An estimate on the parabolic fractal dimension of the singular set for solutions of the

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2 IOP PUBLISHING Nonlinearity 25 (2012) NONLINEARITY doi: / /25/9/2775 An estimate on the parabolic fractal dimension of the singular set for solutions of the Navier Stokes system Igor Kukavica and Yuan Pei Department of Mathematics, University of Southern alifornia, Los Angeles, A 90089, USA kukavica@usc.edu and ypei@usc.edu Received 15 June 2012, in final form 2 August 2012 Published 21 August 2012 Online at stacks.iop.org/non/25/2775 Recommended by K Ohkitani Abstract We estimate the parabolic fractal (or parabolic box-counting) dimension of the singular set for suitable weak solutions of the Navier Stokes equations in a bounded domain D. We prove that the parabolic fractal dimension is bounded by 45/29 improving an earlier result from (Kukavica 2009 Nonlinearity ). Also, we introduce the new (parabolic) λ-fractal dimension, where λ is a parameter, which for λ = 1 agrees with the parabolic fractal and for λ = with the parabolic Hausdorff dimension. We prove that for a certain range of λ, the dimension of the singular set is bounded by 3/2. Mathematics Subject lassification: 35Q30, 76D05, 35K55, 35K15 1. Introduction We consider the partial regularity of solutions of the Navier Stokes equations u 3 t νu + j (u j u) + p = f j=1 u = 0. It is well-known that there exists a weak solution of the above system for divergence-free square integrable initial data under a suitable integrability assumption on the forcing term f (see [F, L, T]). On the other hand, the existence of classical solutions, given smooth data is open. The study of partial regularity for suitable weak solutions, i.e. those weak solutions which satisfy a local energy inequality, was initiated by Scheffer [S]. In the classical paper [KN], affarelli, Kohn and Nirenberg proved that the parabolic Hausdorff dimension of the set of /12/ $ IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA 2775

3 2776 I Kukavica and Y Pei singularities S is at most 1. Alternative proofs of this statement were subsequently found in [K1, Li, V, W] (see also [L, LS]). In [RS2, RS3], Robinson and Sadowski estimated the fractal dimension of the singular set S by 5/3 and used this to prove almost everywhere smoothness of the Lagrangian trajectories. Subsequently, the first author of the present paper proved in [K2] that the parabolic fractal dimension of S is at most 135/82. The purpose of this paper is three-fold. First, we further lower the fractal dimension estimate to 45/29. (To avoid repetition, we omit from here on the adjective parabolic in most places.) The main improvement is in the different treatment of the pressure term in the local energy inequality. Namely, using the L 5/4 -norm of the gradient of the pressure jointly with the L 5/3 -norm of the pressure is advantageous over using the norm of the pressure alone. This necessitates a new treatment of the pressure equation p = ij (u i u j ). The second purpose is a simplification of the proof from [K2]; the main shortcut is elimination of the intermediate radius. The third purpose of the paper is to introduce a new dimension, which we call the λ-fractal dimension where λ is a parameter which is at least one. In the definition of the fractal dimension of a set A, coverings of the set involve balls of equal radii; the λ-fractal dimension is similar to the definition of the fractal dimension but the radii r are allowed to vary between R λ and R. It turns out that this λ-fractal dimension is well-suited for the approach to partial regularity. In theorem 2.2, we prove that the λ-fractal is at most 3/2 as long as λ 21/20. As the λ-fractal dimension is between the Hausdorff and the fractal dimensions (when λ = 1, it agrees with the fractal dimension, while if λ = it coincides with the Hausdorff dimension), it may seem that as λ gets larger, the upper bound for the dimension of S should approach 1. However, we suspect that this is very difficult to prove unless we allow λ to depend on the energy of the solution itself. 2. The main result on the parabolic fractal dimension We start by recalling the definition of a suitable weak solution of the Navier Stokes system u 3 t u + j (u j u) + p = f j=1 u = 0, (2.1) where we have set the viscosity to 1. Let D R 3 R be a bounded domain. We say that (u, p) is a suitable weak solution of the Navier Stokes equations if (i) u L t L 2 x (D) and u L2 t L2 x (D) = L2 (D) with p L 5/3 (D), (ii) f L 10/7 (D) is divergence free, (iii) the Navier Stokes system (2.1) holds in D (D), and (iv) the local energy inequality holds in D, i.e. u 2 φ T dx +2 u 2 φ dx dt ( u 2 (φ t + φ) + ( u 2 +2p)u φ +2(u f)φ ) dx dt (2.2) for all φ 0 (D) such that φ 0inD and for almost all T R. Throughout this paper, we assume that f = 0 for simplicity. The adjustments to the case of the nonzero forcing follow [K2].

4 An estimate on the parabolic fractal dimension of the singular set for solutions of the Navier Stokes system 2777 Now, we recall from [K2] the definition of the parabolic fractal dimension. Let A R 3 R. For r > 0, we denote by N(r) the minimal number of centred parabolic cylinders Q r (x, t) = B r(x) (t r 2,t + r 2 ) needed to cover A, i.e. { } N N(r) = min N N 0 : (x 1,t 1 ),...,(x N,t N ) R 3 R, such that A Q r (x i,t i ). Then the parabolic fractal dimension is defined by dim pf (A) = lim sup r 0 + i=1 (2.3) log N(r) log(1/r). (2.4) Note that the dimension does not change if instead of the centred parabolic cylinders we use the non-centred ones Q r (x, t) = B r (x) (t r 2, 0). We borrow the term fractal dimension from [F, EFNT]; in the literature it is also called the box-counting dimension ([F, RS1]) as well as the capacity dimension ([DG]). Note that the parabolic fractal dimension agrees with the usual fractal dimension when R 3 R is equipped with the parabolic metric. Recall that a point (x, t) D is regular for a suitable weak solution (u, p) if the solution u is bounded in some neighborhood of (x, t), and that a point is singular otherwise. We denote by S the set of singular points. The next statement, which provides an estimate of the fractal dimension of the singular set, is our first main result. Theorem 2.1. For (u, p) as above, we have dim pf (S K) 45/29 for any compact set K D. Now, we introduce the concept of the λ-fractal dimension, where λ [1, ). A R 3 R be bounded. For r 0 (0, 1) and d 0, let { Fr d,λ 0 (A) = inf Ri d : (x 1,t 1 ),...,(x N,t N ) R 3 R such that i A N } Q R i (x i,t i ) with r0 λ R 1,...,R N r 0 i=1 and F d,λ (A) = lim r0 0 Fr d,λ 0 (A). Then the λ-fractal dimension is defined by dim λ-pf (A) = inf{d 0:F d,λ (A) = 0}. Let Note that when λ = 1 the λ-fractal dimension agrees with the fractal dimension, while for λ = it coincides with the parabolic Hausdorff dimension; in the latter case, the condition r λ 0 R 1,...,R N r 0 is interpreted as 0 <R 1,...,R N r 0 (which is the reason why we required r 0 (0, 1) above). The next statement contains our second main result; it provides the bound for the λ-fractal dimension. Theorem 2.2. For λ 21/20, the parabolic λ-fractal dimension of the singular set is less than or equal to 3/2. The proof of theorem 2.1 is given in section 3, while the proof of theorem 2.2 is provided in section 4.

5 2778 I Kukavica and Y Pei 3. The proof of the fractal dimension estimate Theorem 2.1 follows directly from the following statement. Theorem 3.1. Assume that Q ρ D. There exists a sufficiently small universal constant ɛ (0, 1], such that if ρ (0, 1) and if ( u 10/3 + u 2 + p 5/3 + p 5/4) dx dt ɛ ρ 45/29 (3.1) Q ρ then (0, 0) is regular. Now, we describe the test function used in the proofs below. First, fix a function φ 0 (R3 ) such that φ 1onB 3/4 and supp φ B 1. Let 0 <rρ/2 where ρ (0, 1), and denote throughout the paper κ = r/ρ. Also, let ψ (R) be a function such that ψ 1 on [1, ) and supp ψ [0, ). We shall use the test function ( t + ρ φ(x,t) = r 2 G(x, r 2 2 ) ( t + (r/2) 2 ) ( ) x t)ψ ψ φ. ρ 2 (r/2) 2 ρ (3.2) From [K1, K2], we recall the fundamental property φ t (x, t) + φ(x, t) r2 ρ. 5 (3.3) We also have the bounds φ(x,t) 1 r, (x,t) Q r (3.4) φ(x,t) r2 R, 3 (x,t) Q 2R\Q R (3.5) φ(x,t) r2 R, 4 (x,t) Q 2R\Q R (3.6) where r R. Next, denote α (x,t) (ρ) = 1 ρ u 1/2 L t L 2 x (Q ρ(x,t)) β (x,t) (ρ) = 1 ρ 1/2 u L 2 (Q ρ (x,t)) π (x,t) (ρ) = 1 p 1/2 ρ1/2 L 5/4 (Q ρ (x,t)). As usual, we omit the subscript (x, t) when the point is shifted to the origin (0, 0). The next lemma provides estimates for the terms appearing on the right side of the local energy inequality (2.2). Lemma 3.2. We have I 1 = I 2 = u 2 u φdxdt I 3 = 2 pu φdxdt provided Q ρ D. u 2 (φ t + φ) dx dt r2 ρ 3 u 2 L 10/3 (Q ρ ) (3.7) u 5/2 r3/2 L 10/3 (Q ρ ) u 1/2 L 2 (Q ρ (3.8) ) u 5/2 r3/2 L 10/3 (Q ρ ) p 1/2 L 5/3 (Q ρ ) p 1/2 L 2 (Q ρ (3.9) )

6 An estimate on the parabolic fractal dimension of the singular set for solutions of the Navier Stokes system 2779 Proof of lemma 3.2. The estimate (3.7) follows directly from (3.3) and Hölder s inequality. In order to prove (3.8), we decompose the function φ dyadically. Let η(x, t) be a smooth test function, which is identically 1 on Q 3/4 and which is supported in Q 1. Then let η 0 (x, t) = η ( x/2r,t/(2r) 2) and ( ) ( ) x η m (x, t) = η 2 m+1 r, t x η (2 m+1 r) 2 2 m r, t, m N. (2 m r) 2 Note that supp η 0 Q 2r and supp η m Q 2 m+1 r \Q 2 m 1 r for m N. Let m 0 be the largest integer m for which (2 m 1 r) 2 ρ 2, i.e. m 0 = [log(1/κ)/log 2] + 1. Then we have 1 = m 0 η m(t) for (x, t) Q ρ. Now, we write m 0 m 0 m 0 I 2 = u 2 u j j (φη m ) dx dt = ( u 2 θ m (t))u j j (φη m ) dx dt = where θ m (t) is an arbitrary function of time. Note that supp(φη 0 ) Q 2r, and thus by (3.5) and (3.6) (φη 0 ) /r 2 for (x, t) R 3 R. Also, supp(φη 0 ) Q 2 m+1 r\q 2 m 1 r, and thus by (3.5) and (3.6) (φη m ) /2 4m r 2 for all (x, t). Now, for all m {0,...,m 0 }, choose θ m (t) to be the average of u 2 over the region B 2 m+1 r. For each m {0,...,m 0 },wehave J m = ( u 2 θ m (t))u j j (φη m ) dx dt 2 4m r 2 u 2 θ m (t) 10/7 L t L 15/8 x (Q 2 m+1 r ) u L 10/3 t Lx 15/7 (Q 2 m+1 r ) from where, using [K2, lemma 2.6], J m 2 4m r 2 u 3/2 L 10/3 (Q 2 m+1 r ) u 1/2 L 2 (Q 2 m+1 r ) u L 10/3 x (Q 2 m+1 r ) 2 7m/2 u 5/2 r3/2 L 10/3 (Q 2 m+1 r ) u 1/2 L 2 (Q 2 m+1 r ). Therefore, J m 2 7m/2 r 3/2 u 5/2 L 10/3 (Q ρ ) u 1/2 L 2 (Q ρ. Summing up the geometric series, we ) obtain (3.8). For (3.9), we write m 0 I 3 = 2 pu j j φ dx dt = 2 pu j j (φη m ) dx dt m 0 m 0 = 2 (p θ m (t))u j j (φη m ) dx dt = where θ m (t) is the average of p over B 2 m+1 r. Then, for every m {0,...,m 0 },wehave J m 2 4m r p θ m(t) 2 10/7 L t L 15/8 x (Q 2 m+1 r ) u L 10/3 x (Q 2 m+1 r ) 2 4m r p θ m(t) 1/2 p θ m(t) 1/2 2 L 5/4 x (Q 2 m+1 r ) L 5/3 (Q 2 m+1 r ) u L 10/3 t Lx 15/7 (Q 2 m+1 r ) and thus by the Gagliardo Nirenberg inequality J m 2 4m r 2 p 1/2 L 5/4 (Q 2 m+1 r ) p 1/2 L 5/3 (Q 2 m+1 r ) u L 10/3 x (Q 2 m+1 r ) 2 7m/2 p 1/2 r3/2 L 5/4 (Q 2 m+1 r ) p 1/2 L 5/3 (Q 2 m+1 r ) u L 10/3 (Q 2 m+1 r ). Hence, J m 2 7m/2 r 3/2 p 1/2 L 5/4 (Q ρ ) p 1/2 L 5/3 (Q ρ ) u L 10/3 (Q ρ ). Summing up in m, the lemma follows. The next lemma provides an estimate for the gradient of the pressure. J m J m

7 2780 I Kukavica and Y Pei Lemma 3.3. For 0 <r ρ/2, we have provided Q ρ D. p L 5/4 (Q r ) u L 10/3 (Q ρ ) u L 2 (Q ρ ) + ( ) r 12/5 p L 5/4(Qρ) ρ Proof of lemma 3.3. The pressure p satisfies the equation p = ij (u i u j ) which for k {1, 2, 3} gives k p = 2 ij (u i k u j ). Let η be a standard smooth cut-off function, which is identically 1 on Q 3ρ/4 and which vanishes if x ρ or if t ρ 2. One can easily verify that (η k p) = 2 ij (u i k u j η) +2u i k u j ij η 2 i (u i k u j j η) 2 j (u i k u j i η) 2 j ( k p j η) + k pη for k = 1, 2, 3. Denote by N(x) = 1/4π x the Newtonian potential. By inverting the Laplacian, we obtain η k p = 2R i R j (u i k u j η) +2N (u i k u j ij η) 2 i N (u i k u j j η) 2 j N (u i k u j i η) 2 j N ( k p j η) + N ( k pη) where R i is the ith Riesz transform. Denote by q 1,...,q 6 the terms on the right side of the above inequality. For the first term q 1, we have by the alderón Zygmund theorem q 1 L 5/4 (Q r ) i,j u i k u j η L 5/4 (R 3 ( r 2,0)) u L 10/3 (Q ρ ) u L 2 (Q ρ ). For the second term, we write q 2 L 5/4 (Q r ) r 12/5 r12/5 q 2 5/4 L t L x (Q r ) ρ 3 κ 12/5 3 i,j=1 u i k u j L 5/4 t L 1 x (Q ρ) 3 u i k u j L 5/4 (Q ρ ) κ 12/5 u L 10/3 (Q ρ ) u L 2 (Q ρ ). i,j=1 The same inequalities hold for q 3 and q 4 as well. Now, for q 5,wehave q 5 L 5/4 (Q r ) r 12/5 r12/5 q 5 5/4 L t L x (Q r ) p ρ 3 5/4 L t L 1 x (Q ρ) κ12/5 p L 5/4 (Q ρ ) with an analogous treatment for q 6. The proof is concluded by collecting the above estimates. In the proof of theorem 3.1, we also need the following result. Lemma 3.4 ([K1, V]). Let (u, p) be a suitable week solution of the Navier Stokes equation in a domain D. Assume that Q ρ D. Then there exists a sufficiently small constant ɛ (0, 1] such that if η (0, 1) and if then (0,0) is regular. 1 ρ 2/3 u L 3 (Q ρ ) + 1 ρ p 11/10 L 5/4 t L 2 x (Q ρ) ɛ For the proof, see [V, theorem 1] or [K1, remark 6.2.5].

8 An estimate on the parabolic fractal dimension of the singular set for solutions of the Navier Stokes system 2781 Proof of theorem 3.1. Let (u, p) be a suitable weak solution, and assume that (3.1) holds for some ρ (0, 1). By(3.1), we have u L 10/3 (Q ρ ) ɛ 3/10 ρ 27/58 (3.10) u L 2 (Q ρ ) ɛ 1/2 ρ 45/58 (3.11) p L 5/3 (Q ρ ) ɛ 3/5 ρ 27/29 (3.12) p L 5/4 (Q ρ ) ɛ 4/5 ρ 36/29. (3.13) Now, we use (3.2) as a test function in (2.2). Now, let r = ρ 30/29 /2. Using (3.4), we have 1 ( α(r) 2 + β(r) 2) ( u 2 (φ t + φ) + ( u 2 +2p)u φ) dx dt. Using lemma 3.2,weget α(r) 2 + β(r) 2 r2 ρ 3 u 2 L 10/3 (Q ρ ) + u 5/2 r3/2 L 10/3 (Q ρ ) u 1/2 L 2 (Q ρ ) + u 5/2 r3/2 L 10/3 (Q ρ ) p 1/2 L 5/3 (Q ρ ) p 1/2 L 2 (Q ρ ) and thus, by (3.10) (3.13) and r = ρ 30/29 /2, we get α(r) 2 + β(r) 2 ɛ 3/5 ρ 36/29 + ɛ + ɛ 29/20 ρ 81/116 ɛ 3/5. Next, using lemma 3.3, we obtain p L 5/4 (Q r ) ɛ 4/5 ρ 36/29 ρ 63/58 + κ 12/5 ɛ 4/5 ρ 36/29, whence due to κ ρ 1/29 and r = ρ 30/29 /2, we have π(r) = 1 r p 1/2 L 5/4 (Q r ) ɛ 4/5 ρ 21/29 + ɛ 4/5 ρ 117/145 ɛ 4/5 ρ 21/29. Now, if ɛ is sufficiently small, we conclude α(r) 2 + β(r) 2 + π(r) 2 ɛ 0 where ɛ 0 > 0isas small as we wish. With p denoting the average of p over the ball B r for any t ( r 2, 0], we have by the Gagliardo Nirenberg inequality r 1 p p L 5/4 x (Q r ) π(r)2 ɛ 0. Therefore, replacing p p with p (i.e., subtracting a function of time from p), we achieve r 1 p L 5/4 x (Q r ) ɛ 0, which, by Hölder s inequality, implies r 11/10 p L 5/4 t L 2 x (Q r ) ɛ 0. Also, α(r) + β(r) ɛ 0, implies u L 3 (Q r ) ɛ. Using lemma 3.4, we conclude that (0, 0) is regular. 4. The proof of the λ-fractal dimension estimate The proof of theorem 2.2 follows directly from the following statement. Theorem 4.1. Assume that Q ρ D. There exists a sufficiently small constant ɛ (0, 1] such that if ρ (0, 1) and ( u 10/3 + u 2 + p 5/3 + p 5/4) dx dt ɛ R 3/2 (4.1) Q R for all R [ρ 21/20,ρ], then (0, 0) is a regular point. Proof of theorem 4.1. Using (4.1), we have u L 10/3 (Q R ) ɛ 3/10 R 9/20 (4.2) u L 2 (Q R ) ɛ 1/2 R 3/4 (4.3) p L 5/3 (Q R ) ɛ 3/5 R 9/10 (4.4) p L 4/5 (Q R ) ɛ 4/5 R 6/5 (4.5)

9 2782 I Kukavica and Y Pei for all R [ρ 21/20,ρ]. Let r = ρ 21/20 /2. As above, we use (3.2) as a test function in the local energy inequality (2.2). The two terms on the left are estimated from above as in the proof of theorem 3.1, so we only need to estimate the integrals I 1, I 2, and I 3 from section 3. The estimate for the term I 1 remains the same, i.e., I 1 (r 2 /ρ 3 ) u 2 L 10/3 (Q ρ ) to bound I 2, we use the argument as in the proof of (3.8) and obtain m 0 I 2 2 7m/2 u 5/2 r3/2 L 10/3 (Q 2 m+1 r ) u 1/2 L 2 (Q 2 m+1 r ) m 0 2 7m/2 r 3/2 ( ɛ 3/10 (2 m r) 9/20) 5/2( ɛ 1/2 ɛ3/5 (2 m r) 3/4) m0 1/2 = ɛ 2 2m. In order where we used (4.2) and (4.3). Summing up the series, we get I 2 ɛ. Similarly, following the proof of (3.9), we have m 0 I 3 2 7m/2 p 1/2 r3/2 L 5/4 (Q 2 m+1 r ) p 1/2 L 5/3 (Q 2 m+1 r ) u L 10/3 (Q 2 m+1 r ) m 0 2 7m/2 r 3/2 ( ɛ 4/5 (2 m r) 6/5) 1/2( ɛ 3/5 (2 m r) 9/10) 1/2( ɛ 3/10 (2 m r) 9/20) m 0 = by (4.2), (4.4) and (4.5), which gives I 3 ɛ. ollecting the estimates for I 1, I 2, and I 3,we obtain α(r) + β(r) ɛ 3/5 + ɛ. Regarding the pressure, we may use lemma 3.3 in order to obtain ( ) r 12/5 p L 5/4 (Q r ) u L 10/3 (Q ρ ) u L 2 (Q ρ ) + p L 5/4(Qρ) ρ ( ) r 12/5 (ɛ 3/10 ρ 9/20 )(ɛ 1/2 ρ 3/4 ) + (ɛ 4/5 ρ 6/5 ) ρ whence = ɛ 4/5 ρ 6/5 + ɛ 4/5 π(r) = 1 p 1/2 r1/4 L 5/4 (Q r ) r 12/5 ρ 6/5 (ɛ2/5 ρ21/80 ɛ 2 2m ɛ4/5 ρ 6/5 + ɛ 4/5 ρ 33/25 ρ 3/5 + ɛ 2/5 ρ 33/50 ) ɛ 2/5 We thus obtain α(r) + β(r) + π(r) ɛ 2/5, and the regularity follows as in the proof of theorem 3.1. Proof of theorem 2.2. Theorem 2.2 follows immediately from theorem 4.1. In order to complete our estimate on the dimension of the singular set, we need to fill in the gap for λ [1, 21/20] and calculate the λ-fractal dimension directly as an expression of λ. Thus, instead of (4.1), we assume that for R [ρ λ,ρ] ( u 10/3 + u 2 + p 5/3 + p 5/4) dx dt ɛ R ω Q ρ where ω [3/2, 45/29] is to be determined. Then we have u L 10/3 (Q R ) ɛ 3/10 R 3ω/10, u L 2 (Q R ) ɛ 1/2 R ω/2, p L 5/3 (Q R ) ɛ 3/5 R 3ω/5 and p L 4/5 (Q R ) ɛ 4/5 R 4ω/5 for all R [ρ λ,ρ]. With r = ρ δ /2 where δ λ is to be determined, we estimate the integrals I 1, I 2 and I 3 from section 3. First, we have I 1 (r 2 /ρ 3 ) u 2 L 10/3 (Q ρ ɛ3/5 ) r 2 /ρ 3 3ω/5 = ɛ 3/5 r 2+3ω/5δ 3/δ. Let m 0 be as in the proof of (3.8), and let m 1 be the largest integer such that (2 m 1 ρ λ ) 2 ρ 2. Then I 2 = m 1 + m 0 m=m 1 +1 ( u 2 θ m (t))u j j (φη m ) dx dt..

10 An estimate on the parabolic fractal dimension of the singular set for solutions of the Navier Stokes system 2783 The term in the first sum are estimated as in the proof of theorem 3.1 while the terms in the second sum are treated as in theorem 4.1. We get m 1 I 2 + m m/2 (ɛ3/10 r3/2 ρ 3λω/10 ) 5/2 (ɛ 1/2 ρ λω/2 ) 1/2 m=m m/2 (ɛ3/10 r3/2 (2 m r) 3ω/10 ) 5/2 (ɛ 1/2 (2 m r) ω/2 ) 1/2. Summing up both geometric series, we obtain I 2 ɛ ρ λω 3δ/2 +ɛ ρ λω+2δ 7λ/2 /r (λ 1)(7/2 ω), where we used 2 m 1 ρ δ λ. hoosing δ = 30λ/(9 +20λ) and ω = 45/(9 +20λ), we deduce I 2 ɛ. Similarly, we obtain the smallness of I 3 and π(r). We thus conclude that for λ [1, 21/20], the λ-fractal dimension of the singular set is bounded from above by 45/(9+20λ). Acknowledgments The authors thank referees for useful remarks and suggestions. Both authors were supported in part by the NSF grant DMS References [F] onstantin P and Foias 1988 Navier Stokes Equations (hicago Lectures in Mathematics) (hicago, IL: University of hicago Press) [KN] affarelli L, Kohn R and Nirenberg L 1982 Partial regularity of suitable weak solutions of the Navier Stokes equations ommun. Pure Appl. Math [DG] Doering R and Gibbon J D 1995 Applied Analysis of the Navier Stokes Equations (ambridge Texts in Applied Mathematics) (ambridge: ambridge University Press) [EFNT] Eden A, Foias, Nicolaenko B and Temam R 1994 Exponential Attractors for Dissipative Evolution Equations (RAM: Research in Applied Mathematics vol 37) (Paris: Masson) [F] Falconer K 2003 Fractal Geometry Mathematical Foundations and Applications 2nd edn (Hoboken, NJ: Wiley) [K1] Kukavica I 2008 The partial regularity results for the Navier Stokes equations Proc. Workshop on Partial Differential Equations and Fluid Mechanics (Warwick, UK, 2008) [K2] Kukavica I 2009 The fractal dimension of the singular set for solutions of the Navier Stokes system Nonlinearity [L] Lemarié-Rieusset P G 2002 Recent Developments in the Navier Stokes Problem (hapman and Hall/R Research Notes in Mathematics vol 431) (London/Boca Raton, FL: hapman and Hall/R Press) [Li] Lin F 1998 A new proof of the affarelli Kohn Nirenberg theorem ommun. Pure Appl. Math [LS] Ladyzhenskaya O A and Seregin G A 1999 On partial regularity of suitable weak solutions to the threedimensional Navier Stokes equations J. Math. Fluid Mech [RS1] Robinson J and Sadowski W 2007 Decay of weak solutions and the singular set of the three-dimensional Navier Stokes equations Nonlinearity [RS2] Robinson J and Sadowski W 2009 A criterion for uniqueness of Lagrangian trajectories for weak solutions of the 3D Navier Stokes equations ommun. Math. Phys [RS3] Robinson J and Sadowski W 2009 Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier Stokes equations Nonlinearity [S] Scheffer V 1977 Hausdorff measure and the Navier Stokes equations ommun. Math. Phys [T] Temam R 2001 Navier Stokes Equations: Theory and Numerical Analysis (Providence, RI: AMS helsea Publishing) Reprint of the 1984 edition [V] Vasseur A F 2007 A new proof of partial regularity of solutions to Navier Stokes equations NoDEA Nonlinear Diff. Eqns. Appl [W] Wolf J 2008 A direct proof of the affarelli Kohn Nirenberg theorem Parabolic and Navier Stokes equations Part 2 (Banach enter Publ. vol 81) (Warsaw: Polish Acad. Sci. Inst. Math.) pp