CONDITIONS IMPLYING REGULARITY OF THE THREE DIMENSIONAL NAVIER-STOKES EQUATION 4 0 0 STEPHEN MONTGOMERY-SMITH 2 n a Abstract. We obtain logarithmic improvements for conditions J for regularity of the Navier-Stokes equation, similar to those of 0 Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use 1 of a stochastic approach involving Feynman-Kac like inequalities. Aspartoftheourmethods,wegiveadifferentapproachtoapriori ] P estimates of Foia¸s, Guillop´e and Temam. A . h t a m 1. Introduction [ TheversionofthethreedimensionalNavier-Stokesequationwestudy 6 is the differential equation in u = u(t) = u(x,t), where t 0, and v x R3: ≥ 7 ∈ 0 ∂u = ∆u Ldiv(u u), u(0) = u . 2 0 ∂t − ⊗ 1 0 Here L denotes the Leray projection. We will not usually be working 3 with classical solutions. We define u(t), 0 t T, to be a solution of 0 ≤ ≤ the Navier-Stokes equation if, whenever u(t ) is sufficiently regular for / 0 h a mild solution t a m t u(t) = e(t−t0)∆u(t ) e(t−s)∆Ldiv(u(s) u(s))ds 0 : − ⊗ v Zt0 i X to exist for t [t ,t + τ) for some τ > 0, then u(t) is equal to that 0 0 ∈ r mild solution in [t ,t +τ). a 0 0 We also use other ways to describe the three dimensional Navier- Stokes equation. First, let us denote the vorticity by w = w(t) = w(x,t) = curlu. If w is sufficiently smooth then ∂w = ∆w u w +w u, w(0) = curlu . 0 ∂t − ·∇ ·∇ 2000 Mathematics Subject Classification. Primary 35Q30, 76D05, Secondary 60H30, 46E30. Key words and phrases. Navier-Stokes equation, vorticity, Prodi-Serrin condi- tion, Beale-Kato-Majda condition, Orlicz norm, stochastic methods. The author was partially supported by an NSF grant. 1 2 STEPHEN MONTGOMERY-SMITH Another description is given by the so called magnetization variable [4],[16]. Letm = m(t) = m(x,t)beavectorfieldsatisfyinganequation ∂m = ∆m u m m ( u)T, m(0) = u + q 0 0 ∂t − ·∇ − · ∇ ∇ for some scalar field q = q (x). (Here the superscript T denotes the 0 0 transpose.) Then under sufficient smoothness assumptions we have that u is the Leray projection of m. A famous open problem is to prove regularity of the Navier-Stokes equation, that is, if the initial data u is in L and is regular (which 0 2 in this paper we define to mean that it is in the Sobolev spaces Wn,q for some 2 q < and all positive integers n), then the solution u(t) ≤ ∞ is regular for all t 0. Such regularity would also imply uniqueness ≥ of the solution u(t). Currently only the existence of weak solutions is known. Also, it is known that for each regular u that there exists 0 t > 0 such that u(t) is regular for 0 t t . We refer the reader to 0 0 ≤ ≤ [3], [6], [7], [14], [21]. In studying this problem, various conditions that imply regularity have been obtained. For example, the Prodi-Serrin conditions ([17], [19]) state that for some 2 p < , 3 < q with 2 + 3 1 that ≤ ∞ ≤ ∞ p q ≤ T u(t) pdt < k kq ∞ Z0 for all T > 0. If u is a weak solution to the Navier-Stokes equation satisfying a Prodi-Serrin condition, with regular initial data u , then 0 u is regular (see [20]). (Recently Escauriaza, Seregin and Sver´ak [8] showed that the condition when q = 3 and p = is also sufficient.) ∞ This is a long way from what is currently known for the so called Leray-Hopf weak solutions: T u(t) pdt < k kq ∞ Z0 for 2 + 3 3, 2 q 6. p q ≥ 2 ≤ ≤ Another condition is that of Beale, Kato and Majda [1]. They show that regularity follows from the condition T w(t) dt < k k∞ ∞ Z0 for all T > 0. (In fact they proved this for the Euler equation, but the proof works also for the Navier-Stokes equation with only small modifications.) This was strengthened by Kozono and Taniuchi [12] to REGULARITY OF NAVIER-STOKES 3 show that regularity follows from the condition T T u(t) dt w(t) dt < k∇ kBMO ≈ k kBMO ∞ Z0 Z0 for all T > 0, where here BMO denotes the space of functions with bounded mean oscillation. The purpose of this paper is threefold. First, we would like to pro- vide some logarithmic improvements to these conditions. Secondly, we would like to present a stochastic approach to the Navier-Stokes equation, obtaining our conditions using Feynman-Kac like inequali- ties. Thirdly, we would like to present a different process for creating estimates of Foia¸s, Guillop´e and Temam. To this end, the first result of this paper is the logarithmic improve- ment to the Prodi-Serrin conditions. Theorem 1.1. Let 2 < p < , 3 < q < with 2 + 3 = 1. If u is a ∞ ∞ p q solution to the Navier-Stokes equation satisfying T u(t) p k kq dt < 1+log+ u(t) ∞ Z0 k kq for some T > 0, then u(t) is regular for 0 < t T. ≤ We first present a proof of this result (and indeed of a slightly stronger result) that uses a standard approach. Then we present a stochastic approach to the Navier-Stokes equation. This is a kind of Lagrangian coordinates approach to the Navier-Stokes equation, but with a probabilistic twist in that we follow the path of each particle with a stochastic perturbation. A similar approach was adopted by Busnello, Flandoli and Romito [2]. From this we obtain the following Beale-Kato-Majda type condition. For 1 q < , define the function on [0, ) ≤ ∞ ∞ eλ 1 q Φ (λ) = − . q e 1 (cid:18) − (cid:19) Define the Φ -Orlicz norm on any space of measurable functions by the q formula f = inf λ > 0 : Φ ( f(x) /λ)dx 1 . k kΦq q | | ≤ (cid:26) Z (cid:27) (Thus the triangle inequality is a consequence of the fact that Φ is q convex, see [13].) Theorem 1.2. Let 1 < q < , 3 < r < , and T > 0. Suppose that ∞ ∞ u is a solution to the Navier-Stokes equation satisfying 4 STEPHEN MONTGOMERY-SMITH (1) for all T (0,T) 0 ∈ T u(t) dt < , k∇ kΦq ∞ ZT0 and (2) either q < 3, or u(t) < for almost every t [0,T]. k kr ∞ ∈ Then u(t) is regular for 0 < t T. ≤ Note that since c for q > q , we may assume without k·kΦq1 ≤ k·kΦq2 1 2 loss of generality that q > 3/2. Next, if 3/2 < q < 3, since k·kq ≤ (e 1) ,bytheSobolevinequalityweseethatthesecondhypothesis − k·kΦq is automatically satisfied with r = 3q/(3 q). Also, this hypothesis is − always satisfied for Leray-Hopf weak solutions with r = 6. Next we demonstrate how to obtain Theorem 1.1 from Theorem 1.2 using the following result. If u is a solution to the Navier-Stokes equa- tion, we define the sets An,q (λ) = t [T ,T ] : nu(t) λ . T0,T1 { ∈ 0 1 k∇ kq ≥ } Theorem 1.3. Given 3 < q q , and a non-negative integer 1 2 ≤ ≤ ∞ n, there exists constants c ,c ,c > 0 such that if u(t), 0 t T is 1 2 3 2 ≤ ≤ a solution to the Navier-Stokes equation, and if 0 T T , then for 1 2 ≤ ≤ all r (0,√T T ) we have 2 1 ∈ − An,q2 (c r3/q2−n−1) c A0,q1 (c r3/q1−1) . | T1+r2,T2 1 | ≤ 2| T1,T2 3 | Asimilarresultthatonecanobtain(butwedonotprovehere)isthat forpositiveintegersnwehave An,2 (c r1/2−n) c A1,2 (c r−1/2) . | T1+r2,T2 1 | ≤ 2| T1,T2 3 | Corollary 1.4. Under the hypotheses of Theorem 1.3, there exists a constant c > 0 with the following properties. If Θ(λ) is a positive increasing function of λ 0, define ≥ ∞ κ = min (cλ−2 T )+,T dΘ(λ). 0 1 { − } Z0 Then T1 T1 Θ( nu(s) 1/(1+n−3/q2))ds cκ+c Θ(c u(s) 1/(1−3/q1))ds. k∇ kq2 ≤ k kq1 ZT0 Z0 Similarly, T1 T1 Θ( nu(s) 1/(n−1/2))ds cκ+c Θ(c u(s) 2)ds. k∇ k2 ≤ k∇ k2 ZT0 Z0 REGULARITY OF NAVIER-STOKES 5 Since the Leray-Hopf weak solution to the Navier-Stokes equation satisfies T u(t) 2dt < , one can quickly recover the results of 0 k∇ k2 ∞ Foia¸s, Guillop´e and Temam [9] that say that T nu(t) 1/(n−1/2)dt < R 0 k∇ k2 . ∞ R 2. Theorem 1.1 The hypothesis of Theorem 1.1 imply that, given ǫ (0,T), there ∈ exists T (0,ǫ) with u(T ) L . Let T∗ > T be the first point 0 0 q 0 ∈ ∈ of non-regularity for u(t). It is well known that in order to show that T∗ > T, it is sufficient to show an a priori estimate, that is sup u(t) < . This is because it is then possible to T0≤t<min{T∗,T}k kq ∞ extend the regularity beyond T∗ if T∗ T. Without loss of gener- ≤ ality, it is sufficient to consider the case T = T∗ (so as to obtain a contradiction). Proof of Theorem 1.1. Weallowallconstantstoimplicitlydependupon p and q. Let us define quantities v = u u q/2−1, | | 3 2 ∂u A = u q/2−1 i , | | ∂x i,j=1(cid:18) j(cid:19) X 2 3 3 ∂u B = u q/2−3u u k i k | | ∂x j! i,j=1 k=1 X X Note that 3 2 ∂v v 2 := i A+B, |∇ | ∂x ≈ i,j=1(cid:18) j(cid:19) X 3 ∂ ∂u u q−2u i A+B, i ∂x | | ∂x ≈ j j i,j=1 X (cid:0) (cid:1) 3 2 ∂ u q−2u c u q−2 v 2. i ∂x | | ≤ | | |∇ | i,j=1(cid:18) j (cid:19) X (cid:0) (cid:1) We start with the Navier-Stokes equation, take the inner product with u u q−2, and integrate over R3 to obtain | | ∂ u q−1 u = u q−2u ∆udx u q−2u Ldiv(u u)dx. k kq ∂tk kq | | · − | | · ⊗ Z Z 6 STEPHEN MONTGOMERY-SMITH Integrating by parts, we see that 3 ∂ ∂u u q−2u ∆udx = u q−2u i dx v 2, | | · − ∂x | | i ∂x ≈ −k∇ k2 j j Z Z i,j=1 X (cid:0) (cid:1) and 3 ∂ u q−2u Ldiv(u u)dx = u q−2u [L(u u)] dx i j i | | · ⊗ ∂x | | j Z Z i,j=1 X (cid:0) (cid:1) c u q/2−1 v L(u u) ≤ k| | ksk∇ k2k ⊗ kr where r = 1+q/2 and s = (2q+4)/(q 2). Now the Leray projection − is a bounded operator on L , and hence L(u u) u 2 . Also r k ⊗ kr ≈ k k2+q u q/2−1 u q/2−1. Hence k| | ks ≈ k k2+q u q−2u Ldiv(u u)dx c u 1+q/2 v = c v 1+2/q v . | | · ⊗ ≤ k k2+q k∇ k2 k k2+4/qk∇ k2 Z From the Sobolev and interpolation inequalities v c 3/(q+2)v c v (q−1)/(q+2) v 3/(q+2), k k2+4/q ≤ k|∇| k2 ≤ k k2 k∇ k2 and hence u q−2u Ldiv(u u)dx c v 1−1/q v 1+3/q. | | · ⊗ ≤ k k2 k∇ k2 Z NowapplyYoung’sinequality ab ((q 3)a2q/(q−3)+(q+3)b2q/(q+3))/2q ≤ − for a,b 0, to obtain ≥ u q−2u Ldiv(u u)dx c v 2 +c v 2(q−1)/(q−3), | | · ⊗ ≤ 1k∇ k2 2k k2 Z where c may be made as small as required by making c larger. Hence 1 2 ∂ u q−1 u c v 2(q−1)/(q−3), k kq ∂tk kq ≤ k k2 that is, ∂ u c u p+1, ∂tk kq ≤ k kq and so ∂ c u p log(1+log+ u ) k kq . ∂t k kq ≤ 1+log+ u k kq Integrating, we see that for T t < T 0 ≤ T u(s) p log(1+log+ u(t) ) log(1+log+ u(T ) )+c k kq ds, k kq ≤ k 0 kq 1+log+ u(s) ZT0 k kq which provides a uniform bound for u(t) . (cid:3) k kq REGULARITY OF NAVIER-STOKES 7 Remark 2.1. Note that this proof can easily be adapted to show that a sufficient condition for regularity is that T u(s) p k kq ds < , Θ( u(s) ) ∞ Z0 k kq where Θ is any increasing function for which ∞ 1 dx = . xΘ(x) ∞ Z1 3. A Priori Estimates This section is devoted to theproof ofTheorem 1.3 andCorollary 1.4 TheproofisverysimilartotheproofScheffer’sTheorem[18]thatstates that the Hausdorff dimension of the set of t for which the solution u(t) is not regular is 1/2. The main tool is the following result is due to Gruji´c and Kukavica [10] (see also [15]). Theorem 3.1. There exist constants a,c > 0 and a function T : (0, ) (0, ), with T(λ) as λ 0, with the following proper- ties∞. If→u ∞L (R3), then th→ere∞is a so→lution u(t) (0 t T( u )) 0 ∈ q ≤ ≤ k 0kq to the Navier-Stokes equation, with u(0) = u , and u(x,t) is the re- 0 striction of an analytic function u(x+iy,t)+iv(x+iy,t) in the region x+ iy C3 : y a√t , and u( +iy,t)+iv( +iy,t) c u { ∈ | | ≤ } k · · kq ≤ k 0kq for y a√t. | | ≤ Proof of Theorem 1.3. First let us show that there exists a constants c ,c ,c > 0suchthatifu(t),t r2 t t isasolutiontotheNavier- 1 3 4 0 0 − ≤ ≤ Stokes equation, and A0,q1 (c r3/q1−1) < c r2, then nu(t ) < | t0−r2,t0 3 | 4 k∇ 0 kq2 c r3/q2−n−1. 1 To see this, Let us first consider the case when t = 0 and r = 1. By 0 hypothesis, we see that there exists t [ 1, 1+c ]with u(t) < c . ∈ − − 4 k kq1 3 By Theorem 3.1 and the appropriate Cauchy integrals, if c is small 4 enough, then there exists a constant c > 0 such that nu(0) < c . 7 k∇ kq2 1 Now, by replacing u(x,t) by r−1u(r−1x,r−2(t t )), we can relax the 0 − restriction r = 1 and t = 0, and we obtain the statement we asserted. 0 Next, given ǫ > 0, it is trivial to find a finite collection t ,...,t in 1 N A = An,q2 (c r3/q2−n−1) such that the sets [t r2,t ] are disjoint, T1+r2,T2 1 n − n but the sets [t r2 ǫ,t + ǫ] cover A. By the above observation, n n − − A0,q1 (c r3/q1−1) c r2. | t0−r2,t0 3 | ≥ 4 Hence r2 N A Nr2 < c−1 A0,q1 (c r3/q1−1) c−1 A0,q1 (c r3/q1−1) . r2 +2ǫ| | ≤ 4 | tn−r2,tn 3 | ≤ 4 | T1,T2 3 | n=1 X 8 STEPHEN MONTGOMERY-SMITH Since ǫ is arbitrary, the result follows. (cid:3) Proof of Corollary 1.4. We only prove the first inequality. By Theo- rem 1.3, there exist constants c ,c ,c > 0 such that 1 2 3 T1 Θ( nu(s) 1/(1+n−3/q2))ds k∇ kq2 ZT0 ∞ = s [T ,T ] : nu(s) 1/(1+n−3/q2) > λ dΘ(λ) |{ ∈ 0 1 k∇ kq2 }| Z0 ∞ c κ+ s [c λ−2,T ] : nu(s) 1/(1+n−3/q2) > λ dΘ(λ) ≤ 1 |{ ∈ 2 1 k∇ kq2 }| Z0 ∞ c κ+c s [0,T ] : u(s) 1/(1−3/q1) > c λ dΘ(λ) ≤ 1 1 |{ ∈ 1 k kq1 3 }| Z0 T1 = c κ+c Θ(c−1 u(s) 1/(1−3/q1))ds. 1 1 3 k kq1 Z0 (cid:3) 4. A Stochastic Description Let us give a little motivation. Suppose that we defined ϕ (x) to t0,t1 be X(t ), where X satisfies the equation 0 dX(t) = u(X(t),t)dt, X(t ) = x, 1 then ϕ would be the “back to coordinates map” that takes a point t0,t1 at t = t to where it was carried from by the flow of the fluid at time 1 t = t . For the Euler equation, this provides a very effective way to 0 describe the solution, for example, the equation for vorticity can be rewritten in a Lagrangian form: t w(x,t) = w(ϕ (x),0)+ w(ϕ (x),s) u(ϕ (x),s)ds. 0,t s,t s,t ·∇ Z0 Similarly, for the magnetization variable we have t m(x,t) = m(ϕ (x),0) m(ϕ (x),s) ( u(ϕ (x),s))T ds. 0,t s,t s,t − · ∇ Z0 For the Navier-Stokes equation this formula is not true, and the Lapla- cian term can make things complicated. One approach to dealing with this is described in the paper by Constantin [5]. However, we take a different approach using Brownian motion, using a kind of “randomly perturbed back to coordinates map.” Such a method was already dis- cussed in the paper [16], here we make the discussion more rigorous. The author recently found out that a similar approach was followed by Busnello, Flandoli and Romito in [2]. REGULARITY OF NAVIER-STOKES 9 The hypothesis of Theorem 1.2 imply that, given ǫ (0,T), there ∈ exists t′ (0,ǫ) with u(t′) L . Then by known results (for example r ∈ ∈ Theorem 3.1), it follows that there exists 0 < T < ǫ such that u(T ) 0 0 Wn,r′ for all r′ [r, ] and positive integers n. Furthermore, arguin∈g ∈ ∞ as in Section 2, we only need to prove sup u(t) < T0≤t<min{T∗,T}k kr ∞ underthea priori assumptionthatthesolutionisregularfort [T ,T]. 0 If f: R3 R is regular, and T t t < T, define A ∈f(x) = → 0 ≤ 0 ≤ 1 t0,t1 α(x,t ), where α satisfies the transport equation 1 ∂α = ∆α u α, α(x,t ) = f(x). 0 ∂t − ·∇ Since div(u) = 0, an easy integration by parts argument shows that ∂ α(x,t)dx = 0, ∂t Z and hence if f is also in L , then 1 A f(x)dx = f(x)dx. t0,t1 Z Z Since stochastic differential equations traditionally move forwards in time, it will be convenient to consider a time reversed equation. Let b(t) be three dimensional Brownian motion. For T t t < T , 0 0 1 1 define the random function ϕ : R3 R3 by ϕ ≤(x) ≤= X( t ), t0,t1 → t0,t1 − 0 where X satisfies the stochastic differential equation: dX(t) = u(X(t),t)dt+√2db(t), X( t ) = x. 1 − − It follows by the Ito Calculus [11] that if T t t < T, then 0 0 1 ≤ ≤ A f(x) = Ef(ϕ (x)). t0,t1 t0,t1 (Here as in the rest of the paper, E denotes expected value.) Note that if f is also in L , then 1 Ef(ϕ (x))dx = f(x)dx. t0,t1 Z Z Applying the usual dominated and monotone convergence theorems, it quickly follows that the last equality is also true if f is any function in L , or if f is any positive function. 1 Now let us develop the equations for the magnetization variable. (The same approach will also work for the vorticity.) If we set m(T ) = 0 u(T ), then we note that m is the unique solution to the integral equa- 0 tion t m(t) = A u(T ) A (m(s) ( u(s))T)ds (T t < T). T0,t 0 − s,t · ∇ 0 ≤ ZT0 10 STEPHEN MONTGOMERY-SMITH Uniqueness followsquickly bytheusualfixedpointargumentover short intervals, remembering that u(t) is regular for T t < T. 0 ≤ Consider also the random quantity m˜ = m˜(x,t) as the solution to the integral equation for T t < T 0 ≤ t m˜(x,t) = u(ϕ (x),T ) m˜(ϕ (x),s) ( u(ϕ (x),s))T ds. T0,t 0 − s,t · ∇ s,t ZT0 Again, it is very easy to show that a solution exists by using a fixed point argument over short time intervals. It is seen that Em˜ satisfies the same equation as m, and hence Em˜ = m. Next, ϕ (ϕ (x)) = ϕ (x), since both are Y(t ) where Y(t) is t0,t1 t1,t2 t0,t2 0 the solution to the integral equation t Y(t) = ϕ (x)+ u(Y(s),s)ds+√2(b b ). t1,t2 −t − −t1 Zt1 Hence s2 m˜(ϕ (x),s ) m˜(ϕ (x),s ) = m˜(ϕ (x),s) ( u(ϕ (x),s))T ds. s1,t 1 − s2,t 2 s,t · ∇ s,t Zs1 Thus, by Gronwall’s inequality, if T t < T 0 ≤ t m˜(x,t) exp u(ϕ (x),s) ds u(ϕ (x),T ) . | | ≤ |∇ s,t | | T0,t 0 | (cid:18)ZT0 (cid:19) (This is essentially the Feynman-Kac formula.) The goal, then, is to find uniform estimates on the quantity t exp u(ϕ (x),s) ds . s,t |∇ | (cid:18)ZT0 (cid:19) This we proceed to do in the next section. 5. Theorem 1.2 Let us fix q and r satisfying the hypothesis of Theorem 1.2, and allow all constants to implicitly depend upon q and r. We retain the notation from the previous section, in particular the definitions of T , 0 T∗ and T. Proof of Theorem 1.2. Since u(t) < for almost every t [0,T], k kr ∞ ∈ by Theorem 1.3, we see that u(t) < for almost every t [0,T]. k∇ k∞ ∞ ∈ Hence, there exists λ > T−1 such that 0 1 u(t) dt , k∇ kΦq ≤ q ZB