Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations

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1 Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations Kyudong Choi, Alexis F. Vasseur May 6, 20 Abstract We study weak solutions of the 3D Navier-Stokes equations in whole space with L 2 initial data. It will be proved that α u is locally integrable in space-time for any real α such that < α < 3, which says that almost third derivative is locally integrable. Up to now, only second derivative 2 u has been known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-l 4/α+ loc. These estimates depend only on the L 2 norm of initial data and integrating domains. Moreover, they are valid even for α 3 as long as u is smooth. The proof uses a good approximation of Navier-Stokes and a blow-up technique, which let us to focusing on a local study. For the local study, we use De Giorgi method with a new pressure decomposition. To handle non-locality of the fractional Laplacian, we will adopt some properties of the Hardy space and Maximal functions. Mathematics Subject Classification: 76D05, 35Q30. Introduction and main result In this paper, any derivative signs,, α/2, D, and etc denote derivatives in only space variable x R 3 unless time variable t R is clearly specified. We study the 3-D Navier-Stokes equations with L 2 initial data t u + u u + P u = 0 and div u = 0, t 0,, x R 3 u 0 L 2 R 3, div u 0 = 0. 2 Regularity of weak solutions for the 3D Navier-Stokes equations has long history. Leray [27] 930s and Hopf [22] 950s proved existence of a global-time University of Texas at Austin University of Oxford

2 Fractional higher derivatives of weak solutions for Navier-Stokes 2 weak solution for any given L 2 initial data. Such Leray-Hopf weak solutions u lie in L 0, ; L 2 R 3 and u do in L 2 0, ; L 2 R 3 and satisfy the energy inequality: ut 2 L 2 R u 2 L 2 0,t;L 2 R 3 u 0 2 L 2 R 3 for a.e. t <. Until now, regularity and uniqueness of such weak solutions are generally open. Instead, many criteria which ensure regularity of weak solutions have been developed. Among them the most famous one is Ladyženskaja-Prodi-Serrin Criteria [24],[30] and [36], which says: if u L p 0, T ; L q R 3 for some p and q satisfying 2 p + 3 q = and p <, then it is regular. Recently, the limit case p = was established in the paper of Escauriaza, Serëgin and Šverák [6]. We may impose similar conditions to derivatives of velocity, vorticity or pressure. see Beale, Kato and Majda [], Beirão da Veiga [2] and Berselli and Galdi [4] Also, many other conditions exist e.g. see Cheskidov and Shvydkoy [0], Chan [9] and [5]. On the other hand, many efforts have been given to measuring the size of possible singular set. This approach has been initiated by Scheffer [33]. Then, Caffarelli, Kohn and Nirenberg [6] improved the result and showed that possible singular sets have zero Hausdorff measure of one dimension for certain class of weak solutions suitable weak solutions satisfying the following additional inequality t u divu u divup + u 2 u in the sense of distribution. There are many other proofs of this fact e.g. see Lin [28], [42] and Wolf [43]. Similar criteria for interior points with other quantities can be found in many places e.g. see Struwe [40], Gustafson, Kang and Tsai [2], Serëgin [35] and Chae, Kang and Lee [8]. Also, Robinson and Sadowski [3] and Kukavica [23] studied box-counting dimensions of singular sets. In this paper, our main concern is about space-time L p t,x = Lp t L p x estimates of higher derivatives for weak solutions assuming only L 2 initial data. u L 2 0, R 3 is obvious from the energy inequality, and simple interpolation gives u L 0/3. For second derivatives of weak solutions, from standard parabolic regularization theory see Ladyženskaja, Solonnikov and Ural ceva [25], we know 2 u L 5/4 by considering u u as a source term. With different ideas, Constantin [2] showed L 4 3 ɛ for any small ɛ > 0 in periodic setting, and later Lions [29] improved it up to weak-l 4 3 or L 4 3, by assuming u 0 lying in the space of all bounded measures in R 3. They used natural structure of the equation with some interpolation technique. On the other hand, Foiaş, Guillopé and Temam [8] and Duff [5] obtained other kinds of estimates for higher derivatives of weak solutions while Giga and Sawada [9] and Dong

3 Fractional higher derivatives of weak solutions for Navier-Stokes 3 and Du [4] covered mild solutions. For asymptotic behavior, we refer Schonbek and Wiegner [34]. Recently in [4], it has been shown that, for any small ɛ > 0, any integer d and any smooth solution u on 0, T, we have bounds of d u in L 4 d+ ɛ loc, which depend only on L 2 norm of initial data once we fix ɛ, d and the domain of integration. It can be considered as a natural extension of the result of Constantin [2] for higher derivatives. But the idea is completely different in the sense that [4] used the Galilean invariance of transport part of the equation and the partial regularity criterion in the version of [42], which re-proved the famous result of Caffarelli, Kohn and Nirenberg [6] by using a parabolic version of the De Giorgi method [3]. It is noteworthy that this method gave full regularity to the critical Surface Quasi-Geostrophic equation in [7]. The limit non-linear scaling p = 4 d+ appears from the following invariance of the Navier- Stokes scaling u λ t, x = λuλ 2 t, λx: d u λ p L = p λ d u p Lp. 4 In this paper, our main result is better than the above result of [4] in the sense of the following three directions. First, we achieve the limit case weak- L 4 4 d+ or L d+, as Lions [29] did for second derivatives. Second, we make similar bounds for fractional derivatives as well as classical derivatives. Last, we consider not only smooth solutions but also global-time weak solutions. These three improvements will give us that 3 ɛ u, which is almost third derivatives of weak solutions, is locally integrable on 0, R 3. Our precise result is the following: Theorem.. There exist universal constants C d,α which depend only on integer d and real α [0, 2 with the following two properties I and II: I Suppose that we have a smooth solution u of on 0, T R 3 for some 0 < T with some initial data 2. Then it satisfies α 2 d u L p, t 0,T ;L p, K C d,α u 0 2 L 2 R 3 + K p t 0 for any t 0 0, T, any integer d, any α [0, 2 and any bounded open subset K of R 3, where p = 4 d+α+ and = the Lebesgue measure in R3. II For any initial data 2, we can construct a suitable weak solution u of on 0, R 3 such that α 2 d u is locally integrable in 0, R 3 for d =, 2 and for α [0, 2 with d + α < 3. Moreover, the estimate 5 holds with T = under the same setting of the above part I as long as d + α < 3. Let us begin with some simple remarks. 5

4 Fractional higher derivatives of weak solutions for Navier-Stokes 4 Remark.. For any suitable weak solution u, we can define α/2 d u in the sense of distributions D for any integer d 0 and for any real α [0, 2: < α/2 d u; ψ > D,D= d 0, R 3 u α/2 d ψ dxdt 6 for any ψ D = Cc 0, R 3 where α/2 in the right hand side is the traditional fractional Laplacian in R 3 defined by the Fourier transform. Note that α/2 d ψ lies in L t L 2 x. Thus, this definition from 6 makes sense due to u L t L 2 x. Note also 0 = Id. For more general extensions of this fractional Laplacian operator, we recommend Silvestre [37]. Remark.2. Since we impose only 2 to u 0, the estimate 5 is a quantitative regularization result to higher derivatives. Also, in the proof, we will see that u 0 2 L 2 R 3 in 5 can be relaxed to u 2 L 2 0,T R 3. Thus it says that any higher derivatives can be controlled by having only L 2 estimate of dissipation of energy. Remark.3. The result of the part I for α = 0 extends the result of the previous paper [4] because for any 0 < q < p < and any bounded subset Ω R n, we have f Lq Ω C f L p, Ω where C depends only on p, q, dimension n and Lebesgue measure of Ω e.g. see Grafakos [20]. Remark.4. The assumption smoothness in the part I is about pure differentiability. For example, the result of the part I for d and α = 0 holds once we know that u is d-times differentiable. In addition, constants in 5 are independent of any possible blow-time T. Remark.5. p = 4/d+α+ is a very interesting relation as mentioned before. Due to this p, the estimate 5 is a non-linear estimate while many other a priori estimates are linear. Also, from the part II when d + α is very close to 3, we can see that almost third derivatives of weak solutions are locally integrable. Moreover, imagine that the part II for d = α = 0 be true even though we can NOT prove it here. This would imply that this weak solution u could lie in L 4, which is beyond the best known estimate u L 0/3 from L 2 initial data. Before presenting the main ideas, we want to mention that Caffarelli, Kohn and Nirenberg [6] contains two different kinds of local regularity criteria. The first one is quantitative, and it says that if u L 3 Q and P L 3/2 Q is small, then u is bounded by some universal constant in Q/2. The second one says that u is locally bounded near the origin if lim sup r 0 r u 2 L 2 Qr is small. So it is qualitative in the sense that the conclusion says not that u is bounded by a universal constant but that sup u for some local neighborhood is not infinite.

5 Fractional higher derivatives of weak solutions for Navier-Stokes 5 On the other hand, there is a different quantitative local regularity criterion in [42], which showed that for any p >, there exists ɛ p such that if u L t L 2 x Q + u L 2 t L 2 x Q + P L p t L x Q ɛ p, then u in Q /2 7 Recently, this criterion was used in [4] in order to obtain higher derivative estimates. The main proposition in [4] says that if both u P L Q and some other quantity about pressure are small, then u is bounded by at the origin once u has a mean zero property in space. We can observe that u 2 L 2 Q and 2 P L Q have the same best scaling like 4 among all the other quantities which we can obtain from L 2 initial data. However, the other quantity about pressure has a slightly worse scaling. That is the reason that the limit case L 4 d+, has been missing in [4]. Here are the main ideas of proof. First, in order to obtain the missing limit case L 4 d+,, we will see that it requires an equivalent estimate of 7 for p =. Here we extend this result up to p = for some approximation of the Navier- Stokes see the proposition 2.. To obtain this first goal, we will introduce a new pressure decomposition see the lemma 3.3, which will be used in the De Giorgi-type argument. This makes us to remove the bad scaling term about pressure in [4]. As a result, by using the Galilean invariance property and some blow-up technique with the standard Navier-Stokes scaling, we can proceed our local study in order to obtain a better version of a quantitative partial regularity criterion for some approximation of the Navier-Stokes see the proposition 2.2. As a result, we can prove L 4 d+, estimate for classical derivatives α = 0 case. Second, the result for fractional derivatives 0 < α < 2 case is not obvious at all because there is no proper interpolation theorem for L p, loc spaces. For example, due to the non-locality of the fractional Laplacian operator, the fact 2 u L 4 3, loc with 3 u L, loc does not imply the case of fractional derivatives even if we assume u is smooth. Moreover, even though we assume that 2 u L 4 3 R 3 and 3 u L R 3 which we can NOT prove here, the standard interpolation theorem still requires L p R 3 for some p > we refer Bergh and Löfström [3]. To overcome the difficulty, we will use the Maximal functions of u which capture its behavior of long-range part. Unfortunately, second derivatives of pressure, which lie in the Hardy space H L R 3 from Coifman, Lions, Meyer and Semmes [], do not have an integrable Maximal function since the Maximal operator is not bounded on L. In order to handle non-local parts of pressure, we will use some property of Hardy space, which says that some integrable functions play a similar role of the Maximal functionsee 0. Finally, the result II for weak solutions comes from specific approximation of Navier-Stokes equations that Leray [27] used in order to construct a global time weak solution : t u n +u n φ /n u n + P n u n = 0 and div u n = 0 where φ is a fixed mollifier in R 3, and φ /n is defined by φ /n = n 3 φn.

6 Fractional higher derivatives of weak solutions for Navier-Stokes 6 Main advantage for us of adopting this approximation is that it has strong existence theory of global-time smooth solutions u n for each n, and it is well-known that there exists a suitable weak solution u as a weak limit. In fact, for any integer d and for any α [0, 2, we will obtain bounds for u n in the form of 5 with T =, which is uniform in n. Since p = 4/d + α + is greater than for the case d + α < 3, we can know that α 2 d u exists as a locally integrable function from weak-compactness of L p for p >. However, to prove 5 uniformly for the approximation is nontrivial because our proof is based on local study while the approximation is not scaling-invariant with the standard Navier-Stokes scaling: After the scaling, the advection velocity u φ /n depends the original velocity u more non-locally than before. Moreover, when we consider the case of fractional derivatives of weak solutions, it requires even Maximal of Maximal functions to handle non-local parts of the advection velocity which depends the original velocity non-locally. The paper is organized as follows. In the next section, preliminaries with the main propositions 2. and 2.2 will be introduced. Then we prove those propositions 2. and 2.2 in sections 3 and 4, respectively. Finally we will explain how the proposition 2.2 implies the part II of the theorem. for α = 0 and for 0 < α < 2 in subsections 5.2 and 5.3 respectively while the part I will be covered in the subsection 5.4. After that, the appendix contains some missing proofs of technical lemmas. 2 Preliminaries, definitions and main propositions We begin this section by fixing some notations and reminding some well-known results on analysis. After that we will present definitions of two approximations and two main propositions. In this paper, any derivatives, convolutions and Maximal functions are with respect to space variable x R 3 unless time variable is specified. Notations for general purpose We define Br = the ball in R 3 centered at the origin with radius r, Qr = r 2, 0 Br, the cylinder in R R 3 and Bx; r = the ball in R 3 centered at x with radius r. To the end of this paper, we fix φ C R 3 satisfying: R 3 φxdx =, suppφ B, 0 φ

7 Fractional higher derivatives of weak solutions for Navier-Stokes 7 φx = for x 2 and φ is radial. For real number r > 0, we define functions φ r C R 3 by φ r x = r 3 φ x r. Moreover, for r = 0, we define φ r = φ 0 = δ 0 as the Dirac-delta function, which implies that the convolution between φ 0 and any function becomes the function itself. From the Young s inequality for convolutions, we can observe f φ r Lp Ba f Lp Ba+r 8 due to suppφ r Br for any p [, ], for any f L p loc and for any a, r > 0. L p, weak-l p and Sobolev spaces W n,p Let K be a open subset K of R n. For 0 < p <, we define L p K by the standard way with quasi norm f L p K = K f p dx /p. From the Banach- Alaoglu theorem, any sequence which is bounded in L p K for p, has a weak limit from some subsequence due to the weak-compactness. Also, for 0 < p <, the weak-l p K space or L p, K is defined by L p, K = {f measurable in K R d : sup α p { f > α} K < } α>0 with quasi norm f L p, K = sup α>0 α { f > α} K. /p From the Chebyshev s inequality, we have f L p, K f L p K for any 0 < p <. Also, for 0 < q < p <, L p, K L q K once K is bounded refer the remark.3 in the beginning. For any integer n 0 and for any p [, ], we denote W n,p R 3 and W n,p Br as the standard Sobolev spaces for the whole space R 3 and for any ball Br in R 3, respectively. The Maximal function M and the Riesz transform R j The Maximal function M in R d is defined by the following standard way: Mfx = sup fx + y dy. r>0 Br Also, we can express this Maximal operator as a supremum of convolutions: Mf = C sup δ>0 χ δ f where χ = { x <} is the characteristic function of the unit ball, and χ δ = /δ 3 χ /δ. One of properties of the Maximal function is that M is bounded from L p R d to L p R d for p, ] and from L R d to L, R d. In this paper, we denote M and M t as the Maximal functions in R 3 and in R, respectively. Br

8 Fractional higher derivatives of weak solutions for Navier-Stokes 8 For j 3, the Riesz Transform R j in R 3 is defined by: R j fx = i x j x ˆfx for any f S the Schwartz space. Moreover we can extend such definition for functions L p R 3 for < p < and it is well-known that R j is bounded in L p for the same range of p. The Hardy space H The Hardy space H in R 3 is defined by HR 3 = {f L R 3 : sup P δ f L R 3 } δ>0 where P = C + x 2 2 is the Poisson kernel and P δ is defined by P δ = δ 3 P /δ. A norm of H is defined by L norm of sup δ>0 P δ f. Thus H is a subspace of L R 3 and f L R 3 f HR3 for any f H. Moreover, the Riesz Transform is bounded from H to H. One of important applications of the Hardy space is the compensated compactness see Coifman, Lions, Meyer and Semmes []. Especially, it says that if E, B L 2 R 3 and curle = div B = 0 in distribution, then E B HR 3 and we have E B HR3 C E L2 R 3 B L2 R 3 for some universal constant C. In order to obtain some regularity of second derivative of pressure, we can combine compensated compactness with boundedness of the Riesz transform in HR 3. For example, if u is a weak solution of the Navier-Stokes, then a corresponding pressure P satisfies 2 P L 0, ;HR 3 C u 2 L 2 0, ;L 2 R 3 9 see Lions [29] or the lemma 7 in [4]. Now it is well known that if we replace the Poisson kernel P with any function G C R 3 with compact support, then we have a constant C depending only on G such that sup δ>0 G δ f L R 3 C sup P δ f L R 3 = C f HR 3 0 δ>0 where G δ = G /δ/δ 3. see Fefferman and Stein [7] or see Stein [39], Grafakos [20] for modern texts. Due to the supremum and the convolution in 0, we can say that even though the Maximal function sup δ>0 χ δ f of any non-trivial Hardy space function f is not integrable, there exist at least Gδ integrable functions sup δ>0 f, which can capture non-local data as

9 Fractional higher derivatives of weak solutions for Navier-Stokes 9 Maximal functions do. However, note the position of the absolute value sign in 0, which is outside of the convolution while it is inside of the convolution for the Maximal function. It implies that 0 is slightly weaker than the Maximal function in the sense of controlling non-local data. This weakness is the reason that we introduce certain definitions of ζ and h α in the following. Some notations which will be useful for fractional derivatives α/2 The following two definitions of ζ and h α will be used only in the proof for fractional derivatives. We define ζ by ζx = φ x 2 φx. Then we have ζ C R 3, suppζ B2, ζx = 0 for x 2 and ζ x 2 j = for x 2k for any integer k. j=k In addition, we define function h α for α > 0 by h α x = ζx/ x 3+α. Also we define h α δ and d h α δ by h α δ x = δ 3 h α x/δ and d h α δ x = δ 3 d h α x/δ for δ > 0 and for positive integer d, respectively. Then they satisfy h α δ C R 3, supph α δ B2δ Bδ/2, and x 3+α ζ x 2 j = 2 j α hα 2 j x for any integer j. 2 The definition of the fractional Laplacian α/2 For 3 < α 2 and for f SR 3 the Schwartz space, α 2 f is defined by the Fourier transform: α 2 fξ = ξ α ˆfξ 3 Note that 0 = Id. Especially, for α 0, 2, the fractional Laplacian can also be defined by the singular integral for any f S: α fx fy 2 fx = Cα P.V. dy. 4 x y 3+α We introduce two notions of approximations to Navier-Stokes. The first one or Problem I-n is the approximation Leray [27] used while the second one or Problem II-r will be used in local study after we apply some certain scaling to Problem I-n. Definition of Problem I-n: the first approximation to Navier-Stokes R 3

10 Fractional higher derivatives of weak solutions for Navier-Stokes 0 Definition 2.. Let n be either an integer or the infinity, and let 0 < T. Suppose that u 0 satisfy 2. We say that u, P [C 0, T R 3 ] 2 is a solution of Problem I-n on 0, T for the data u 0 if it satisfies and t u + u φ u + P u = 0 n 5 div u = 0 t 0, T, x R 3 ut u 0 φ in L2 -sense as t 0. 6 n Remark 2.. When n =, 5 is the Navier-Stokes on 0, T R 3 with initial value u 0. Remark 2.2. If n is not the infinity but an positive integer, then for any given u 0 of 2, we have existence and uniqueness theory of Problem I-n on 0, with the energy equality ut 2 L 2 R u 2 L 2 0,t;L 2 R 3 = u 0 φ n 2 L 2 R 3. 7 for any t < and it is well-known that we can extract a sub-sequence which converges to a suitable weak solution u of and 3 with the initial data u 0 of 2 by limiting procedure on a sequence of solutions of Problem I-n see Leray [27], or see Lions [29], Lemarié-Rieusset [26] for modern texts. Remark 2.3. As mentioned in the introduction section, we can observe that this notion Problem I-n is not invariant under the standard Navier-Stokes scaling ut, x ɛuɛ 2 t, ɛx due to the advection velocity u φ /n unless n is the infinity. Definition of Problem II-r: Stokes the second approximation to Navier- Definition 2.2. Let 0 r < be real. We say that u, P [C 4, 0 R 3 ] 2 is a solution of Problem II-r if it satisfies t u + w u + P u = 0 div u = 0, t 4, 0, x R 3 8 where w is the difference of two functions: wt, x = w t, x w t, t 4, 0, x R 3 9 which are defined by u in the following way: w t, x = u φ r t, x and w t = φyu φ r t, ydy. R 3

11 Fractional higher derivatives of weak solutions for Navier-Stokes Remark 2.4. This notion of Problem II-r gives us the mean zero property for the advection velocity w: R 3 φxwt, xdx = 0 on 4, 0. Also this w is divergent free from the definition. Moreover, by multiplying u to 8, we have t u divw u divup + u 2 u 2 2 = 0 20 in classical sense because our definition needs u to be C. Remark 2.5. We will introduce some specially designed ɛ-scaling which is a bridge between Problem I-n and Problem II-r it can be found in 96. This scaling is based on the Galilean invariance in order to obtain the mean zero property for the velocity u: R 3 φxut, xdx = 0 on 4, 0. Moreover, after this ɛ-scaling is applied to solutions of Problem I-n, the resulting functions will satisfy not conditions of Problem II- n but those of Problem II- nɛ it can be found 97. These things will be stated precisely in the section 5. Remark 2.6. When r = 0, the equation 8 is the Navier-Stokes on 4, 0 R 3 once we assume the mean zero property for u. Now we present two main local-study propositions which require the notion of Problem II-r. These are kinds of partial regularity theorems for solutions of Problem II-r. The main difficulty to prove these two propositions is that η and δ > 0 should be independent of any r in [0,. We will prove this independence very carefully, which is the heart of the section 3 and 4. The first local study proposition for Problem II-r The following one is a quantitative version of partial regularity theorems which extends that of [42] up to p =. The proof will be based on the De Giorgi iteration with a new pressure decomposition lemma 3.3 which will appear later. Proposition 2.. There exists a δ > 0 with the following property: If u is a solution of Problem II-r for some 0 r < verifying both u L 2,0;L 2 B P L 2,0;L B + u L 2 2,0;L 2 B 5 4 δ and M u L 2 2,0;L 2 B2 δ, then we have ut, x on [ 3 2, 0] B 2. The above proposition, whose proof will appear in the section 3, contains two bad scaling terms u L t L 2 and P x L, while the following proposition 2.2 t L x does not have those two. Instead, the proposition 2.2 will assume the mean-zero

12 Fractional higher derivatives of weak solutions for Navier-Stokes 2 property on u with the additional terms. We will see later that these additional ones have the best scaling like u 2 also, see 4. The second local study proposition for Problem II-r Proposition 2.2. There exists a η > 0 and there exist constants C d,α depending only on d and α with the following property: If u is a solution of Problem II-r for some 0 r < verifying both 0 4 B2 R 3 φxut, xdx = 0 for t 4, 0 and 2 u 2 t, x + 2 P t, x + M u 2 t, x dxdt η, 22 then d u C d,0 on Q 3 = 3 2, 0 B 3 for every integer d 0. Moreover if we assume further 0 MM u 2 + M M u q 2/q 4 B2 + M u q 2/q + d+4 δ>0 m=d sup m h α δ 2 P dxdt η 23 for some integer d and for some real α 0, 2 where q = 2/α + 6, then α 2 d u C d,α on Q 6 for such d, α. Remark 2.7. For the definitions of h α and m h α δ, see around 2. The proof will be given in the section 4 which will use the conclusion of the previous proposition 2.. Moreover we will use an induction argument and the integral representation of the fractional Laplacian in order to get estimates for the fractional case. The Maximal function term of 22 is introduced to estimate non-local part of u while the Maximal of Maximal function terms of 23 is to estimate non-local part of w which is already non-local. On the other hand, because 2 P has only L integrability, we can not have L Maximal function of 2 P. Instead, we use some integrable functions, which is the last term of 23. This term plays the role which captures non-local information of pressure see 0. These will be stated clearly in sections 4 and 5. 3 Proof of the first local study proposition 2. This section is devoted to prove the proposition 2. which is a partial regularity theorem for Problem II-r. Remember that we are looking for δ which

13 Fractional higher derivatives of weak solutions for Navier-Stokes 3 must be independent of r. In the first subsection 3., we present some lemmas about the advection velocity w and a new pressure decomposition. After that, two big lemmas 3.4 and 3.5 in the subsections 3.2 and 3.3, which give us a control for big r and small r respectively, follow. Then the actual proof of the proposition 2. will appear in the last subsection 3.4 where we can combine those two big lemmas. 3. A control on the advection velocity w and a new pressure decomposition The following lemma says that convolution of any functions with φ r can be controlled by just one point value of the Maximal function with some factor of /r. Of course, it is useful when r is away from 0. Lemma 3.. Let f be an integrable function in R 3. Then for any integer d 0, there exists C = Cd such that for any 0 < r <. d f φ r L B2 C r d + 4 r 3 inf Mfx x B2 Proof. Let z, x B2. Then d f φ r z = f d φ r z = d φ r L Bz,r d φ L r d+3 r r Bz,r fy dy = d φ L Bx,r+4 r d+3 fy d φ r z ydy Bz,r fy dy fy dy C r d + 4 r 3 Mfx. We used Bz, r Bx, r + 4. Then we take sup in z and inf in x. Recall that φ is the fixed function in this paper. The following corollary is just an application of the previous lemma to solutions of Problem II-r. Corollary 3.2. Let u be a solution of Problem II-r for 0 < r <. Then for any integer d 0, there exists C = Cd such that and if d. w L 2 4,0;L B2 C + 4 r 3 M u L 2 Q2 d w L 2 4,0;L B2 C r d + 4 r 3 M u L 2 Q2

14 Fractional higher derivatives of weak solutions for Navier-Stokes 4 Proof. Recall R 3 wt, yφydy = 0 and suppφ B. Thus for z B2 wt, z = wt, zφydy wt, yφydy R 3 R 3 wt, L B2 z y φydy R 3 C u φ r t, L B2 φydy R 3 C + 4 r 3 inf x B2 M u t, x. For last [ inequality, we used the lemma 3. to u. For d, use d w = d u φ r ]. To use De Giorgi type argument, we require more notations which will be used only in this section. For real k 0, define B k = the ball in R 3 centered at the origin with radius 2 +, 23k T k = k, Q k = [T k, 0] B k and s k = distance between Bk c and B k 5 6 = 2 3k Also we define s = 0. Note that 0 < s < 4 strictly decreasing to zero as k goes to. 24 and the sequence {s k} k= is For each integer k 0, we define and fix a function ψ k C R 3 satisfying: ψ k = in B k 2 3, ψ k = 0 in B C k ψ k x, ψ k x C2 3k and 2 ψ k x C2 6k for x R 3. This ψ k plays role of a cut-off function for B k. 25 To prove the proposition 2., We need the following important lemma about pressure decomposition. Here we decompose our pressure term into three parts: a non-local part which depends on k, a local part which depends on k and a non-local part which does not depend on k and will be absorbed into the velocity component later.

15 Fractional higher derivatives of weak solutions for Navier-Stokes 5 Lemma 3.3. There exists a constant Λ > 0 with the following property: Suppose A ij L B 0 i, j 3 and P L B 0 with P = ij i j A ij in B 0. Then, there exist a function P 3 with P 3 B 23 L such that, for any k, we can decompose P by and they satisfy P = P,k + P 2,k + P 3 in B 3, 26 P,k L B k 3 + P,k L B k 3 Λ 2 2k ij A ij L B 6, 27 P 2,k = ij i j ψ k A ij in R 3 and 28 P 3 L B 23 Λ P L B 6 + ij A ij L B Note that Λ is a totally independent constant. Proof. The product rule and the hypothesis imply ψ P = ψ P 2 div ψ P + P ψ = ψ i j A ij 2 div ψ P + P ψ = P,k P 2,k P 3 where P,k, P 2,k and P 3 are defined by ij P,k = ij P 2,k = ij i j ψ ψ k A ij i j ψ k A ij P 3 = ij j [ i ψ A ij ] ij i [ j ψ A ij ] + ij i j ψ A ij 2 div ψ P + P ψ. P,k and P 3 are defined by the representation formula f = 4π x f while P 2,k by the Riesz transforms. Since ψ = on B, we have P = ψ P on B. Thus 26 holds. 3 3 By definition of P 2,k, 28 holds.

16 Fractional higher derivatives of weak solutions for Navier-Stokes 6 For 27 and 29, it follows the proof of the lemma 3 of [42] directly. For completeness, we present a proof here. Note that ψ ψ k is supported in B B 6 k 2 and ψ is supported in B B. Thus for x B k, 3 P,k x = i j ψ ψ k A ij ydy 4π B 6 B k 2 x y 3 ij sup 2 y x y A ij x dy y B C k 2 3 C sup y B C k 2 3 ij B 6 x y 3 A ij L B 6 C 2 9k A ij L B 6. ij We used integration by parts and facts x y 2 3k and ψ ψ k. In the same way, for x B k 3, P,k x C 2 2 2k A ij L B 6. ij ij For x B 2 3, [ P 3 x = y 4π B 6 B 3 x y j [ i ψ A ij ] ij ij + i j ψ A ij 2 div ψ P + P ψ ]dy ij C 3 A ij L B 6 + P L B 6. ij i [ j ψ A ij ] These prove 27 and 29 and we keep the constant Λ = maxc, C 2, C 3 for future use. Before presenting De Giori arguments for large r and small r, we need more notations. In the following two main lemmas 3.4 and 3.5, P 3 will be constructed from solutions u, P for Problem II-r by using the previous lemma 3.3 and it will be clearly shown that P 3 has L t L x bound. Thus we can define E k t = t 2 2 k + P 3 s, L B 23 ds, for t [ 2, 0] and for k 0. 30

17 Fractional higher derivatives of weak solutions for Navier-Stokes 7 Note that E k depends on t. We also define followings like in [42] v k = u E k +, E k { u Ek } d k = u u 2 + v k u u 2 and U k = sup v k 2 dx + d k 2 dxdt t [T k,0] B k Q k = v k 2 L T k,0;l 2 B k + d k 2 L 2 Q k. In this way, P 3 will be absorbed into v k, which is the key idea of proof of this proposition De Giorgi argument to get a control for large r The following big lemma says that we can obtain a certain uniform non-linear estimate in the form of W k C k W β k when r is large. Then an elementary lemma can give us the conclusion we will see the lemma 3.6 later. On the other hand, for small r, we have the factor of /r 3 which blows up as r goes to zero. This weak point implies that we still need some extra work after this lemma. it will be the next big lemma 3.5. Lemma 3.4. There exist universal constants δ > 0 and C > such that if u is a solution of Problem II-r for some 0 < r < verifying both u L 2,0;L 2 B P L 2,0;L B + u L 2 2,0;L 2 B 5 4 δ and M u L 2 2,0;L 2 B2 δ, then we have U k { C k U 7 6 k, for any k if r s r C 3 k U 7 6 k, for any k if r < s. Remark 3.. s is a pre-fixed constant defined in 24 such that 0 < s < /4, and δ, C is independent of any 0 < r <. It will be clear that the exponent 7/6 is not optimal and we can make it close to 4/3 arbitrarily. However, any exponent bigger than is enough for our study. Proof. We assume δ <. First we claim that there exists a universal constant Λ 2 such that w u L2 2,0;L 3/2 B 6 Λ 2 δ for any 0 < r <. 3 In order to prove the above claim 3, we need to separate it into a large r case and a small r case:

18 Fractional higher derivatives of weak solutions for Navier-Stokes 8 I-large r case. From the corollary 3.2 if r s, then Likewise, w L 2 4,0;L B2 C + 4 s 3 M u L 2 Q2 C M u L 2 Q2 Cδ. 32 w L2 4,0;L B2 Cδ. 33 With Holder s inequality and B 6 B 0 = B B 5 4 B2, w u L 2 2,0;L 3/2 B 6 C u L 2,0;L 2 B 5 4 w L2 4,0;L B2 so we obtained 3 for r s. C δ 2 C δ. II-small r case. On the other hand, if r < s, then w L2 4,0;L B2 C + 4 r 3 M u L2 Q2 C r 3 M u L 2 Q2 C r 3 δ 34 and w L2 4,0;L B2 C r 3 δ. 35 However, it is not enough to prove 3 because r factor blows up as r goes 3 to zero. So, instead, we use the idea that w and u are similar if r is small: and u L 4 2,0;L 3 B 0 C u L 2,0;L 2 B 0 + u L 2 2,0;L 2 B 0 Cδ w L4 2,0;L 3 B 6 = u φ r L4 2,0;L 3 B 6 u L4 2,0;L 3 B 0 Cδ because u φ r in B depends only on u in B 0. recall that r s 6 and s is the distance B0 c and B and refer 8. For w, 6 w L 2,0;L B2 = w L t 2,0 = φyu φ r ydy L t R 3 C u φ r L x B L t C u L x B 5 4 L t 2,0 2,0 2,0 36 C u L 2,0;L 2 B 5 4 Cδ

19 Λ δ + 9 Λ 2 δ 0 Λ Λ 2 δ 2. 4 Fractional higher derivatives of weak solutions for Navier-Stokes 9 because w is a constant in x, suppφ B and u φ r in B depends only on u in B + s which is a subset of B 5 4. As a result, we have w u L2 2,0;L 3/2 B 6 C u L 4 2,0;L 3 B w L 4 2,0;L 3 B 6 so that we obtained 3 for r s. Hence, taking Cδ w + w L 4 2,0;L 3 B 6 37 C δ 2 C 2 δ Λ 2 = maxc, C 2,, 38 we have 3 and Λ 2 is independent of 0 < r < as long as δ <. From now on, we assume δ < sufficiently small to be 0 Λ Λ 2 δ /2 Recall that Λ comes from the lemma 3.3. Thanks to the lemma 3.3 and 3, by putting A ij = w i u j we can decompose P by P = P,k + P 2,k + P 3 in [ 2, 0] B 3 for each k with following properties: P,k + P,k L 2 2,0;L B k 3 Λ 2 2k ij w i u j L 2 2,0;L B 6 9 Λ Λ 2 δ 2 2k 2 2k for any k, 39 P 2,k = ij i j ψ k w i u j in [ 2, 0] R 3 for any k and 40 P 3 L 2,0;L B 23 Λ P L 2,0;L B + ij w i u j L2 2,0;L B Note that the above 4 enables E k to be well-defined and it satisfies 0 E k see the definition of E k in 30. In the following remarks , we gather some easy results, which were obtained in [42], without proof. They can be found in the lemmas 4, 6 and the remark of the lemma 4 of [42]. Note that any constants C in the following remarks do not depend on k.

20 Fractional higher derivatives of weak solutions for Navier-Stokes 20 Remark 3.2. For any k 0, the function u can be decomposed by u = u v k u + u v k u. Also we have u v k u v k, u u d k, u Ek u d k, v k d k and uv k 42 3dk. u Remark 3.3. For any k and for any q, vk >0 Lq Q k C2 0k 5 3q U 3q k and vk >0 L T k,0;l q Q k C2 2k Remark 3.4. For any k, v k L 0 3 Qk CU 2 k. q U q k. From the above remarks , we have for any p 0 3, v k L p Q k = v k vk >0 L p Q k v k L 0 3 Qk v k >0 L / p 3 0 Q k 0k v k 0 L 3 C2 3 p 3 0 U 5 3 p 3 0 Qk k C2 7k 3 U 5 3p k. 43 Likewise, for any p 2, v k L T k,0;l p B k C2 k U p k 44 and d k Lp Q k C2 5k 3 U 5 3p 3 k. 45 Second, we claim that for every k, functions v k verifies: vk 2 t 2 + divw v2 k 2 + d2 k v2 k 2 + divup,k + P 2,k + v k u u P,k + P 2,k 0 46 in 2, 0 B 2 3. Remark 3.5. Note that the above inequality 46 does not contain the P 3 term. We will see that this fact comes from the definition of E k t in 30.

21 Fractional higher derivatives of weak solutions for Navier-Stokes 2 Indeed, observe that v2 k 2 = u v2 k u 2 2 and note that E k does not depend on space variable but on time variable. Thus we can compute, for time derivatives, t v2 k u 2 = v k t v k u t u = v k t u v k t E k u t u 2 = u v k u tu v k t E k = u v k u tu v k P 3 t, L B 23 while, for any space derivatives α, α v2 k u 2 = u v k 2 u αu. Then we follow the same way as the lemma 5 of [42] did: First, we multiply 8 by u v k u, and then we sum the result and 20. We omit the details which can be found in the proof of the lemma 5 of [42]. As a result, we have 0 t v 2 k 2 + divw v2 k 2 + d2 k v2 k 2 + v k P 3 t, L B 23 + divup + v k u P u = t v 2 k 2 + divw v2 k 2 + d2 k v2 k 2 + v k P 3 t, L B 23 + v k u u P 3 + divup,k + P 2,k + v k u u P,k + P 2,k. For the last equality, we used the fact P = P,k + P 2,k + P 3 in B 3 and divup 3 + v k u u P 3 = v k u u P Thus we proved the claim 46 due to v k P 3 t, L B 23 + v k u u P 3 0 on 2, 0 B 2 3. For any integer k, we introduce a cut-off function η k x C R 3 satisfying η k = in B k, η k = 0 in B C k, 0 η k, 3 η k C2 3k and 2 η k C2 6k, for any x R 3. Multiplying 46 by η k and integrating [σ, t] R 3 for T k σ T k t 0, R3 η k x v kt, x 2 t dx + η k xd 2 2 ks, xdxds σ R 3 R3 η k x v kσ, x 2 dx 2 t R3 + η k xws, x v ks, x 2 t R3 dxds + η k x v ks, x 2 dxds σ 2 σ 2 t η k x divup,k + P 2,k + v k σ R u u P,k + P 2,k s, xdxds. 3

22 Fractional higher derivatives of weak solutions for Navier-Stokes 22 Integrating in σ [T k, T k ] and dividing by T k T k = 2 k+, sup t [T k,] R3 η k x v kt, x 2 Tk 2 t dx + η k xd 2 ks, xdxds T k R 3 R3 2 k+ η k x v kσ, x 2 dx T k 2 0 R3 + η k xws, x v ks, x 2 0 R3 dx ds + η k x v ks, x 2 dx ds T k 2 T k η k x divup,k + P 2,k + v k R u u P,k + P 2,k s, xdx ds. 3 T k From η k = on B k, U k sup t [T k,] 2 Thus we have where sup t [T k,] R3 η k x v kt, x dx + R3 η k x v kt, x 2 dx + 2 T k t T k η k xd 2 ks, xdxds R 3 η k xd 2 ks, xdxds. R 3 U k I + II + III + IV 48 I = C2 6k Q k v k s, x 2 dxds, II = η k x ws, x v k s, x 2 dxds, Q k 0 III = 2 η k x divup,k + v k T k R u u P,k s, xdx ds 3 0 IV = 2 η k x divup 2,k + v k R u u P 2,k s, xdx ds. 3 T k and 49 For I, by using 43, for any 0 < r <, I = C2 6k v k 2 L 2 Q k C20k U 5 3 k. 50

23 Fractional higher derivatives of weak solutions for Navier-Stokes 23 For II with r s, by using 32 and 44, II C2 3k w L2 4,0;L B2 v k 2 L2 T k,0;l B k C2 3k δ v k L T k,0;l 6 5 B k v k L 2 T k,0;l 6 B k C2 4k δ U 5 6 k v k L T k,0;l 2 B k + v k L 2 T k,0;l 2 B k C2 4k δ U 5 6 k U 2 k C2 4k δ U 4 3 k C2 4k U 4 3 k. 5 For r < s, follow the above steps using 34 instead of using 32 then we get For III non-local pressure term, observe that II C r 3 24k U 4 3 k. 52 divup,k + v k u u P,k = v k u u P,k because everything is smooth. Thus, by using 39 and 43, for any 0 < r <, III C v k u u P,k L Q k C v k P,k L Q k v k L2 T k,0;l B k P,k L2 T k,0;l B k vk >0 L 2 T k,0;l 2 B k v k L T k,0;l 2 B k 2 2k C2 43k 5 3 U 6 k U 2 k C2 43k 4 3 U 3 k. For IV local pressure term, as we did for III, observe 53 By definition of P 2,k, we have divup 2,k + v k u u P 2,k = v k u u P 2,k. P 2,k = i j ψ k w i u j = i j ψ k w i u j + ψ k j w i u j ij ij = i j ψ k w i u j v k u + v k jψ k w i u j u ij + ψ k j w i u j v k u + ψ v k k j w i u j u and P 2,k = ij i j ψ k w i u j v k u + v k jψ k w i u j u + ψ k j w i u j v k u + ψ k j w i u j v k u.

24 Fractional higher derivatives of weak solutions for Navier-Stokes 24 Thus we can decompose P 2,k by the Riesz transform into P 2,k = G,k + G 2,k + G 3,k + G 4,k where G,k = ij G 2,k = ij G 3,k = ij G 4,k = ij i j ψ k w i u j v k u, i v k j ψ k w i u j, u i ψ k j w i u j v k u i v k ψ k j w i u j. u and From L p -boundedness of the Riesz transform with the fact suppψ k B k 5/6 B k, we have G 2,k L2 T k,0;l 2 R 3 C2 3k w L2 T k,0;l B k v k L T k,0;l 2 B k, G 4,k L 2 T k,0;l 2 R 3 C w L 2 T k,0;l B k v k L T k,0;l 2 B k. For any < p <, G,k L 2 T k,0;l p R 3 C p 2 3k w L 2 T k,0;l B k G 3,k L 2 T k,0;l p R 3 C p w L 2 T k,0;l B k. Therefore, by using 33 and 35 G 2,k + G 4,k L 2 T k,0;l 2 R 3 and, for any < p <, { G,k + G 3,k L2 T k,0;l p R 3 { and C 2 3k U 2 k, if r s C 2 3k r U 3 2 k, if r < s C p 2 3k, if r s C p 2 3k r, 3 if r < s. Thus, by using the above estimates and 43, for r s and p > 5, IV C v k u u P 2,k L Q k C v k P 2,k L Q k C v k G,k + G 3,k L Q k + C v k G 2,k + G 4,k L Q k v k L 2 T k,0;l p G p,k + G 3,k L B k 2 T k,0;l p B k + v k L 2 T k,0;l 2 B k G 2,k + G 4,k L 2 T k,0;l 2 B k C C p 2 6k 4p 5 3 U 3p k.

25 Fractional higher derivatives of weak solutions for Navier-Stokes 25 By the same way, for r < s and p > 5, Thus, by taking p = 0, IV IV C C p r 3 2 6k { 4p 5 3 U 3p k. C 2 6k 3 U 7 6 k, if r s C r 2 6k 3 3 U 7 6 k, if r < s. 54 Finally, combining 50, 5, 52, 53 and 54 gives us I + II + III + IV { C k U 7 6 k, if r s r C k U k, if r < s. 3.3 De Giorgi argument to get a control for small r The following big lemma makes us be able to avoid the weak point of the previous lemma 3.4 when we handle small r including the case r = 0. Recall the definition of s k in 24 first. It is the distance between Bk c and B k 5 and s k is strictly decreasing to zero as k. For any 0 < r < s 6 we define k r as the integer such that s kr+ < r s kr. Note that k r is integer-valued, k r and is increasing to as r goes to zero. For the case r = 0, we simply define k r = k 0 =. Lemma 3.5. There exist universal constants δ 2 and C 2 > such that if u is a solution of Problem II-r for some 0 r < s verifying both u L 2,0;L 2 B P L 2,0;L B + u L2 2,0;L 2 B 5 4 δ 2 and M u L2 4,0;L 2 B2 δ 2, then we have U k C 2 k U 7 6 k for any integer k such that k k r. Remark 3.6. Note that δ 2 and C 2 are independent of any r [0, s and the exponent 7/6 is not optimal and we can make it almost 4/3. Remark 3.7. This lemma says that even though r is very small, we can make the above uniform estimate for the first few steps k k r. Moreover, the number k r of these steps is increasing to the infinity with a certain rate as r goes to zero. In the subsection 3.4, we will see that this rate is enough to obtain a uniform estimate for any small r once we combine two lemmas 3.4 and 3.5.

26 Fractional higher derivatives of weak solutions for Navier-Stokes 26 Proof. In this proof, we can borrow any inequalities in the proof of the previous lemma 3.4 except those which depend on r and blow up as r goes to zero. Let 0 r < s and take any integer k such that k k r. Like δ of the previous lemma 3.4, we assume δ 2 so small that δ 2 <, 0Λ Λ 2 δ 2 2. We begin this proof by decomposing w by w = u φ r = u v k u φ r + u v k u φ r = w, + w,2. Thus the advection velocity w has a new decomposition: w = w w = w, + w,2 w = w, w + w,2. We will verify that w, w is bounded and w,2 can be controlled locally. First, for w,, w, t, x = u v k u φ r t, x u v k u t, L R 3 55 for any 4 t and any x R 3. From 36, we still have Combining above two results, w L 2,0;L B2 C δ C. 56 w, + w L 2,0;L B2 C. 57 For w,2, we observe that any L p norm of w,2 = u v k u φ r in B k 5 is less 6 than or equal to that of v k in B k because r s kr s k and s k is the distance between Bk c and B k 5 see 8. Thus we have, for any p, 6 w,2 L p T k,0;l p B k 5 = u v k φ r L 6 u p T k,0;l p B k = v k φ r L p T k,0;l p B k 5 v k L p Q k. 6 So, by using 43, we have w,2 L p T k,0;l p B k 5 C2 7k 5 3 U 3p 0 k, for any p Remark 3.8. The above computations says that, for any small r, the advection velocity w can be decomposed into one bounded part w, w and the other part w,2, which has a good contribution to the power of U k. Recall that the transpost term estimate 3 is valid for any 0 < r <. Moreover, the argument around 37 says that 3 holds even for the case r = 0. Thus, for any r [0, s, we have the same pressure estimates 39, 40 and

27 Fractional higher derivatives of weak solutions for Navier-Stokes Thus we can follow the proof of the previous lemma 3.4 up to 48 without any single modification. It remains to control I IV. For I, 50 holds here too because 50 is independent of r. For II, by using 57 and 59 with the fact suppη k B k 3 B k 5 6, we have II = η k w v k 2 L Q k C2 3k w, + w v k 2 L Q k + w,2 v k 2 L T k,0;l B k 5 6 C2 3k v k 2 L 2 Q k + C23k w,2 L 0 3 Tk,0;L 0 3 B k 5 6 v k 2 L 0 7 Qk 5 6 C2 23k 5 3 U 3 k + C20k U 5 3 k C2 0k U 5 3 k. 60 For IIInon-local pressure term, we have 53 here too since 53 is independent of r. For IV local pressure term, by definition of P 2,k and decomposition of w, P 2,k = i j ψ k w i u j v k u + ψ v k kw i u j u ij = i j ψ k w, i w i u j v k u + ψ kw,2 i u j v k u ij + ψ k w, i w i v k u j u + ψ kw,2 v k i u j. u Thus we can decompose P 2,k by where By using P 2,k, = ij P 2,k,2 = ij P 2,k = P 2,k, + P 2,k,2 + P 2,k,3 + P 2,k,4 i j ψ k w, i w i u j v k u, i j ψ k w,2 i u j v k u, P 2,k,3 = i j ψ k w, i w v k i u j and u ij P 2,k,4 = i j ψ k w,2 v k i u j. u ij u v k u and the fact ψk is supported in B k 5 with 57, 6 P 2,k, Lp T k,0;l p R 3 C p, for < p < 6

28 Fractional higher derivatives of weak solutions for Navier-Stokes 28 and, with 59, P 2,k,2 Lp T k,0;l p R 3 C p ψ k w,2 Lp T k,0;l p R 3 CC p 2 7k 5 3 U 3p k for p Observe that for i =, 2, div ug i + vk u u G i = div u v k u G i G i div For P 2,k,, by using 42, 43, 45, 63 and 6 with p = 0 0 η k x divup 2,k, + v k R u u P 2,k, s, xdx ds 3 T k C 3k v k P 2,k, L Q k + 3 d k P 2,k, L Q k uvk. 63 u C 3k v k L 0 9 Qk P 2,k, L 0 Q k + 3 d k L 0 9 Qk P 2,k, L 0 Q k C2 6k 3 3 U 2 k + C2 5k 7 3 U 6 k C2 6k 7 3 U 6 k. 64 Likewise, for P 2,k,2, by using 62 instead of 6 0 η k x divup 2,k,2 + v k R u u P 2,k,2 s, xdx ds 3 T k C2 23k 5 3 U 3 k + C24k U 4 3 k C2 23k 4 3 U 3 k. 65 From definitions of P 2,k,3 and P 2,k,4 with divw = 0, we have P 2,k,3 + P 2,k,4 = v k i j ψ k w i u j u ij = v k j i ψ k w i u j u + ψ v k kw i i u j u. ij Then we use the fact w = w, w + w,2 so that we can decompose P 2,k,3 + P 2,k,4 = H,k + H 2,k + H 3,k + H 4,k where H,k = ij H 2,k = ij H 3,k = ij H 4,k = ij j i ψ k w, i w i v k u j u j i ψ k w,2 v k i u j, u j ψ k w, i w i i u j v k u j ψ k w,2 v k i i u j u., and

29 Fractional higher derivatives of weak solutions for Navier-Stokes 29 By using u + v k, 0 T k η k x divup 2,k,3 + P 2,k,4 + v k R u u P 2,k,3 + P 2,k,4 dx ds 3 C 3k Q k + v k P 2,k,3 + P 2,k,4 s, x + P 2,k,3 + P 2,k,4 dxds C 3k P 2,k,3 L Q k + v k P 2,k,3 L Q k + P 2,k,4 L Q k + v k P 2,k,4 L Q k + H,k L Q k + H 2,k L Q k + H 3,k L Q k + H 4,k L Q k From 43 and 57 with the Riesz transform, P 2,k,3 L Q k C P 2,k,3 0 L 9 Tk,0;L 0 C v 7k k 9 R 3 0 L 9 C2 Qk Likewise U 2 k. 67 and H,k L Q k C2 6k 3 3 U 2 k 68 v k P 2,k,3 L Q k v k L 2 Q k P 2,k,3 L 2 Q k Using 43, 59, 42 and 45, we have C2 7k 5 3 U 6 k C2 7k 5 3 U 6 P 2,k,4 L Q k C2 4k 3 3 U 2 k C2 4k 5 3 U 3 k. 69 k, 70 and H 2,k L Q k C2 23k 3 3 U 2 v k P 2,k,4 L Q k C2 2k k, U 3 H 3,k L Q k C2 5k 3 U 7 6 k k, H 4,k L Q k C2 4k U 7 6 k. 74 Combining 64, 65 and 66 together with 67,, 74, we obtain IV C2 23k 7 3 U 6 k. 75 Finally we combine 60 and 75 together with 50 and 53 in the previous lemma in order to finish the proof of this lemma 3.5.

30 Fractional higher derivatives of weak solutions for Navier-Stokes Combining the two De Giorgi arguments First we present one small lemma. Then the actual proof of the proposition 2. will follow. The following small lemma says that certain non-linear estimates give zero limit if the initial term is sufficiently small. This fact is one of key arguments of De Giorgi method. Lemma 3.6. Let C > and β >. Then there exists a constant C 0 such that for every sequence verifying both 0 W 0 < C 0 and we have lim k W k = 0. 0 W k C k W β k for any k, Proof. It is quite standard or see the lemma in [42]. Finally we are ready to prove the proposition 2.. Proof of proposition 2.. Suppose that u is a solution of Problem II-r for some 0 r < verifying u L 2,0;L 2 B P L 2,0;L B + u L 2 2,0;L 2 B 5 4 δ and M u L 2 4,0;L 2 B2 δ where δ will be chosen within the proof. From two big lemmas 3.4 and 3.5 by assuming δ minδ, δ 2, we have C k U 7 6 k, for any k if r s. U k r C 3 k U 7 6 k, for any k if 0 < r < s. 76 C 2 k U 7 6 k for k =, 2,, k r if 0 r < s. Note that k r = if r = 0. Thus we can combine the case r = 0 with the case r s into one estimate: U k C 3 k U 7 6 k for any k if either r s or r = 0. where we define C 3 = max C, C 2. We consider now the case 0 < r < s. Recall that s k = D 2 3k where D = > and s kr+ < r s kr for any r 0, s. It gives us r D 2 3kr+. Thus if k k r and if 0 < r < s, then the second line in 76 becomes U k r 3 C k U 7 6 k 29kr+ D 3 C k U 7 6 k 2 9k+ C k U 7 6 k 2 8 C k U 7 6 k. 77

31 Fractional higher derivatives of weak solutions for Navier-Stokes 3 So we have for any r 0, s, U k { 2 8 C k U 7 6 k, for any k k r. C 2 k U 7 6 k for k =, 2,, k r. Define C = max2 8 C, C 2, C 3 = max2 8 C, C 2. Then we can combine all three cases r = 0, 0 < r < s and s r < into one uniform estimate: U k C k U 7 6 k for any k and for any 0 r <. Finally, by using the recursive lemma 3.6, we obtain C0 such that U k 0 as k 0 whenever U 0 < C 0. This condition U 0 < C0 is achievable once we C0 assume δ so small that δ 2 because 2. U 0 u L 2,0;L 2 B P L 2,0;L B + u L 2 2,0;L 2 B 5 4 C0 Thus we fix δ = min 2, δ, δ 2 which does not depend on any r [0,. Observe that for any k, sup 3 2 t 0 B 2 ut, x 2 +dx U k from E k and 3 2, 0 B 2 Q k. Due to the fact U k 0, the conclusion of this proposition 2. follows. 4 Proof of the second local study proposition 2.2 First we present technical lemmas, whose proofs will be given in the appendix. In the subsection 4.2, it will be explained how to apply the previous local study proposition 2. in order to get a L -bound of the velocity u. Then, the subsections 4.3 and 4.4 will give us L -bounds for classical derivatives d u and for fractional derivatives α/2 d u, respectively. 4. Some lemmas The following lemma is an estimate about higher derivatives of pressure which we can find a similar lemma in [4]. However they are different in the sense that here we require n th order norm of v to control nth derivatives of pressure see 78 while in [4] we require one more order, i.e. nth order. This fact follows the divergence structure and it will be useful for a bootstrap argument in the subsection 4.3 when large r is large we will see 84.

32 Fractional higher derivatives of weak solutions for Navier-Stokes 32 Lemma 4.. Suppose that we have v, v 2 C B 3 with div v = div v 2 = 0 and P C B which satisfy on B R 3. P = div divv 2 v Then, for any n 2, 0 < b < a < and < p <, we have the two following estimates: n P L p Bb C a,b,n,p v 2 W n,p 2 Ba v W n,p Ba 78 + P L Ba where p = p + p 2, and n P L Bb C a,b,n v 2 W n, Ba v W n, Ba 79 + P L Ba Note that such constants are independent of any v, v 2 and P. Also, is allowed for p and p 2. e.g. if p =, then p 2 = p. Proof. See the appendix. The following is a local result by using a parabolic regularization. It will be used in the subsection 4.3 to prove 82 and 84. Lemma 4.2. Suppose that we have smooth solution v, v 2, P on Q =, 0 B of t v + divv 2 v + P v = 0 divv = 0 and divv 2 = 0. Then, for any n, 0 < b < a <, < p < and < p 2 <, we have n v L p b 2,0;L p 2 Bb C a,b,n,p,p 2 v 2 v L p a2,0;w n,p 2 Ba + v L p a2,0;w n,p 2 Ba + P L a 2,0;L Ba 80 where v 2 v is the matrix whose i, j component is the product of j-th component v 2,j of v 2 and i-th one v,i of v and divv 2 v = j jv 2,j v,i. Note that such constants are independent of any v, v 2 and P. i

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