2^−j|α| sup

|β|=|α|

∂^βf

p. Here the constantCis independent off andj.

We conclude this section by a proof of the inequality (1.3). Using Lemma 2.2, we have ₋₁u _∞C u ₂,

and forj0,

2^j _ju _∞C _j∇u _∞.

Due to the Biot–Savart law [23],∇ucan be written as

∇u(x)=Cw(x)+K∗w(x), w=curlu,

whereCis a constant matrix, andKis a matrix valued function with homogeneous of degree−3. So, we get that for j 0,

2^j ju _∞C jw _∞, where we used the fact that

j(Tf )

pC jf p, forj0, 1p∞,

ifT is a singular integral operator of convolution type with smooth kernel [28]. Then the inequality (1.3) is concluded from the definition of Besov space.

3. Proofs of theorems

This section is devoted to the proof of Theorems 1.2 and 1.4.

3.1. The proof of Theorem 1.2

The proof is based on the following decomposition lemma which may be independent of interest.

Lemma 3.1.Assume thatu∈L^q(0, T;B_p,^r _∞)with ²_q+_p³ =1+r, ₁₊³_r < p∞,r∈(0,1], and(p, r)=(∞,1).

Thenucan be decomposed as

u=u^l+u^h withu^l∈L¹(0, T;Lip)andu^h∈L^q˜

0, T;L^p˜ for somep,˜ q˜satisfying ²_q˜+³_p˜ =1,p >˜ 3.

Proof. FixN∈Nto be determined later on. We set u^l=S_Nu, u^h=u−u^l.

By the definition ofS_Nand Lemma 2.2, we have ∇u^l

∞C

jN−1

2^{j (1+p3)} _ju _pC2^2(1−1q)N u _Bp,^r_∞. (3.1) Due to the conditions on(p, q, r), we can choosep˜such that

˜

p >max(3, p) and 3 p −3

˜

p−r <0.

Thus, by Lemma 2.2 u^h

˜

p

jN

2^(p3−p3˜)j _ju _pC2^{(3p−p3˜−r)N} u _Bp,^r_∞. (3.2) Now we choose

N= q

2log₂

e+ u _Bp,∞^r +1.

Then by (3.1), we have T

0

∇u^l(t )

∞dtC T 0

e+u(t )

B_p,^r_∞

q

dt <+∞. (3.3)

On the other hand, from (3.2) we get that

T 0

u^h(t )^q˜_˜

pdtC

T 0

e+u(t )

B_p,^r_∞

q

dt <+∞. (3.4)

Hence, we complete the proof of Lemma 3.1 by (3.3) and (3.4). 2

Lemma 3.2.Letu, vbe as in Theorem1.2. Setw=u−v. Then for anyt∈ [0, T], there holds u(t ), v(t )

+2 t 0

∇u,∇vdt= u0, v0 + t 0

w· ∇u, wdt.

Proof. Lemma 3.1 ensures that the trilinear form F (u, v, w):=

T 0

R³

u· ∇w·v dx dt

is continuous from (LT)²×L^q(0, T;B_p,^r _∞)to R. Then the lemma can be proved by following the argument of Lemma 4.4 in [13]. Here we omit the details. 2

Now we are in position to prove Theorem 1.2. Sinceuandvare Leray–Hopf weak solutions, there hold u(t )²

2+2 t 0

∇u(t)²

2dt u₀ ²₂, v(t )²

2+2 t 0

∇v(t)²

2dt v₀ ²₂. On the other hand, Lemma 3.2 yields that

u(t ), v(t ) +2

t 0

∇u,∇vdt= u0, v0 + t 0

w· ∇u, wdt.

Combining the above inequalities, we obtain w(t )²

2+2 t 0

∇w(t)²

2dt=u(t )²

2+v(t )²

2−2u, v(t )+2 t 0

∇u(t)²

2dt

+2 t 0

∇v(t)²

2dt−4 t 0

∇u,∇v(t) dt

u0−v0 2 2−2

t 0

w· ∇u, wdt. (3.5)

We decomposeu=u^l+u^has in Lemma 3.1 and rewrite t

0

w· ∇u, wdt= t 0

w· ∇u^l, w dt+

t 0

w· ∇u^h, w dt.

We get by Hölder inequality that

t 0

w· ∇u^l, w dt

t 0

w(t)²

2∇u^l(t)

∞dt. (3.6)

Integration by parts, we get t

0

w· ∇u^h, w dt= −

t 0

w· ∇w, u^h dt,

from which and the Gagliardo–Nirenberg inequality, it follows that

t 0

w· ∇u^h, w dt

t 0

∇w 2 w 2p˜

˜ p−2

u^h

˜ pdt

C

t 0

∇w ₂ w ¹⁻

3˜ p

2 ∇w

3˜ p

2u^h_˜

pdt

C

t 0

w(t)²

2u^h(t)^q˜

˜ pdt

¹

˜

qt

0

∇w(t)²

2dt 1−_q¹_˜

C

t 0

w(t)²

2u^h(t)^q˜

˜ pdt+

t 0

∇w(t)²

2dt.

This together with (3.5) and (3.6) gives w(t )²

2+ t 0

∇w(t)²

2dt u₀−v₀ ²+C t 0

w(t)²

2∇u^l(t)

∞+u^h(t)^q˜

˜ p

dt.

This jointed with the Gronwall inequality produces that w(t )²

2+ t 0

∇w(t)²

2dt u0−v0 2

exp

C t 0

∇u^l(t)

∞+u^h(t)^q˜

˜ p

dt

u₀−v₀ ²₂exp

C t 0

e+u(t)

B_p,^r_∞

q

dt

.

This finishes the proof of Theorem 1.2.

3.2. The proof of Theorem 1.4

Assume thatuandvare two weak solutions of (1.1) on(0, T )with the initial datau0. Letw=u−v,wsatisfies the equation in the sense of distribution

w_t−w+w· ∇u+v· ∇w+ ∇ ˜p=0, (3.7)

for some pressurep. We get by taking the operation˜ _j on both sides of (3.7) that

∂_tjw−_jw+_j(w· ∇u)+_j(v· ∇w)+ ∇_jp˜=0. (3.8)

Multiplying (3.8) by_jw, we get by Lemma 2.2 forj−1 that 1

2 d

dt_jw(t )²

2+ca_j2^2j_jw(t )²

2−

_j(w· ∇u), _jw

−

_j(v· ∇w)−v· ∇_jw, _jw

, (3.9)

witha₋₁=0 anda_j=1 forj0. Here we used the fact that _j(v· ∇w), _jw

=

_j(v· ∇w)−v· ∇_jw, _jw .

Case 1.uandvsatisfy the assumption (a).

Due tor₁+r₂>−1, one ofr₁andr₂must be bigger than−¹₂. Without loss of generality, we assume thatr₁>−¹₂. Step 1.Estimate of_j(w· ∇u), _jw.

Using the Bony’s decomposition (2.4), we have _j(w· ∇u)=_j

T_wi∂_iu +_j

T_∂iuwⁱ

+_jR wⁱ, ∂_iu

.

Considering the support of the Fourier transform of the termT_wi∂iu, we have _j

T_wi∂_iu

=

|j−j|4

_j

S_j−1wⁱ∂_iju

. (3.10)

This gives by Lemma 2.2 that j

T_wi∂iu

2

|j−j|4

2^j

kj−2

kw 2p1

p1−2 _ju p₁

|j−j|4

2^j

kj−2

2^kp31 kw 2 _ju p₁

2^{j (1−r1)} u _B^r1 p1,∞

jj+2

2^j

3

p1 _jw 2. (3.11)

Similarly, we have _j

T_∂iuwⁱ

=

|j−j|4

_j

S_j−1(∂_iu)_jwⁱ

. (3.12)

Applying Lemma 2.2 to (3.12) yields that _j

T_∂iuwⁱ

2

|j−j|4

kj−2

2^k _ku _{∞ j}w ₂ 2^{j (1−r1+}

3 p1)

u _B^r1 p1,∞

|j−j|4

_jw ₂. (3.13)

Since divw=0, we have _jR

wⁱ, ∂_iu

=

j,jj−3; |j−j|1

∂_ij

_jwⁱ_ju

, (3.14)

from which and Lemma 2.2, it follows that _jR

wⁱ, ∂_iu 2p1

p1+2

j,jj−3; |j−j|1

2^j _jw _{2 j}u _p1 2^j u _B^r1

p1,∞

jj−3

2^−jr1 _jw 2. (3.15)

Summing up (3.11)–(3.15), we obtain _j(w· ∇u), _jw2^{j (1−r1)} u _B^r1

p1,∞

jj

2^j

3

p1 _jw _{2 j}w ₂

+2^{j (1+p31)} u _B^r1 p1,∞

jj

2^−jr1 _jw _{2 j}w ₂. (3.16)

Step 2.Estimate of_j(v· ∇w)−v· ∇_jw, _jw. Using the Bony’s decomposition (2.4), we write

_j(v· ∇w)=_j T_vi∂_iw

+_j T_∂iwvⁱ

+_jR vⁱ, ∂_iw

, v· ∇_jw=T_vi∂_ijw+T_∂

ijwvⁱ. Then we have

j(v· ∇w)−v· ∇jw= j, T_vi

∂iw+j

T∂_iwvⁱ

+jR vⁱ, ∂iw

−T_∂

i_jwvⁱ. Similar arguments as in deriving (3.11) and (3.15), we have

_j

T_∂iwvⁱ

22^{−j r2} v _B^r2 p2,∞

jj+2

2^j(1+

3 p2)

_jw ₂, (3.17)

jR

vⁱ, ∂iw 2p2

p2+2 2^j v _B^r2 p2,∞

jj−3

2^−jr2 _jw 2. (3.18)

In view of the definition ofT_∂

i_jwvⁱ, T_∂

i_jwvⁱ=

jj−2

S_j+2j∂iw_jvⁱ,

and note thatS_j+2_jw=_jwforj> j, we get T_∂

i_jwvⁱ, _jw

=

j−2jj

S_j+2_j∂_iw_jvⁱ, _jw ,

from which and Lemma 2.2, it follows that T_∂

i_jwvⁱ, jw2^{j (1+}

3 p2−r2)

v _B^r2

p2,∞ jw ²₂. (3.19)

Now, we turn to estimate[T_vi, _j]∂_iw. In view of the definition of_j, we write T_vi, _j

∂_iw=

|j−j|4

S_j−1vⁱ, _j

∂_ijw

=

|j−j|4

2^3j

R³

h

2^j(x−y)

S_j−1vⁱ(x)−S_j−1vⁱ(y)

∂i_jw(y) dy

=

|j−j|4

2^4j

R³

1 0

y· ∇S_j−1vⁱ(x−τy) dτ ∂_ih 2^jy

_jw(x−y) dy, (3.20) from which and the Minkowski inequality, we deduce that

T_vi, _j

∂_iw

2

|j−j|4

∇S_j−1v _{∞ j}w ₂ 2^{j (1+}

3 p2−r₂)

v Bp^r22,∞

|j−j|4

_jw ₂. (3.21)

Summing up (3.17)–(3.21), we obtain

_j(v· ∇w)−v· ∇_jw, _jw2^{−j r2} v _B^r2 p2,∞

jj

2^j(1+

3 p2)

_jw 2 _jw 2

+2^{j (1+}

3 p2)

v _B^r2 p2,∞

jj

2^−jr2 _jw _{2 j}w ₂. (3.22)