Partial regularity for suitable weak solutions to Navier-Stokes equations

Partial regularity for suitable weak solutions to Navier-Stokes equations Yanqing Wang Capital Normal University Joint work with: Quansen Jiu, Gang Wu

Contents 1 What is the partial regularity? 2 Review partial regularity theory to the Navier-Stokes equations 3 A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier-Stokes equations 4 Partial regularity of weak solutions on time to the 3d fractional Navier-Stokes equations

Partial regularity of elliptic and parabolic problems In 1957, De Giorgi solved the 19th Hilbert problem via proving that the weak solutions of the second order of divergence form equation with discontinuous coefficients is regular. The proof strongly rest on a key iteration technique named after De Giorgi. In 1968, De Giorgi found a counterexamples to show that the weak solution of the second order of divergence form elliptic systems may be not continuous when most mathematicians wanted to extend De Giorgi s work to system. (n 3) At the same time, Morrey proved that the measure of set on the irregular point of weak solution to non-linear elliptic systems is zero, which may be the first result on partial regularity theory. Morrey s theorem was generalized by Giusti, Miranda for elliptic case and Giaquinta, Giusti for parabolic case via an indirect powerful approach called the blow-up technique originated from De Giorgi.

The goal of partial regularity Having in mind De Giorgi s counterexamples, full regularity of the weak solutions to system may not always be expected. However, thanks to the anti-sobolev inequality u 2 C dx (R r) 2 B(r) B(R) u 2 dx, which is valid for weak solutions of the quasilinear elliptic systems of divergence form. The regularity of the weak solutions is not too bad. At least we can consider the partial regularity of the weak solutions. As pointed by Duzaar and Grotowski, partial regularity theory involves obtaining estimates on the size of singular point set S(u) (i.e. showing that S(u) has zero n-dimensional Lebesgue measure or better, controlling the Hausdorff dimension of S(u), and showing higher regularity on R(u).

Navier-Stokes equations Physicist C. Navier (1785-1836) and G. Stokes (1819-1903) derived Navier-Stokes equations to describe the motion of Newtonian fluid. u t u + u u + p = f, (x, t) Ω (0, T ), divu = 0, (x, t) Ω (0, T ), u(x, t) Ω = 0, (1) where Ω R n (n 2) is a bounded regular domain. If (u(x, t), p(x, t)) solves N-S, then λ > 0 is also a solution to N-S. (λu(λx, λ 2 t), λ 2 p(λx, λ 2 t)) Quantity x i t u p xi t Dimension 1 2-1 -2-1 -2 Existence and smoothness of Navier-Stokes solutions on R n (n 3)?

Leray-Hopf weak solutions Leray-Hopf weak solutions and their property Let u 0 L 2 (Ω) with divu = 0 and f L 2 (0, T ; L 2 (Ω)). Then there exists a function u such that (1) u L (0, T ; L 2 (Ω)) L 2 (0, T ; W 1,2 (Ω)), (2) (u, p) solves N-S in Ω (0, T ) in the sense of distributions, (3) (u, p) satisfies the energy inequality for t < T u(t) 2 L 2 (Ω) + 2 t 0 u(τ) 2 L 2 (Ω)dτ u(0) 2 L 2 (Ω) + t 0 f, u dτ. 1 Eventual regularity of the weak solutions Leray show that every weak solution becomes smooth after a large time. 2 Leray proved that one dimension Lebesgue measure of the set of the possible time irregular points for the weak solutions to the 3d Navier-Stokes equations is zero. (1/2 dimension Hausdorff measure)

Leray-Hopf weak solutions Leray-Hopf weak solutions and their property Let u 0 L 2 (Ω) with divu = 0 and f L 2 (0, T ; L 2 (Ω)). Then there exists a function u such that (1) u L (0, T ; L 2 (Ω)) L 2 (0, T ; W 1,2 (Ω)), (2) (u, p) solves N-S in Ω (0, T ) in the sense of distributions, (3) (u, p) satisfies the energy inequality for t < T u(t) 2 L 2 (Ω) + 2 t 0 u(τ) 2 L 2 (Ω)dτ u(0) 2 L 2 (Ω) + t 0 f, u dτ. 1 Eventual regularity of the weak solutions Leray show that every weak solution becomes smooth after a large time. 2 Leray proved that one dimension Lebesgue measure of the set of the possible time irregular points for the weak solutions to the 3d Navier-Stokes equations is zero. (1/2 dimension Hausdorff measure)

Leray-Hopf weak solutions Leray-Hopf weak solutions and their property Let u 0 L 2 (Ω) with divu = 0 and f L 2 (0, T ; L 2 (Ω)). Then there exists a function u such that (1) u L (0, T ; L 2 (Ω)) L 2 (0, T ; W 1,2 (Ω)), (2) (u, p) solves N-S in Ω (0, T ) in the sense of distributions, (3) (u, p) satisfies the energy inequality for t < T u(t) 2 L 2 (Ω) + 2 t 0 u(τ) 2 L 2 (Ω)dτ u(0) 2 L 2 (Ω) + t 0 f, u dτ. 1 Eventual regularity of the weak solutions Leray show that every weak solution becomes smooth after a large time. 2 Leray proved that one dimension Lebesgue measure of the set of the possible time irregular points for the weak solutions to the 3d Navier-Stokes equations is zero. (1/2 dimension Hausdorff measure)

Suitable weak solutions Suitable weak solutions(scheffer; Caffrelli, Kohn, Nirenberg; F. Lin) (i) u L (0, T ; L 2 (Ω)) L 2 (0, T ; W 1,2 (Ω)), p L 3/2 ((0, T ); L 3/2 (Ω)). (ii) (u, p) solves N-S in Ω (0, T ) in the sense of distributions. (iii) (u, p) satisfies the local energy inequality in the sense of distributions t ( u 2 2 ) + div ( u u 2 2 ) + div(up) + u 2 u 2 2 fu. (2) Ω s r 2 s u(s, x) 2 φdx + 2 Ω u 2 φdxdt r 2 Ω ) s u ( 2 t φ + φ dxdt + ( u 2 + 2 p )u φdxdt. r 2 Anti-Sobolev Inequality! Ω

Scheffer s Work The Hausdorff dimension of the times singular set of the weak solutions of the 3d Navier-Stokes is at most 1/2. (1976) The Hausdorff dimension of the space-times singular set of the weak solutions of the 3d Navier-Stokes (f = 0) is at most 5/3, (C. M. P. 1980) whose proof is based on t ( u 2 2 ) + div ( u u 2 2 ) + div(up) + u 2 u 2 2 0. 3d-Hausdorff of the space-times singular set of the weak solutions of the 4d Navier-Stokes space-times singular set is finite. (C. M. P. 1978) 1d-Hausdorff of the space-times singular set at the boundary of the weak solutions of the 3d Navier-Stokes space-times singular set is finite. (C. M. P. 1982)

CKN theorem Prop.1(Caffrelli, Kohn, Nirenberg, Comm. Pure. Appl. Math. 1982) ε 1, ε 2 and C 1 > 0. Suppose(u, p) is a suitable weak solution in Q 1 with force f L q, for some q > 5, suppose further that 2 0 ( u 3 + u p ) + ( p dx) 5 4 dt ε1 and f q ε 2. Then Q(1) 1 B(1) Q(1) u(x, t) C 1 for a.e.(x, t) Q(1/2).(H 5/3 (S) = 0) Prop. 2 (CKN theorem) There is an absolute constant ε 3. If (u,p) is Suitable weak solution and lim sup r 0 1 r Q(r) then (0,0) is a regular point. (H 1 (S) = 0) u 2 dxdt ε 3,

Blow-up method (natural method) Blow-up ( Fanghua Lin, Comm. Pure. Appl. Math. 1998) There are two positive constants ε 0 and C 0 such that u 3 + p 3/2 dxdt ε 0 Q(1) implies u(x, t) C α (Q (1/2) ) C 0 for some α > 0. Blow-up is originated from partial regularity of elliptic system and parabolic system, which relies on Campanato s Lemma on Hölder continuity. For more transparent proof with no zero force, See Ladyzenskaja and Seregin. (1999, J. Math. Fluid Mech) Near the boundary, Seregin (2002, J. Math. Fluid Mech) It seems that if u L 3+τ t L 3+τ x ( τ > 0), this method works. However, we only have u L 3 t L 3 x in R 4.

De Giorgi iteration I De Giorgi (Alexis Vasseur, NoDEA 2007) For every p > 1, there exists a universal constant ε, such that the solution u in [ 1, 1] B(1) verifying [ 0 ] 2 p sup u 2 dx + u 2 dxdt + ( p dx) p dt ε, t [ 1,0] B(1) 1 B(1) Q(1) is bounded by 1 on [ 1 2, 1] B( 1 2 ). To overcome the difficulty of applying De Giorgi iteration technique to the Navier-Stokes equations (system), Vasseur introduced a useful split on the velocity field u as follow ( u = u 1 v ) k + u v k u v, k 0, where v k = [ u (1 2 k )] +, and the sequences

De Giorgi iteration II U k := sup v k (x, t) 2 dx + d k (x, t) 2 dxdt, t [T k,0] B k Q k d 2 k = (1 2 k )ϕ 1,k u 2 + v k u u u 2, and B s = B ( 1 2 (1 + 2 3s ) ), fors R +, T k = 1 2 ( 1 2 k ), Q k = [T k, 0] B k. Vasseur exploited the classical inequality U k+1 C k U ζ k (ζ > 1), which stems form the local energy inequality.

Stationary Navier-Stokes equations According to the dimensional analysis of the NavierõStokes equations, u t u + u u + p = f, div u = 0, (x, t) R n (0, T ), u + u u + p =f, div u = 0, x R n+2. (3) For x R 4, the regularity of the weak solutions was established by Gerhardt (Math. Z. 1979) and Giaquinta, Modica ( J. Reine Angew. Math 1982). Is the weak solutions of (3) regular when n + 2 5?

Partial regularity to Stationary Navier-Stokes equations u + u u + p = f, x R n x R 5, H 1 (S) = 0. M. Struwe, (Comm. Pure. Appl. Math. 1988) Elliptic technology. Near the boundary, x R 5, H 1 (S) = 0. K. Kang, ( J. Math. Fluid Mech. 2004) Blow up method. x R 6, H 2 (S) = 0. H. Dong and R. Strain (Indiana 2012) Decay estimate; Bootstrap; Campanato s Lemma. The proof heavily relied on the local energy inequality blow div ( u u 2 ) + div(up) + u 2 u 2 2 2 fu.

4d Navier-Stokes equations Scheffer proved that P 3 (S) = 0. ( Comm. Math. Phys. 1978) H. Dong and D. Du ( Comm. Math. Phys. 2007) Let T be the first blow up time of the solution to the 4d N-S. There is a constant ε satisfying the following property. Assume that a point z 0 Ω T the inequality lim sup r 0 1 r 2 Q(z 0,r) u 2 dxdt ε. Then z 0 is a regular point. (H 2 (S space at time T ) = 0) The difficulty of the 4d N-S caused by the interpolation inequality L t L 2 x L 2 t W 1,2 x L 3 t L 3 x, x R 4. It is an open problem to show the existence of the suitable weak solutions to the 4d N-S. Naturally, we expect to obtain the partial regularity of suitable weak solutions to the 4d Navier-Stokes equations, which is analogous to the CKN theorem.

Common characteristic of these equation I The L loc bound of weak solutions guarantees regularity for these equations. It is well known that the Leray-Hopf weak solutions of time-dependent Navier-Stokes equations are regular if u L p t L q x with 2 p + n q 1. The L n loc bound of weak solutions of the stationary Navier-Stokes equations also gives regularity, which is proved by G. Galdi. Of course, with the L loc bound in hand, by the boostraping argument, higher regularity can be obtained immediately.

Common characteristic of these equation II Non-stationary case 0 u(0, x) 2 φdx + 2 u 2 φdxdt Ω } {{ T Ω } u L 3 t,x, x R n (n 4) 0 ) 0 u ( 2 t φ + φ dxdt + ( u 2 + 2 p )u φdxdt. T } Ω T {{ Ω } u L 3 t,x Stationary case u 2 φdx Ω } {{ } u L 3 x Ω u 2 φdx + Ω ( u 2 + 2 p )u φdx, x R n, n 6. } {{ } u L 3 x which means that left hand controls the right hand, which allows us to establish the inequality U k C k U 3/2 k 1 (Ideal)

Navier-Stokes equations in R n (n=2,3,4) Theorem 1.1 ( J. Diff. Equa. 2014) Suppose that the pair (u, p) is a suitable weak solution to (1), and for some q > n+2 (n = 2, 3, 4). There exists an absolute constant ε 2 01 > 0, 0 ( 3/2 u 2 dx+ u 2 dxdt+ p dx) dt+ f q dxdt ε 0 sup t [ 1,0] B(1) Q(1) 1 Then u can be controlled by 1 a.e. on [ 1 2, 0] B( 1 2 ). (H(n2 4)/n (S) = 0) Theorem 1.2 ( J. Diff. Equa. 2014) Suppose (u, p) is a suitable weak solution to (1) and for a universal constant ε 1 > 0, 1 lim sup u 2 dxdt < ε r 0 r n 2 1, n = 2, 3, 4. Q(r) B(1) Then (0, 0) is a regular point for u(x, t). (H n 2 (S) = 0) Q(1)

Stationary Navier-Stokes equations in R n (n=2,3,4,5,6) Theorem 2.1 ( J. Diff. Equa. 2014) Let the pair (u, p) be a suitable weak solution of (3). Then u can be bounded by 1 a.e. on B( 1 ) provided the following condition holds, 2 u 2 dx + u 2 dx + p dx + f q dx ε 02, B(1) B(1) B(1) for an absolute constant ε 02 > 0, and some q > n 2 Theorem 2.2 ( J. Diff. Equa. 2014) B(1) (n = 2, 3, 4, 5, 6). Suppose (u, p) is a suitable weak solution to (3). Then 0 is regular point for u(x) if the following conditions holds, 1 lim sup u 2 dx < ε r 0 r n 4 2, n = 2, 3, 4, 5, 6, B(r) for a universal constant ε 2 > 0. (H n 4 (S) = 0)

Remark We first obtain the partial regularity of the suitable weak solutions to the 4d non-stationary Navier-Stokes equations. (Independently and simultaneously, Dong and Gu obtain the similar result.) Here, we recover some classical results of these equations in low dimensions. The nonzero external force f is considered and treated in the same way. The assumption on the force f is weaker than others in high dimensions. In fact, these conditions on the external force are optimal under the definition of regular point here. A unified concise proof! Near the boundary, Dong and Gu showed the partial regularity theory of 4d non-stationary Navier-Stokes and 6d stationary Navier-Stokes. (See also Liu, Wang and Xin)

Main idea The strategy is to show that the local energy inequality yields the equality U k C k U γ k 1.(γ > 1) In genera, it is difficult to apply De Giorgi iteration to the system. To this end, we apply the decomposition utilized by Vasseur ( u = u 1 v ) k + u v k u u, (4) where v k = [ u (1 2 k )] +, k 0. We also denote U k := sup v k (x, t) 2 dx + d k (x, t) 2 dxdt, t [T k,0] B k Q k d 2 k = (1 2 k )ϕ 1,k u 2 + v k u u u 2, and B s = B ( 1 2 (1 + 2 3s ) ), fors R +, T k = 1 2 ( 1 2 k ), Q k = [T k, 0] B k.

De Giorgi iteration process I Thanks to both the function [ x (1 2 k ) ] 2 + g(x) = x 2, 2 and its derivative ( [ x (1 2 k ) ] ) g + (x) = 1 x, x are Lipschitz function, according to the local energy inequality and the equations, which allow us to apply the chain rule to show that v k satisfies the following inequality: t ( v k 2 2 ) + div(u v ( ) k 2 2 ) + div(up)+u vk u 1 p in the sense of distribution. v k 2 2 + d 2 k fu v k u, (5)

De Giorgi iteration process II Choosing the proper test function in the above inequality, we deduce that U k C2 6k v k 2 dz + C 2 3k v k 3 dz + η k (x) f uv k u dz + η k (x) [ ( ) vk div(up) + u 1 u p ] dz. C 2 6k+k U 3/2 k 1 + C2(2β 1)k f L qu β k 1 (Tchebichev s inequality) + η k (x) [ ( ) vk div(up) + u 1 u p ] dz. C k U 3/2 k 1 + C k U β k 1 + C k U γ k 1, (γ > 1). Remark: The non stationary case and stationary case are almost the same.

Localize the pressure Part 1 To conclude, we need to deal with the pressure terms. n p = i j (u i u j ). Indeed, yields that i=1 i i (pφ k ) = φ k i j (u j u i ) + 2 i φ k i p + p i i φ k. p(x) := P 1,k (x) + P 2,k (x) + P 3,k (x), x B k 1/3, where P 1,k (x) = i j Γ (φ k (u j u i )), P 2,k (x) = 2 i Γ ( j φ k (u j u i )) Γ ( i j φ k u j u i ), P 3,k (x) = 2 i Γ ( i φ k p) Γ ( i i φ k p). Moreover, P 2,k L 3/2 (T k 1,0;L (B k 1/3 )) 2 3k(n+1) C(U 0 ), x R n, n = 2, 3, 4; P 3,k L 3/2 (T k 1,0;L (B k 1/3 )) 2 3k(n+1) C p L 3/2 (T k 1,0;L 1 (B k 1 )), x R n, n = 2, 3,

Localize the pressure Part 2 We decompose P 1,k into three parts P 1,k (x) = i j Γ (φ k (u j u i )) ( ) v k = i j Γ φ k u i u u v k j u ( v k 2 i j Γ φ k u i u u j ( i j Γ [φ k u i 1 v k u ( 1 v k u )) ) u j ( 1 v k u :=P 13,k (x, t) + P 12,k (x, t) + P 11,k (x, t). We need to deal with P 13,k [ u i v k u j v k P 13,k (x) = i j Γ φ k u u := P 131,k + P 132,k + P 133,k. )] ] u i v k + 2φ k u (u jv k u )

The Calderón-Zygmund, the Tchebichev inequality, and Hölder inequality can yield U k C( p 3/2 L, U t L 1 0, f q ) 2 νk (U 3/2 k 1 + U 7/6 k 1 + U 5/4 k 1 + U 11/10 k 1 + U β k 1 ). x Recalling U 0 = sup B(1) u 2 dx + Q(1) u 2 dxdt and U k as k, we t [ 1,0] find that 0 U k+1 C k U γ k C k+(k 1)γ U γ2 k 1 C k+(k 1)γ+(k 2)γ2 + +γ k 1 U γk+1 0 C γ k+1 (γ 1) 2 U γk+1 0 = (C 1 (γ 1) 2 U 0 ) γk+1. Clearly, if C 1 (γ 1) 2 U 0 < 1 then U k 0 as k.

Fractional Navier-Stokes equations In 1960s, J. Lions introduced the following equations with fractional dissipation { ut + u u + νλ 2α u + p = 0, div u = 0, u 0 = u 0 (x). Λ 2α f (ξ) = ξ 2αˆf (ξ), where ˆf (ξ) = 1 (2π) n R n f (x)e iξ x dx in R n. Λ 2α f (k) = k 2αˆf (k), where ˆf (k) = 1 (2π) n T n f (x)e ik x dx, k Z n on the torus. Note that if u(x, t) is a solution to (6), then (6) u λ := λ 2α 1 u(λx, λ 2α t) (7) for any λ > 0 also solves (6). The corresponding energy is E(u λ ) = ess sup u λ 2 dx + R 3 Λ α u λ 2 dxdt R 3 = λ 4α 5 ess sup u 2 dx + R 3 Λ α u 2 dxdt. R 3

Previous related result Lions showed the global regular solution to the hyper-dissipative Navier-Stokes equations with α n+2 4. ( 1969) N.H. Katz and N. Pavlović proved that if T is the time of first breakdown for generalized Navier-Stokes equations with 5/4 > α > 1 then the Hausdorff dimension of the singular set at time T is at most 5 4α. ( Geom. Funct. Anal. 2002) The existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces were established by Wu. (Dynamics of PDE, 2004 ; Commun. Math. Phys. 2006 ) Tao obtained the global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equations. (Analysis & PDE 2009)

Hausdorff dimensional on time singular points 2014 For 5/6 α < 5/4, there exists a constant T > 0 such that every weak solution to (6) is a strong solution on (T, ). (α < 5/6? Open) For 5/6 α < 5/4, the (5 4α)/2α dimensional Hausdorff measure of the possible time singular points of the weak solutions of (6) on (0, ) is zero, namely, H 5 4α 2α (IR) = 0. This result not only is an interpolation between the Scheffer s (Leray s) Hausdorff dimension of the possible time singular set of the weak solutions to the Navier-Stokes equations (α = 1) and Lions s global solvability for the hyper-dissipative case (α 5/4), but also further generalizes Scheffer s (Leray s) classic work in the sense that 5/6 α < 1. It is worth noting that when the dissipation index α varies from 5/6 to 5/4, the corresponding Hausdorff dimension is from 1 to 0. So, it seems that this result is optimal.

The case 1 α < 5/4 may be expected. When α < 1, we have to consider the BKM criteria. It is not obvious in the regime α < 1. In his seminal paper, Leray proved the global small solution in the sense u(0) 1/2 L u(0) 1/2 2 L being sufficiently small and the local solvability in H 1. 2 In modern view, it is worth mentioning that u(0) Ḣ1/2 C u(0) 1 2 L 2 u(0) 1 2 L 2 (8) is invariant under the scaling (7). A key observation is an analogous inequality of (8) u Ḣ 5 4α 2 C u 6α 5 2α L 2 Λ α u 5 4α 2α L, 5/6 α 5/4. 2 In fact, H α is the critical or subcritical space to this system when α 5/6. In other words, adopting bootstrap argument, one could improve the regularity of the solution constructed in H α with α > 5/6 has higher regularity provided the initial data is more regular.

The key point is to exploit some new estimate 1 d 2 dt Λα u(t) 2 L 2 (R 3 ) Cν 2α 5 6α 5 Λ α u(t) 2(8α 5) 6α 5 L 2 (R, 3 ) which gives the local well-posedness of H α (R 3 )(5/6 α 5/4) and global well-posedness of H 5/4 (R 3 ). What s more, we could deduce that the blow-up rate Λ α u(t) L 2 5 2α Cν 4α (t 0 t) 6α 5 4α, t < t 0, 5/6 < α < 5/4, for t 0 to be a possible irregular including interval [0, T ). 1 d 2 dt Λα u(t) 2 L 2 (R 3 ) + (ν C u(t) 6α 5 2α L 2 (R 3 ) Λα u(t) 5 4α 2α ) Λ 2α u(t) 2 L 2 (R 3 ) L 2 (R 3 ) 0, which yields the global well-posedness of H α (R 3 )(5/6 < α < 5/4) for small initial data and the eventually regularity of the weak solutions.

I would like to express my sincere gratitude to Professor Quansen Jiu and Changxing Miao for many helpful discussions and constant support.

3d N-S on time H 1 = 0 3d N-S on time H 1/2 = 0 3d N-S on space-time P 5/3 = 0 3d N-S on space-time P 1 = 0 5d SNS H 1 = 0 4d N-S on space-time P 3 = 0 3d Blow-up 3d De Giorgi 3d GNS at first blow-up H 5 4α = 0 4d N-S at first blow-up on space H 2 = 0 6d SNS H 2 = 0 A unified proof of NS & SNS 3d GNS on time H (5 4α)/2α = 0 Leray Struwe Caffarelli, Kohn and Nirenberg Katz and Pavlović Dong and Strain Scheffer Lin (Ladyzenskaja and Seregin) Du and Dong Vasseur Our result Some results on partial regularity to Navier-Stokes equations