亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        ARTIAL REGULARITY FOR STATIONARY NAVIER-STOKES SYSTEMS BY THE METHOD OF A-HARMONIC APPROXIMATION*

        2020-08-03 13:10:54YichenDAI戴祎琛

        Yichen DAI (戴祎琛)

        School of Mathematical Sciences, Xiamen University, Fujian 361005, China

        E-mail: yichendai0613@163.com

        Zhong TAN (譚忠)

        School of Mathematical Science and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Fujian 361005, China

        E-mail: tan85@xmu.edu.cn

        Abstract In this article, we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition. The proof is based on the A-harmonic approximation technique. In this article,we extend the result of Shuhong Chen and Zhong Tan [7] and Giaquinta and Modica [18] to the stationary Navier-Stokes system with subquadratic growth.

        Key words Stationary Navier-Stokes systems; controllable growth condition; partial regu-larity; A-harmonic approximation

        1 Introduction and Statement of the Result

        Throughout this article, on a domain, where is a bounded with Lipschitz boundary in Rnwith dimension n ≥ 2, we consider weak solutions u:Ω→RNof stationary Navier-Stokes systems of the type

        1.1 Assumptions on the structure functions

        for all x ∈?, u ∈RN, and ξ, ζ ∈RNn, where ν, L>0 are given constants.

        There exist β ∈(0,1) and K :[0,∞)?→[1,∞) monotone nondecreasing such that

        Furthermore,we assume that there is a function ω :[0,∞)×[0,∞)?→[0,∞)with ω(t,0)=0 for all t such that t →ω(t,s) is monotone nondecreasing for fixed s, where sω(t,s) is concave and monotone nondecreasing for fixed t, and such that

        for all x ∈?, u ∈RN, and ξ,∈RNn.

        1.2 Assumptions on the inhomogeneity Bi

        For the inhomogeneity Bi, we assume that Bisatisfies the controllable growth condition

        for all x ∈?, u ∈RN, and Du ∈RNn, where

        Moreover, because of the growth assumptions above, it is easily seen that the classical Navier-Stokes system

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

        Let us consider the Helmholtz-Weyl decomposition of the space Lp. Refer to the study of the relevant properties of these spaces by [17]:

        Set

        for p ∈[1,∞). We denote by Hp(?) the completion of D in the norm of Lpand put

        Let ω be either a bounded or an exterior C2-smooth domain or a half space in Rn, n ≥2.

        Then, Gp(ω) and Hp(ω) are orthogonal subspaces in Lp(ω). Moreover,

        where ⊕denotes direct sum operation.

        Furthermore,the validity of decomposition(1.2)implies the existence of a unique projection operator

        that is, of a linear, bounded, idempotent (= Pω) operator having Hp(ω) as its range and Gp(ω) as its null space.

        As a result, we obtain, for all ? ∈W1,p0 (ω,RN),

        The point of the technique of harmonic approximation is to show that a function is‘a(chǎn)pproximately-harmonic’. That is, a function g, for which

        ? Dg·D?dx is sufficiently small for any test function ?, lies L2-close to some harmonic function. The harmonic approximation lemma can be found in Simon’s proof[24]. Allad[2]and de Giorgi[11]developed the regularity theory of minimal surfaces (see [11]). Until now, the technique of harmonic approximation was developed further and adapted to various settings in the regularity theory (see [16]), for a survey on the numerous applications of harmonic type approximation lemmas. The harmonic approximation method allows the author to simplify the original ε-regularity theorem(see[25])because of Schoen-Uhlenbeck (see [26]).

        In this article, we extend the result of Shuhong Chen and Zhong Tan [7] and Giaquinta and Modica [18] to the stationary Navier-Stokes system with subquadratic growth. However,Shuhong Chen and Zhong Tan [7] has proven the partial regularity under the controllable condition (B1) with polynomial growth rate p = 2. This controllable growth condition has not been used widely, even in the parabolic systems system (see [6, 10]. In this article, we fill this gap in the theory and prove the partial regularity with subquadratic growth under the controllable growth condition (B1).

        Our work is organized as follows. First of all, we collect some preliminary material forms in Section 2, which will be useful in our proof. The first step of our proof is to establish a Caccioppoli type inequality in Section 3. Next, we derive decay estimate in Section 5,which characterizes the singular set, by using the A-harmonic approximation lemma stated in Section 2.

        Next, we will state our main result as follows.

        Theorem 1.1We assume that u ∈W1,p(?,RN) with ?·u = g is a weak solution of systems (1.1) under the conditions (A1) to (A4), the controllable growth condition (B1), and g ∈L2(?).

        Then, there exists an open subset ?u?? with

        We point out that the H?lder exponent β of the gradient of the solution is the optimal one so that no higher regularity of the solution can be expected.

        Moreover, we have the following characterization of the singular set.

        Proposition 1.2In the situation of the preceding theorem, the singular set satisfies,moreover,

        where

        and

        In the case of a structure function A(x,u,ξ) ≡A(x,ξ) that does not depend on u, the above statement remains true if we replace the setby

        Furthermore, we have

        and

        2 Preliminary Material

        2.1 The function V and W

        We define

        for all A ∈Rk, where k ∈N, which immediately yields

        Then, we state some standard inequalities for later reference [1, 5].

        Lemma 2.1For any 1 < p < 2, A,B ∈Rk, and V as defined above, we have the followings:

        Immediately, we draw a conclusion that for |B ?A|≤1,

        For |B ?A|>1, we have

        Lemma 2.2For every 1

        for any A,B ∈Rk.

        2.2 A-harmonic approximation

        Here, we state an A-harmonic approximation lemma. Its proof can be found in [14]. We consider bilinear forms A on RNnthat are positive and bounded in the sense

        for all ξ, η ∈RNn.

        Definition 2.3A map h ∈W1,1(BR(x0),RN) is called A-harmonic if it satisfies the following linear parabolic system

        Lemma 2.4Let ν and L be two positive constants. Assume that A is a bilinear form on RNnwith the properties (2.3) and∈W1,p(Bρ(x0),RN) is approximately A-harmonic in the sense

        Then, for any ε>0, there is a δ >0, depending on p, n, N, ν, L, and ε, and an A-harmonic map h such that

        and

        Next, we recall a simple consequence of the a prior estimates for solutions of linear elliptic systems of second order with constant coefficients;see[5](Proposition 2.10)for a similar result.

        Lemma 2.5Assume that h ∈W1,1(Bρ(x),RN) satisfy

        where the constant C1depends only on n, N, κ, and K.

        2.3 Poinc′are-type inequality

        We state a Poinc′are-type inequality involving the function V,which can be found in[5,14].

        Lemma 2.6(Poinc′are-type inequality)

        Let 1

        where p′=2n/(n ?p).

        In particular, the previous inequality is valid with p′replaced by 2.

        Next, we state a useful elementary lemma. Its proof can be found in [19].

        Lemma 2.7For R0< R1, we assume that f : [R0,R1] →[0,∞) is a bounded function and, for all R0<σ <ρ

        for nonnegative constants A, B, α, and ? ∈(0,1). Then, we have the estimate

        for all R0<σ0<ρ0

        Finally, we state two lemmas which will be used to estimate the Hausdroff dimension of the singular set.

        Lemma 2.8([22, 4.2]) Let u ∈Wθ,p(Br(x0),Rn), where p ≥1, θ ∈(0,1), and Br(x0)?RN. Then, we have, for a constant c=c(n,q),

        Lemma 2.9([22, Section 4]) Let ? be a open set in Rn. Let λ be a finite, non-negative,and increasing function defined on the family of open subsets of ? which is also countably superadditive in the following sense that

        whenever {Oi}i∈Nis a family of pairwise disjoint open subsets of ?. Then, for 0 < α < n, we have dimH(Eα)≤α, where

        3 A Caccioppoli Type Inequality

        The first step in the proof of partial regularity is to establish a Caccioppoli type inequality,prepared for decay estimate in Section 5. The precise statement is as follows.

        For x0∈?, u0∈RN, and p0∈RNn, we define P =P(x)=u0+p0(x?x0).

        Theorem 3.1Let u ∈W1,p(?,RN) with ?·u = g be a weak solution of system (1.1),satisfying (A1) to (A3) and the controllable growth condition (B1).

        Then, for arbitrary ρ and R with 0<ρ

        where σ =max{2p/(p ?2β),p/(p ?1 ?β)} and 0<β

        Here, the constant c depends only on n, N, p, L, λ, and ν.

        ProofLet 0 < ρ ≤s < t ≤R and choose a standard cut off function ? ∈(Bt(x0))with ? ≡1 in Bs(x0), 0 ≤? ≤1 and |D?|≤. Let ? and ψ be two test functions satisfying

        where v =u(x,t)?P(x)=u ?u0?p0(x ?x0), such that

        Because supp Dψ ?BtBs, we have

        A straightforward calculation shows that, using the ellipticity condition (A1) stated in the Section 1 and Lemma 2.2, respectively, we have

        Then, we infer that

        where

        Next, we successively deal with the estimation of the terms I1to I5.

        3.1 Estimate for I1

        Here, we consider that

        so that

        Combining assumption (A3), estimates (3.2), (3.5) and Young’s inequality, we imply that

        3.2 Estimate for I2

        From (A3), we arrive at

        In order to estimate it, we split the domain of integration into four parts as

        which are written by I2≤I21+I22+I23+I24.

        For the first term I21, we infer by Young’s inequality

        Secondly, we arrive at

        Similarly,

        Combining (3.7) to (3.11), we deduce that

        where σ =max{2,2p/(p?2β),p/(p ?1),p/(p ?1 ?β)}=max{2p/(p?2β),p/(p ?1 ?β)} and 0<β

        3.3 Estimate for I3

        Similarly as I2, we arrive at

        3.4 Estimate for I4

        The term I4can be estimated by H?lder’s,Sobolev’s,and Young’s inequalities,respectively.

        3.5 Estimate for I5

        Finally, we deal with I5as

        First of all, we decompose the term I51into

        For the term I511, we use assumption (A2) and Lemma 2.2, that is,

        where Du?(1?θ1)Dψ =Du?Dψ+θ1Dψ =(Du?Dψ)+θ1(Du?(Du?Dψ)):=A+θ1(B?A).

        Recalling estimate (3.5), we use the fact that supp Dψ ?BtBsand Young’s inequality

        Similarly, the term I512, we use assumption (A2) and Lemma 2.2 to obtain

        where p0+θ2D?=p0+θ2(D?+p0?p0):=A+θ2(B ?A).

        The terms I513, I514, and I52are similar as I2, I3, and I4, respectively. Recalling (3.4),(3.6), and (3.12)–(3.17), we arrive at

        where we used Lemma 2.1 to imply

        4 Approximate A-Harmonicity by Linearization

        In this Section, we apply a linearization argument which will be useful to prove that the weak solutions of systems (1.1) are approximately A-harmonic provided their excess is small.This result is also prepared for decay estimate in Section 5, where we used the A-harmonic approximation lemma.

        Definition 4.1We define excess functionals

        Lemma 4.2Let u ∈W1,p(?,RN)with ?·u=g be a weak solution of (1.1). We consider ρ<1 and

        Furthermore, we fix p0in RNnand set, then,

        ProofNote that

        Then, we have

        4.1 Estimate for I1

        For the first term I1, we distinguish the case of |Du ?p0| ≤1 and |Du ?p0| > 1. We employ assumption (A2), (A4), and Lemma 2.2 with the result

        where we used the H?lder’s inequality and Jensen’s inequality.

        4.2 Estimate for I2

        Next, we estimate I2with the help of the continuity assumption(A3), the Lemma 2.1,and Young’s inequality by

        4.3 Estimate for I3

        Similarly, we split the domain of integration into four parts as

        We can derive by Lemma 2.1

        4.4 Estimate for I4

        Finally, for the term I4, we use a fact that, H?lder’s, Sobolev’s, and Young’s inequalities, then obtain

        Putting together (4.2)–(4.5) and recalling the definition of H(t), we infer from (4.1) that

        5 A Decay Estimate

        In this Section, we establish an initial excess-improvement estimate by assuming that the excess Φ(ρ)is initially sufficient small. And we characterize the singular set of the solution with the help of the crucial decay estimate.

        Lemma 5.1(Excess-improvement estimate) We assume that the hypotheses listed in Section 1 are in force and M > 0. For any weak solution u ∈W1,p(?,RN)∩L∞(?,RN) of(1.1) with ?·u=g, if it satisfies the followings:

        then, we have

        Proof Step 1We set

        and

        Then, we take δ ∈(0,1) to be corresponding constant from the A-harmonic approximation lemma, that is Lemma 2.4. According to (2.2) and Lemma 2.1, we imply

        Moreover,we can derive, by Lemma 4.2 and the smallness condition (5.2) and (5.3), that

        With these two estimates (5.7) and (5.8) that satisfies the hypotheses of the A-harmonic approximation stated in Lemma 2.4, there exists an A-harmonic function h ∈W1,p(Bρ(x0),RN)with

        and

        Now, with the help of (5.10) and Lemma 2.1, we similarly distinguish the cases |Dh| > 1 and|Dh|≤1 to infer

        Furthermore, recalling the assumption of (5.1) and Lemma 2.5, we arrive at

        The estimates (5.10)–(5.12) will be useful in the next step of proof.

        Step 2We can deduce, with the help of Lemma 2.1, that

        We note that

        where we used Lemma 2.1 and decomposed Bτρ(x0) into

        and

        Furthermore,we draw a conclusion from(5.13), Lemma 2.1,and the fact that |V(A)|=V(|A|)and t →V(t) is monotone increasing, that is,

        Applying Theorem 3.1,that is,Caccioppoli type inequality on B2τρ(x0)with ux0,ρa(bǔ)nd(Du)x0,ρ+γDh(x0) in place of u0and p0, respectively, we have

        5.1 Estimate for I1

        To estimate the first term I1, we use Lemma 2.1 to obtain

        Here, we draw a conclusion from (2.2), (5.10), and Lemma 2.1,

        For the second term I12, applying Taylor’s theorem to h on B2τρ(x0), Lemma 2.1, Lemma 2.5,and (5.11), then we have

        5.2 Estimate for I2

        Next, we deal with the term I2by (5.12),

        5.3 Estimate for I3

        Finally, we use assumption (5.4) and Sobolev’s inequality to obtain

        Combining all the above estimates from (5.15) to (5.20), we arrive at

        As a result, we deduce (recalling that (5.14)) that

        The regularity result then follows from the fact that this excess-decay estimate for any x in a neighborhood of x0. By this estimate and Campanato’s characterization of H?lder continuous[4], we conclude that V(Du) has the modulus of continuity ρ →Φ(x0,ρ) by a constant times ρ2β. Furthermore, this modulus of continuity carries over to Du.

        Finally, we prove that the Hausdorff dimension of the singular set is less than n ?p, which follows from [15, Proposition2.1]. We consider a function u ∈W1,p(Q,RN) and a set-function λ defined by

        on every subset O ?Q. Obviously, all the assumptions on λ in Lemma 2.9 are fulfilled. To estimate the Hausdorff dimensions ofand, we define

        Now let ε>0. Then, Lemma 2.9 implies Hn?p+ε()=0. By Lemma 2.8, we conclude that if x0∈, then x0∈, and therefore,and Hn?p+ε()=0.

        Again, it follows that Hn?p+ε(SΣu2) = 0 from Lemma 2.9. To prove, we consider centres x0∈QSΣu2and radii R <1 such that BR(x0) ?Q. Then, we use Jensen’s inequality and Lemma 2.8 to estimate that

        for every k ∈N0sufficiently large. Summing up these terms finally yields

        Hence, because ε0∈(0,ε) is chosen arbitrarily,we obtain Hn?p+ε()=0. So, the Hausdorff dimension of the singular set is less than n ?p.

        男女性杂交内射女bbwxz| 强d乱码中文字幕熟女1000部| 成人全部免费的a毛片在线看| 欧美成人家庭影院| 国产激情久久久久影院老熟女免费| 国产nv精品你懂得| 久久久精品人妻一区二区三区| 国产哟交泬泬视频在线播放| av免费在线观看网站大全| 蜜桃精品人妻一区二区三区| 久久无码专区国产精品s| 国产精品视频久久久久| 国产精品国产三级国产专区51区 | 越南女子杂交内射bbwbbw| 国产精品黄色片在线观看| 亚洲中文字幕一区av| 中文字幕人妻在线中字| 男女18禁啪啪无遮挡| 久久久久久国产福利网站| 顶级高清嫩模一区二区| 免费va国产高清大片在线| 欧美三级不卡视频| 人妻精品人妻一区二区三区四五| 国产成人综合久久大片| 亚洲av成人无码一区二区三区在线观看 | 中文字幕一区二三区麻豆| 久久国产免费观看精品3| 国产免费久久精品国产传媒| 日本变态网址中国字幕| 国产精品一二三区亚洲| 国产精品毛片无遮挡| 精品国产一区二区三区久久久狼| 毛片av中文字幕一区二区| 亚洲人成在久久综合网站| 欧美国产精品久久久乱码| 久久夜色精品国产亚洲噜噜| 精品福利一区二区三区| 九九久久99综合一区二区| 欧美国产亚洲日韩在线二区| av在线免费观看你懂的| 日本不卡的一区二区三区中文字幕 |