On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations

On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations G. Seregin, V. Šverák Dedicated to Vsevolod Alexeevich Solonnikov Abstract We prove two sufficient conditions for local regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. One of them implies smoothness of L 3, -solutions as a particular case. Introduction In this paper, we address to local properties of weak solutions to the Navier- Stokes equations. To be precise, let us consider the Navier-Stokes equations, describing the motion of a viscous incompressible fluid in Q, i.e.: t v + div v v v = p div v = 0 } in Q. Here, Q = B ], 0[ is the unit space-time cylinder. The question is under what conditions the point z = 0 is a regular point of v. The latter means that there exists a nonempty neighborhood of the origin such that v is Hölder continuous in the intersection of this neighborhood and the closure of Q. Our first assumption on the velocity v and the pressure p is that they form the so-called suitable weak solution to the Navier-Stokes equations in Q. The definition of it is as follows. Definition. Let ω be a open set in R 3. We say that a pair u and q is a suitable weak solution to the Navier-Stokes equations on the set ω ] T, T [

if it satisfies the conditions: u L 2, (ω ] T, T [) L 2 ( T, T ; W 2 (ω)); (.) q L 3 (ω ] T, T [); (.2) 2 u and q satisfy the Navier-Stokes equations in the sense of distributions; u and q satisfy the local energy inequality ϕ(x, t) u(x, t) 2 dx + 2 ω ω ] T,t[ ω ] T,t[ ϕ u 2 dxdt ( u 2 ( ϕ + t ϕ) + u ϕ( u 2 + 2q)) dxdt (.3) (.4) for a.a. t ] T, T [ and for all nonnegative functions ϕ C 0 (R 3 R ), vanishing in a neighborhood of the parabolic boundary Q ω {t = T } ω [ T, T ] of Q. For discussions about the notion of suitable weak solutions, we refer the reader to papers [8], [], [7], [6], [0], [], and [5]. For the reader convenience, we first formulate known results from the the so-called ε-regularity theory of suitable weak solutions. Explanations and proofs of the them can be found in the above cited papers. Lemma.2 Consider two functions v and p defined in the space-time cylinder Q(z 0, R) = B(x 0, R) ]t 0 R 2, t 0 [, where B(x 0, R) R 3 stands for the ball of radius R with the center at the point x 0. Assume that v and p form a suitable weak solution to the Navier-Stokes equations in Q(z 0, R). There exists a universal positive constant ε such that if R 2 Q(z 0,R) ( ) v 3 + p 3 2 dz < ε, (.5) then, for any k = 0,,..., the function z k v(z) is Hölder continuous in the closure of the set Q(z 0, R/2) and, moreover, for some universal positive constant C k. sup k v(z) < C k R k (.6) z Q(z 0,R/2) 2

Lemma.3 Assume that v and p form a suitable weak solution to the Navier-Stokes equations in Q(z 0, R). There exists an absolute positive constant ε 0 such that if sup 0<r R r 2 Q(z 0,r) v 3 dz < ε 0, (.7) then there exists a nonempty neighborhood O z0 of the point z 0 such that the function z v(z) is Hölder continuous in O z0 Q(z 0, R). One of the goals of our paper is somehow to weaken condition (.7). It might seem that it would be enough to replace this condition with just boundedness of the left hand in (.7). But we cannot prove that. Our result in this direction is as follows. Theorem.4 Let v and p be a suitable weak solution to the Navier-Stokes equations in Q Q(0, ). Suppose that v satisfies the condition: M 0 sup v 3 dz < +. (.8) 0<R< R 2 Q(R) Here, Q(R) B(R) ] R 2, 0[ so that Q() = Q. Assume that there exist numbers β 0 α 0, T 0 >, and ρ 0 > 0 such that the following two conditions hold: where where lim sup R 0 ρ 2 0R 2 Q(Rz,ρ 0 R) ( ) v 3 + p 3 2 dz < ε, z A 0, (.9) A 0 {z = (x, t) x α 0, T 0 t 0}, for each x B 0, there is ρ = ρ (x ) > 0 such that lim sup v(x, 0) 2 dx = 0, (.0) R 0 ρ R B(Rx,ρ R) B 0 {x R 3 x n β 0 }. Then the function v is Hölder continuous in the closure of the set Q(r) for some r > 0. 3

However, because of condition (.9), Theorem.4 still looks like another statement of the ε-regularity theory. Our next result includes smoothness of local L 3, -solutions proved in [2], [2], [3], and [5] as a particular case. Theorem.5 Let v and p be a suitable weak solution to the Navier-Stokes equations in Q. Suppose that v satisfies the condition M 0 sup sup v 3 dz < +. (.) R 2 z 0 Q(/2) 0<R /4 Q(z 0,R) Assume also that there are numbers β 0 and T > 3 such that condition (.0) holds and M 2 lim sup a + lim sup R 0 R 2 0 T R 2 B(Ra) v 3 dz < +. (.2) Then the function v is Hölder continuous in the closure of the set Q(r) for some r > 0. 2 Auxiliary Results We start with a lemma on a kind of decay estimate for the pressure p. Lemma 2. We let C(r) r 2 v 3 dz, D(r) r 2 p 3 2 dz. Q(r) Q(r) Assume that v and p form a suitable weak solutions to the Navier-Stokes equations and satisfy condition (.8). Then, for each α ]0, [, a constant c = c (α) exists such that { } D(r) c r α D() + M 0 (2.) for any r ]0, ]. Here, M 0 is the constant in (.8). 4

Proof. In [9], the following estimate was established D(r) c 2 {( r ρ ) ( ρ ) 2C(ρ) } D(ρ) +, (2.2) r which is valid for all 0 < r < ρ and for some universal constant c 2. Fix α ]0, [ and choose τ = τ(α) ]0, [ so that By (.8), (2.2), and (2.3), we have c 2 τ α. (2.3) D(τρ) τ α D(ρ) + c 2 τ 2 C(ρ) τ α D(ρ) + c 2 τ 2 M 0 for any ρ ]0, ]. We may iterate the latter estimate. As a result, we find D(τ k ) τ αk D() + c 2 τ M 2 0 (2.4) τ α for any k =, 2,... Required estimate (2.) can be easily deduced from (2.4). Lemma 2. is proved. Now, we are in a position to prove the main blow-up lemma. Lemma 2.2 Assume that v and p form a suitable weak solutions to the Navier-Stokes equations and satisfy condition (.8). Assume that z = 0 is a singular point of v. There exists a pair of functions u and q defined on Q = R 3 ], 0[ with the following properties. For any a > 0, they form a suitable weak solution to the Navier-Stokes equations in Q(a). In addition, u C([ a 2, 0]; L β (B(a))) (2.5) for any a > 0 and for any β [, 2[, u 3 dz ε 0 > 0, (2.6) Q() where ε 0 is the constant of Lemma.3, ρ 2 ( u 3 + q 3 2 ) dz lim sup R 0 ρ 2 R 2 Q(z,ρ) 5 Q(Rz,ρR) ( v 3 + p 3 2 ) dz (2.7)

for any z Q and for any ρ > 0, and ( ) u(x, 0) 5 4 ( 5 ρ 7/4 4 dx c 3 lim sup R 0 ρr B(x,ρ) B(Rx,ρR) ) v(x, 0) 2 2 dz (2.8) for any x R 3, for any ρ > 0, and for some universal positive constant c 3. Proof By Lemma 2. and condition (.8), we show that ( ) v 3 + p 3 R 2 2 dz M < + (2.9) Q(R) for any R ]0, ]. Since the origin is a singular point, there exists a sequence {R k > 0} k=, tending to zero as k +, such that (see Lemma.3) v 3 dz ε 0 > 0 (2.0) Rk 2 Q(R k ) for any k =, 2,... We extend v and p to zero outside Q and denote these extensions by v and p. For any e = (y, s) R 3 R, we let u k (y, s) = R k v(r k y, R 2 ks), q k (y, s) = R 2 kp(r k y, R 2 ks). We can state that the pair u k and q k is a suitable weak solution to the Navier- Stokes equations in the cylinder Q(/R k ). In particular, it satisfies the local energy inequality: φ(y, s) u k (y, s) 2 dy + 2 φ u k 2 dyds B(/R k ) B(/R k ) ] /Rk 2,s[ [ ] u k 2 ( t φ + φ) + u k φ( u k 2 + 2q k ) dyds (2.) B(/R k ) ] /R 2 k,s[ for a.a. s ] /Rk 2, 0[ and for all non-negative functions φ C 0 (R 3+ ) vanishing in a neighborhood of the parabolic boundary of the cylinder Q(/R k ). Moreover, by our scaling, we have a 2 Q(a) ( u k 3 + q k 3 2 ) de = a 2 R 2 k Q(aR k ) 6 ( ) v 3 + p 3 2 dz M (2.2)

if ar k <. Thanks to (2.), (2.2) and the known multiplicative inequality, we may find subsequences of u k and q k still denoted in the same way such that, for any a > 0, u k u in L 3 (Q(a)), q k q in L 3 (Q(a)), 2 u k u in L 2 (Q(a)), u k u in L 2, (Q(a)), u k u in L 0 (Q(a)). 3 To evaluate the derivative of v in t, we use the Navier-Stokes equations written in the following form: [ ] u k t w dyds = (u k u k u k ) : w + q k div w dyds Q(a) Q(a) for any w L 3 ( a 2, 0; W 3(B(a))). This gives us t u k t u in L 3 ( a 2, 0; ( W 3(B(a))) ) 2 and thus, by compactness arguments, u k u in L 3 (Q(a)). (2.3) Letting k, we pass to the limit in the Navier-Stokes equations and the local energy inequality and show that the pair of functions u and q, defined on Q = R 3 ], 0[, is a suitable weak solution to the Navier-Stokes equations in Q(a) for any a > 0. This is a nontrivial solution since the limit function satisfies (2.6). We know that the convective term div u k u k is bounded, for example, in L 5 (Q(a)) for each R k a <. Using an appropriated choice of a cut-off 4 function, the coercive estimates for Stokes system, multiplicative inequalities and duality arguments, we can show that the following norms are bounded: for R k a <. This implies that t u k 5 4,Q(a), 2 u k 5 4,Q(a), qk 5 4,Q(a) u k u in C([ a 2, 0]; L 5 (B(a))) 4 7

for any a > 0. Relation (2.5) can be obtained now by the interpolation. Now, our goal is to prove (2.7) and (2.8). This is an easy task. Indeed, by the scaling, we have the following two relations: ρ 2 Q(z,ρ) for any z Q, and ( ρ 7/4 0 B(x,ρ 0 ) ( u k 3 + q 3 2 ) dz = ρ 2 Rk 2 u k (x, 0) 5 4 dx ) 4 5 ( c 3 ρ 0 R k Q(R k z,ρr k ) ( = (ρ 0 R k ) 7/4 B(R k x,ρ 0 R k ) B(R k x,ρ 0 R k ) ) v(x, 0) 2 2 dz ( v 3 + p 3 2 ) dz v(x, 0) 5 4 dx ) 4 5 for any x R 3 and for some universal positive constant c 3. Taking the limit as k +, we find (2.7) and (2.8). Lemma 2.2 is proved. 3 Proof of Theorem.4 We see that, by (.9) and (2.7), ρ 2 0 Q(z,ρ 0 ) ( u 3 + q 3 2 ) dz < ε for any z A 0. According to Lemma.2, we can state that, for any k = 0,, 2,..., the function k u is Hölder continuous in the set and, moreover, Q = (R 3 \ B(α 0 )) [ T 0, 0] sup( u(z) + u(z) < M (3.) z Q for some positive constant M. From (3.) it follows that the vorticity ω = u satisfies the inequality t ω ω M( ω + ω ) (3.2) 8

at least in Q = (R 3 \ B( 3 2 α 0)) ] 2 (T 0 + ), 0[. On the hand, condition (.0) and inequality (2.8) say that u(, 0) = 0 in B 0 and, therefore, ω(, 0) = 0 in B 0. By the backward uniqueness result (see [2] and [3]), we state that ω(x, t) = 0, x n > β 0, < t < 0. The unique continuation through spatial boundaries implies (see, for example, [3] and references there) ω 0 in (R 3 \ B(β 0 )) ], 0[. The same arguments as in [2] and in [5] allows us to state that in fact ω 0 in R 3 ], 0[. In turn, this means that, for a.a t ], 0[, u(, t) is bounded harmonic function defined in the whole R 3. So, as a result, we have u(x, t) = a(t), t ], 0[. Here, the function a can be found by solving the Navier-Stokes equations and, therefore, But we know that a (t) = q(x, t) q(x, t) = a (t)x + b(t). ρ 2 0 Q(z,ρ 0 ) q 3 2 dz < ε for all z A 0. Obviously, the last two relations are in a contradiction if a 0. So, we have proved that u is a constant in R 3 ], 0[. But u is a Hölder continuous function far away from the origin and equal to zero for x n > β 0 and t = 0. So, u is identically equal to zero in R 3 ], 0[ which contradicts with (2.6). Theorem.4 is proved. It is curious that conditions (.8) and (.9) are fulfilled if v(x, t) 2 + p(x, t) 3 2 c x + t for (x, t) Q and for some constant c. However, it is not clear whether L 3, -solutions satisfy condition (.9). 9

4 Proof of Theorem.5 Our first observation is that Lemma 2.2 can be strengthened in the following way. Lemma 4. We let C(z 0, r; v) r 2 v 3 dz, D(z 0, r; p) r 2 p 3 2 dz. Q(z 0,r) Q(z 0,r) Assume that v and p form a suitable weak solutions to the Navier-Stokes equations and satisfy condition (.). Then, for each α ]0, [, a constant c = c (α) exists such that { } D(z 0, r; p) c r α D() + M 0 (4.) for any r ]0, /4] and for any z 0 Q(/2). Here, M 0 is the constant in (.). The proof is the same as the proof of Lemma 2.. Now, assume that z = 0 is a singular point. Since (.8) holds, we can use arguments of the proof of Lemma 2.2. First of all, we note that all statements of Lemma 2.2 are valid for blow-up functions u and q as well. Moreover, for scaled functions u k, we have the identity 0 u k 3 dz = R 2 k 0 v 3 dz T B(a) T Rk 2 B(R k a) for any a > 0. Thanks to (2.3), passage to the limit as k + gives us: 0 T B(a) u 3 de lim sup k + 0 Rk 2 T Rk 2 B(R k a) v 3 dz lim sup R 0 R 2 0 T R 2 B(Ra) v 3 dz. 0

It remains to pass to the limit as a + and take into account condition (.2). As a result, we find 0 T Now, we introduce the notation: E(e 0, r) r 2 R 3 u 3 de M 2. (4.2) Q(e 0,r) ( u 3 + q 3 2 ) de. For D(e 0, r; q), we have the decay estimate similar to (2.2) ( ) 2C(e0 D(e 0, r; q) c 2 {rd(e 0, ; q) +, ; u)} r (4.3) for all 0 < r and for some absolute constant c 2. From (4.3), it follows that { ( ) 2C(e0 E(e 0, r) ( + c 2 ) rd(e 0, ; q) +, ; u)} (4.4) r for the same r as in (4.3). Now, our goal is to show that D(e 0, ; q) L < +, e 0 R 3 ], 0], (4.5) where a positive constant L depends on M 0 and D() only. To this end, we first observe that D(e 0, ; q) lim inf D(e 0, ; q k ). (4.6) k + On the other hand, D(e 0, ; q k ) = D(z k, R k ; p), z k = (R k y 0, R 2 ks 0 ). (4.7) For the definition of the scaled pressure q k and its properties, we refer the reader to Lemma 2.2. For sufficiently large k, we have Then, by Lemma 4., R k e 0 Q(/2), R k /4. D(z k, R k ; p) c { } Rk α D() + M 0. (4.8)

Clearly, that (4.5) follows from (4.6) (4.8). Combining (4.4) and (4.5), we find { ( ) 2C(e0 E(e 0, r) ( + c 2 ) rl +, ; u)} r (4.9) for any 0 < r and for any e 0 R 3 ], 0]. Now, let ε be the number of Lemma.2. We may fix a positive number r ]0, ] so that ( + c 2 )rl < ε/3. (4.0) For this fixed r, according (4.2), there exists a number R > 0 such that 0 T R 3 \B(R/2) u 3 de εr 2 3( + c 2 ). (4.) Obviously, (4.) implies the estimate ( ) 2C(e0 ( + c 2 ), ; u) < ε/3 r whenever e 0 (R 3 \ B(R)) ] T /2, 0[ is. Combining the latter inequality with (4.9) and (4.0), we show E(e 0, r) < ε (4.2) for any e 0 (R 3 \ B(R)) ] T /2, 0[. It remains to repeat arguments of Theorem.4 and complete the proof by getting the same contradiction. Theorem.5 is proved. References [] Caffarelli, L., Kohn, R.-V., Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., Vol. XXXV (982), pp. 77 83. [2] Escauriaza,L., Seregin, G., Šverák, V., On backward uniqueness for parabolic equations, Zap. Nauchn. Seminar. POMI, 288(2002), 00 03. 2

[3] Escauriaza,L., Seregin, G., Šverák, V., On backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 69(2003)2, 47 57. [4] Escauriaza,L., Seregin, G., Šverák, V., Backward uniqueness for the heat operator in half space, Algebra and Analyis, 5 (2003), no., 20 24. [5] Escauriaza,L., Seregin, G., Šverák, V., On L 3, -Solutions to the Navier- Stokes equations and backward uniqueness, Uspekhi Matematicheskih Nauk,v. 58, 2(350), pp. 3 44. English translation in Russian Mathematical Surveys, 58(2003)2, pp. 2-250. [6] Ladyzhenskaya, O. A., Seregin, G. A., On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. math. fluid mech., (999), pp. 356-387. [7] Lin, F.-H., A new proof of the Caffarelly-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 5(998), no.3, pp. 24 257. [8] Scheffer, V., Hausdorff measure and the Navier-Stokes equations, Commun. Math. Phys., 55(977), pp. 97 2. [9] Seregin, G. A. On the number of singular points of weak solutions to the Navier-Stokes equations, Comm. Pure Appl. Math., 54(200), issue 8, pp. 09-028. [0] Seregin, G.A., Local regularity of suitable weak solutions to the Navier- Stokes equations near the boundary, J.math. fluid mech., 4(2002), no., 29. [] Seregin, G.A., Differentiability properties of weak solutions to the Navier-Stokes equations, Algebra and Analysis, 4(2002), No., pp. 93-237. [2] Seregin, G., Šverák, V., The Navier-Stokes equations and backward uniqueness, Nonlinear Problems in Mathematical Physics II, In Honor of Professor O.A. Ladyzhenskaya, International Mathematical Series II, 2002, pp. 359 370. G. Seregin Steklov Institute of Mathematics at St.Petersburg, 3

St.Peterburg, Russia V. Šverák School of Mathematics, University of Minnesota, Minneapolis, USA 4