Lianhua HE

School of Mathematical Sciences,Xiamen University,Xiamen 361005,China;School of Mathematics Science,Guizhou Normal University,Guiyang 550001,China

E-mail:hlh1981@126.com

Zhong TAN

School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scienti fi c Computing,Xiamen University,Xiamen 361005,China

E-mail:tan85@xmu.edu.cn

Abstract In this article,we consider the partial regularity of stationary Navier-Stokes system under the natural growth condition.Applying the method of A-harmonic approximation,we obtain some results about the partial regularity and establish the optimal H?lder exponent for the derivative of a weak solution on its regular set.

Key words partial regularity;Navier-Stoke systems;natural growth condition;A-harmonic

1 Introduction

In this paper,we shall consider the partial regularity of the stationary Navier-Stokes systems which satisfy the following type

Moreover,due to the assumptions aboutand Biit is easily seen that the classical Navier-Stokes system

in its weak formulation is included in(1.1)provided n≤4.

In this article,we consider the partial regularity of weak solutions of system(1.1)by the method of harmonic approximation.However,for example,potential theory and means of representation formulate[2–4]also can be used to deal with the problem of the partial regularity.Here the technique of harmonic approximation is a new method that derives from Simon’s proof[5]of the regularity theorem of Allard[6],which shows that a function g for

whichR

?Dg ·D?dx is sufficiently small for all test functions ?,lies L2-close to some harmonic function.This technique has been successfully applied to obtain partial regularity results of general elliptic systems[1,7–9],parabolic systems and Navier-Stokes systems[10–12].Chen and Tan[10]proved the partial regularity under controllable growth condition.The natural growth condition was not used widely,see[13–15].In this paper,we fi ll this gap in the theory and extend the result of Duzaar[7]to prove the partial regularity under the natural growth condition(H4).

Our work is organized as follow.In Section 2,we introduce some preliminary lemmas which will be very usefully in our proof.The fi rst step of our proof is to establish a Caccioppoli type inequality in Section 3 which plays an important role in the proof of the decay estimate.Next,we apply the Caccioppoli type inequality in Section 3 and A-harmonic approximation lemma in Section 2 to obtain the decay estimation in Section 4.Finally,by iterating the decay estimation,we can infer the our main result about optimal regularity as follow.

Theorem 1.1We assume that u ∈ H1,2(?,RN)∩L∞(?,RN)with ?·u=g for g ∈ L2(?)is a weak solution of systems(1.1)under the natural growth condition and the conditions(H1)–(H3)satisfying k u kL∞≤ M.Then ?1is a open in ? and u ∈ C1,β(?1,Rn).Further???1?Σ1∪Σ2where

and

And in particular,Ln(? ? ?1)=0

2 Preliminary Lemmas

Let us consider the operator divergence Lu= ? ·u:(?,RN)→ L2(?).Clearly it is a continuous operator and its adjoint L?:L2(?)→ H?1(?,RN)is given by L?f= ?f in the sense of distributions.If(?,RN)is equipped with the scalar product

then the canonical isomorphism between H10and H?1is de fi ned by the Laplace operator

Obviously,we have

Moreover,see for example Ladyzhenskaya[4],Temam[16],we have

Lemma 2.1Range of L?=R(L?)= △(KerL⊥).In particular,R(L?)is closed.

It is well known that the condition “R(L)is closed”,L being a linear and continuous operator between two Hilbert spaces,can be stated in many di ff erent ways;we shall need the following equivalent statement

(1)R(L)is closed.

(2)R(L?)is closed.

(3)R(L)=(KerL?)⊥.

(4)There exists a positive constant C such that k u k≤C k Lu k for all u∈KerL⊥.

(5)There exists a positive constant C such that k f k≤ C k L?f k for all f ∈ KerL?⊥.

Lemma 2.1 implies:if T ∈ H?1(?,RN)and hT,?i=0 for all ? ∈ KerL,then there exists a function p ∈ L2(?)such that L?p=T and it permits to interpret problem(1.1).In fact,due to the growth conditions,if u ∈ H1(?,RN)is a weak solution to problem(1.1)then?Dα(x,u,?u)?Bi(x,u,?u)∈ H?1and therefore there exists a function p ∈ L2(?)such that?Dα?Bi=(L?p)i,i.e.,weak solutions to problem(1.1)are solutions in the sense of distributions to

for some p∈ L2(?).Moreover,from statement(5),it follows

In the sequel we shall refer to u or(u,p)as the weak solution to systems(1.1).Obviously,if(u,p)∈C2×C1is a weak solution and??∈C1,1,then(u,p)satis fi es(2.1)in a classical sense.The opposite is also true,since the closure of the space of smooth solenoidal vector fi elds in(?,RN)coincides with

(see[4,16]).

Now for all open sets ω ? ?,we denote by

the orthogonal projection of(ω,RN)into the orthogonal complement to V(ω).For all φ ∈(ω,RN),we have φ ? Pωφ ∈ V(ω).Then we can de fi ne the weak solution of system(1.1)as the following type.

De fi nition 2.2A function u ∈ H1(?,RN)with ? ·u=g is a weak solution to system(1.1)if and only if for all open sets ω ? ? and for all φ ∈(ω,RN),the following equality holds

From statements(3)and(4),it follows

Lemma 2.3Let ω be a bounded domain with Lipschitz boundary.Let f be a function

Because constants in(2.2),(2.5),(2.6)and statements(4)–(5)are obtained from the open mapping theorem,we do not know the explicit dependence on the domain.The following remark will be useful for us.For fi xed ? denote such constant by c(?).We immediately see that c(?)is invariant under translations,orthogonal transformation,or dilatation.In particular,if the domain ? in(2.2)or Lemma 2.4 is as following

then the constant c(?)does not rely on x0and R.

Now we present the A-harmonic approximation lemma,the Poincar′e inequality and a result due to Campanato.We restrict ourselves here to noting that the lemma is in fact true if condition(2.8)is replaced by the Legendre-Hadammard condition.Such a version has already been applied by Duzaar and Grotowski(see[7]Lemma 2.1)

De fi nition 2.5For A∈RnN,we de fi ne u∈H1,2(?,RN)is A-harmonic if it satis fi es

Lemma 2.6(A-harmonic approximation lemma) Consider fi xed positive λ and L,and n,N ∈ N with n ≥ 2.Then for any given ε>0 there exists δ= δ(n,N,λ,L,ε) ∈ (0,1]with the following property:for any A∈Bil(RnN)satisfying

and

for any g∈ H1,2(Bρ(x0),RN)(for some ρ>0,x0∈ Rn)satisfying

satisfying

Lemma 2.7(Poincar′e inequality) There exists cpdepending only on n,without loss of generality cp≥1,such that every u∈H1,2(Bρ(x0))satis fi es

The proof of this lemma can be read in the textbook[17](Section 7.8).From(7.45)in that book we note that the above result follows with cp=22n.

Our fi nal tool is a standard estimate for the solutions to homogeneous second order elliptic systems with constant coefficients,due originally to Campanato[18](Theorem 9.2).The result follows from Caccioppoli’s inequality for h and its derivatives of any order,Sobolev’s inequality,and Poinca′e inequality.Note that the original result is given for equations,but extends immediately to systems.for convenience we state the estimate in a slightly more general form than that given in[18].

Lemma 2.8Consider A,λ and L in Lemma 2.4.Then there exists c0depending only to n,N,λ and L(without loss of generality,we take c0≥1)such that any A-harmonic function h on Bρ(x0)satis fi es

Finally,we introduce an important lemma which is used in latter proof.

Lemma 2.9Let f(t)be a nonnegative bounded function de fi ned for 0≤T0≤t≤T1.Suppose that for T0≤t

where A,B,α,β,θ are constants and θ<1.Then there exists a constant c1=c1(θ,α,β)such that for T0≤ρ

3 Caccioppoli Type Inequality

In the section,our aim is to prove the Caccicoppoli type inequality.

Theorem 3.1Consider p0fi xed in RnN,u0fi xed in RN. Let u ∈ H1,2(?,RN)∩L∞(?,RN)with ? ·u=g be a weak solution of system(1.1)satisfying kukL∞ ≤ M and conditions(H1)–(H4)be satis fi ed.Then for all x0∈ ? and arbitrary ρ and R with 0< ρ

with c2depending on λ,L,n,N;c3depending on λ, β,L,n,N and c4depending on λ,L,n,N, ?.

From H?lder’s,Young’s,Sobolev’s inequalities and Lemma 2.4,we can gain

4 The Proof of Main Theorem

In this section,we proceed to the proof of the partial regularity result.

Now,we proceed to the proof of the partial regularity result.

For ε>0 to be determined later,we take δ= δ(n,N,λ,L,ε)∈ (0,1]to be the corresponding constant from A-harmonic approximation Lemma 2.6.Writing

where we use(4.34)and(4.37)in obtaining the second-last inequality,and(4.33)for the fi nal inequality.

From(4.37)and(4.38),assumption(4.35)holds for all j∈N.Now it is standard to deduce from(4.37)that ρ → Φ(x0,ρ)can be dominated,for ρ ∈ (0,ρ0],by a constant times ρ2β.Since(4.34)will continue to hold for any x sufficiently close to x0.From[18],we can infer the desired partial regularity result(cf.[17]).