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Logarithmic improvement of regularity criteria for the Navier–Stokes equations in terms of pressure
Applied Mathematics Letters 58 (2016) 62–68
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Logarithmic improvement of regularity criteria for the Navier–Stokes equations in terms of pressure Chuong V. Tran a , Xinwei Yu b,∗ a
School of Mathematics and Statistics, University of St. Andrews, St Andrews KY16 9SS, United Kingdom Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada b
article
abstract
info
Article history: Received 15 December 2015 Received in revised form 9 February 2016 Accepted 9 February 2016 Available online 17 February 2016
In this article we prove a logarithmic improvement of regularity criteria in the multiplier spaces for the Cauchy problem of the incompressible Navier–Stokes equations in terms of pressure. This improves the main result in Benbernou (2009). © 2016 Elsevier Ltd. All rights reserved.
Keywords: Navier–Stokes Global regularity Pressure Prodi–Serrin Multiplier spaces Logarithmic improvement
1. Introduction At the center stage of mathematical fluid mechanics are the incompressible Navier–Stokes equations ut + u · ∇u = −∇p + ν△u, div u = 0,
(x, t) ∈ Ω × (0, ∞)
(x, t) ∈ Ω × (0, ∞)
u(x, 0) = u0 (x),
x∈Ω
(1) (2) (3)
with appropriate boundary conditions. Here Ω ⊆ Rd is a domain with certain regularity, u : Ω → Rd is the velocity field, p : Ω → R is the pressure, and ν > 0 is the (dimensionless) viscosity. The system (1)–(3) on one hand describes the motion of viscous Newtonian fluids, while on the other hand serves as the starting point of mathematical modeling of many other types of fluids, such as non-Newtonian fluids, magnetic fluids, electric fluids, and ferro-fluids. In this article we focus on the Cauchy problem of (1)–(3), where Ω = Rd . ∗ Corresponding author. E-mail addresses: [email protected] (C.V. Tran), [email protected] (X. Yu).
http://dx.doi.org/10.1016/j.aml.2016.02.003 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
C.V. Tran, X. Yu / Applied Mathematics Letters 58 (2016) 62–68
63
As (1)–(3) serve as the foundation of the modern quantitative theory of incompressible fluids, it is important to have complete mathematical understanding of these equations. However the achievement of this goal is still out of the question. In particular, there is still no satisfactory answer to the question of well-posedness of the Cauchy problem of (1)–(3). The first systematic study of this well-posedness problem (for the case d = 3) was carried out by Jean Leray in [1], where it is shown that for arbitrary T ∈ (0, ∞] there is at least one function u(x, t) satisfying the following: i. ii. iii. iv.
u ∈ L∞ (0, T ; L2 (Rd )) ∩ L2 (0, T ; H 1 (Rd )); u satisfies (1) and (2) in the sense of distributions; u takes the initial value in the L2 sense: limt↘0 ∥u(·, t) − u0 (·)∥L2 = 0; u satisfies the energy inequality ∥u(·, t)∥2L2 + 2ν
t
∥∇u(·, τ )∥2L2 dτ 6 ∥u0 ∥2L2
(4)
0
for all 0 6 t 6 T . Such a function u(x, t) is called a Leray–Hopf weak solution for (1)–(3) in Rd × [0, T ). It is easy to show that if a Leray-Hopf weak solution is smooth, then it satisfies (1)–(3) in the classical sense. One could further show that such a smooth Leray-Hopf solution must be unique. Therefore the wellposedness problem would be settled if all Leray-Hopf solutions could be shown to be smooth. However such a general result has not been established up to now. On the other hand, various additional assumptions guaranteeing the smoothness of Leray-Hopf solutions have been discovered. For example, it has been shown that if a Leray-Hopf solution u(x, t) further satisfies u ∈ Lr (0, T ; Ls (Rd ))
with
2 d + 6 1, d < s 6 ∞, r s
(5)
then u(x, t) is smooth and is thus a classical solution, see e.g. [2–4]. The borderline case u ∈ L∞ (0, T ; L3 ) is much more complicated and requires a totally different approach. It was settled much later by Escauriaza, Seregin, and Sverak in [5]. Many generalizations and refinements of (5) have been proved, see e.g. [6–11]. If we formally take the divergence of (1) we obtain the following relation between u and p: − △p = div(div(u ⊗ u))
(6)
where u ⊗ u is a d × d matrix with i-j entry ui uj . Thus intuitively we have p ∼ u2 . Transforming (5) via this relation, we expect that p ∈ Lr (0, T ; Ls (Rd ))
with
2 d d + 6 2,
(7)
should guarantee the smoothness of u. This is indeed the case and was confirmed in [12,13]. Many efforts have been made to refine (7), see e.g. [14–21]. It is worth mentioning that the relation (6) has also played crucial roles in the proofs of other regularity criteria not of the Prodi-Serrin type. For example, 2 in [22] it is used to show that Leray-Hopf weak solutions are regular as long as either | u | +2p is bounded above or p is bounded below. Among generalizations of (7), it is shown in [14] that u is smooth as long as p ∈ L2/(2−r) (0, T ; X˙ r (Rd )d ) for 0 < r 6 1 where X˙ r (Rd ) is the multiplier space. Multiplier spaces are defined for 0 6 r < d/2 and functions f ∈ L2loc (Rd ) through the norm ∥f ∥X˙ r :=
sup ∥g∥H˙ r 61
∥f g∥L2 < ∞,
(8)
C.V. Tran, X. Yu / Applied Mathematics Letters 58 (2016) 62–68
64
where H˙ r (Rd ) is the completion of the space D(Rd ) with respect to the norm ∥u∥H˙ r = ∥(−△)r/2 u∥L2 , see e.g. [23] for properties of such spaces. Among its properties the following are most relevant in the present context. • Ld/r ⊂ X˙ r for 0 6 r < d/2, • This inclusion is strict. For example by the Hardy inequality for fractional Laplacians (see e.g. [24,25]) −r we have | x | ∈ X˙ r (Rd ). Thus the criterion p ∈ L2/(2−r) (0, T ; X˙ r (Rd )d ) refines (7). In this article we will present the following logarithmic improvement of this criterion. Theorem 1. Let u0 ∈ L2 (Rd ) ∩ Lq (Rd ) for some q > d, and ∇ · u0 = 0. Let u(t, x) be a Leray–Hopf solution of NSE in [0, T ). If the pressure p satisfies
2/(2−r)
∥p∥X˙
T
r
log(e + ∥p∥W m,∞ )
0
dt < ∞
(9)
for some m ∈ N ∪ {0} and r ∈ (0, 1], then u(t, x) is smooth up to T and could be extended beyond T . 2. Proof of theorem Without loss of generality, we take ν = 1 in (1) to simplify the presentation. We apply the following result from [26,27] to guarantee short-time smoothness of the solution and thus relieving us from worrying about the legitimacy of the various integral and differential manipulations below. Theorem 2. Let u0 ∈ Ls (Rd ), s > d. Then there exist T > 0 and a unique classical solution u ∈ BC(0, T ; Ls (Rd )). Moreover, let (0, T∗ ) be the maximal interval such that the solution u stays in C(0; T∗ ; Ls (Rd )), s > d. Then for any t ∈ (0, T∗ ), ∥u(·, t)∥Ls >
C (T∗ − t)
(10)
s−3 2s
where the constant C is independent of T∗ and s. We also recall that (6) implies p=
d
Ri Rj (ui uj )
(11)
i,j=1
where Rj , j = 1, . . . , d are the Riesz transforms. As a consequence of the standard theory of singular integrals, the following holds: For any s ∈ (1, ∞) and m ∈ N, α ∈ (0, 1), ∥p∥Ls 6 C∥u∥2L2s ,
∥p∥C m,α 6 C
max
i,j=1,2,...,d
∥ui uj ∥C m,α
(12)
where the constant C depends on s, m, α but not on p or u. Proof of Theorem 1. Assume the contrary. Let T ∗ 6 T be the first “blow-up” time. By Theorem 2 we must have lim supt↗T ∗ ∥u(·, t)∥Ls = ∞ for all s > d. In the following we will prove in two steps that under such assumption ∥u(·, t)∥H k stays bounded up to T ∗ for some k > d2 +m, thus reaching contradiction as H k ↩→ Ls . Note that again by Theorem 2 we can assume u to be smooth in (0, T ∗ ) and freely manipulate all functions in integration and differentiation.
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65
(1) Ls estimate. Pick any s > max{4, d}. We multiply (1) by |u|s−2 u· and integrate in Rd to obtain ∥u∥s−1 Ls
d ∥u∥Ls = − dt =
|u|
s−2
|u|s−2 u · △udx
u · ∇pdx +
Rd
Rd
pu · ∇(|u|s−2 )dx + |u|s−2 u · △udx Rd s−2 = (s − 2) p|u| (ˆ u · ∇|u|)dx + |u|s−2 u · △udx Rd
Rd
(13)
Rd
u (if u = 0, just define u ˆ = 0 too). where u ˆ := |u| Recalling the identity
u · △u = ∇ · (|u|∇|u|) − |∇u|2 ,
(14)
we easily derive
|u|s−2 u · △udx = −
Rd
4(s − 2) ∥∇|u|s/2 ∥2L2 − ∥|∇u||u|s/2−1 ∥2L2 , s2
(15)
and reach the following estimate d ∥u∥sLs + ∥|∇u||u|s/2−1 ∥2L2 + ∥∇|u|s/2 ∥2L2 . dt
p|u|s−2 (ˆ u · ∇|u|)dx.
(16)
Rd
From here on we will use A . B to denote A 6 cB for some constant c whose value does not depend on u or p. Since s−2 s−2 s−2 6 2 2 ∇|u| dx, |p||u| p|u| (ˆ u · ∇|u|)dx (17) |u| d d R
R
application of Young’s inequality turns (16) into d ∥u∥sLs + ∥|∇u||u|s/2−1 ∥2L2 + ∥∇|u|s/2 ∥2L2 . dt
|p|2 |u|s−2 dx.
(18)
Rd
Now let w := |u|s/2 . From (18) it follows that d 4 + s2 ∥u∥sLs + ∥∇w∥2L2 . dt s2
|p|2 |w|2(1−2/s) dx.
(19)
Rd
We have Rd
|p|2 |w|2(1−2/s) dx 6 ∥pw∥L2 ∥p∥Ls/2 ∥w1−4/s ∥L2s/(s−4) 1−4/s
6 ∥p∥X˙ r ∥w∥H˙ r ∥u∥2Ls ∥w∥L2 = ∥p∥X˙ r ∥w∥H˙ r ∥w∥L2 (2−r)
6 ∥p∥X˙ r ∥w∥L2 2/(2−r)
6 C∥p∥X˙
r
∥w∥rH˙ 1
1 ∥w∥2L2 + ∥w∥2H˙ 1 2
where we have applied H¨ older’s inequality and the definition of X˙ r norm (8).
(20)
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From (20) we conclude 2/(2−r)
2 ∥p∥X˙ d 2−r r ∥u∥Ls . ∥p∥X log(e + ∥p∥W m,∞ )∥u∥Ls . ˙ r ∥u∥Ls = dt log(e + ∥p∥W m,∞ )
(21)
Now let k > d2 + m be arbitrary. By Sobolev embedding theorems we have ∥u∥C m,α . ∥u∥H k for some α > 0. Application of (12) now gives ∥p∥W m,∞ 6 ∥p∥C m,α .
max
i,j=1,2,...,d
∥ui uj ∥C m,α . ∥u∥2C m,α . ∥u∥2H k .
(22)
Consequently (21) yields 2/(2−r)
∥p∥X˙ d r ∥u∥Ls . log(e + ∥u∥H k )∥u∥Ls . dt log(e + ∥p∥W m,∞ )
(23)
Let ϵ > 0 be small and T ∗ − ϵ < t < T ∗ . Integrating (23) from T ∗ − ϵ to t we see that ∥u∥Ls (t) 6 ∥u0 ∥Ls exp
T ∗ −ϵ
2/(2−r)
T∗
∥p∥X˙
r
log(e + ∥p∥W m,∞ )
dt max log(e + ∥u∥H k ) .
(24)
[0,T ]
Thanks to the integrability assumption (9), for any δ > 0, we can take ϵ > 0 small enough to have ∥u(t)∥Ls 6 C(ϵ)(e +
max
t′ ∈[T ∗ −ϵ,T1 )
∥u(t′ )∥H k )δ
(25)
for all T1 ∈ (T ∗ − ϵ, T ∗ ) and t ∈ [T ∗ − ϵ, T1 ). Note that by our assumption C(ϵ) −→ ∞ as δ ↘ 0. In the following we will see that it is possible to take a fixed positive value of δ and thus exclude this possibility. 2d (2) H k estimate. Let 2 < q < d−2 and p > s be such that 1q + p1 = 12 . Let k > d2 + m be a large natural number to be specified later. Let Λ := (−△)1/2 . We multiply (1) by Λ2k u and integrate: d 1 2 2 2k ∥u∥H˙ k + ∥u∥H˙ k+1 . u · ∇u · Λ udx dt 2 d R k−1 k+1 = Λ ∇ · (u ⊗ u) · (Λ u)dx d R
. ∥u∥Lp ∥Λk u∥Lq ∥Λk+1 u∥L2
(26)
Here we have used the product estimate ∥Λs (f g)∥Lp 6 ∥f ∥Lp1 ∥g∥H s,p2 + ∥f ∥H s,p3 ∥g∥Lp4 ,
(27)
which holds for p, p2 , p3 ∈ (1, +∞) and 1/p = 1/p1 + 1/p2 = 1/p3 + 1/p4 , from Lemma 2.1 of [28]. Now we interpolate ∥u∥Lp . ∥u∥θLs ∥Λk+1 u∥1−θ L2 , k
∥Λ u∥Lq .
∥u∥µLs ∥Λk+1 u∥1−µ L2
(28) (29)
C.V. Tran, X. Yu / Applied Mathematics Letters 58 (2016) 62–68
67
1−d/2+d/q 1 where θ := k+1−d/2+d/p k+1−d/2+d/s , µ := k+1−d/2+d/s . Note that since q > 2, µ < k+1 when k is large enough. From now on we fix k > d2 + m to be such a large natural number. Substituting (28) and (29) into (26) yields
d 1 k+1 . ∥u∥2H˙ k + ∥u∥2H˙ k+1 . ∥u∥θ+µ u∥3−θ−µ Ls ∥Λ L2 dt 2
(30)
As s > d we have θ + µ > 1. Application of Young’s inequality now gives d ∥u∥2H˙ k . ∥u∥γLs dt
(31)
for some positive number γ. Now we fix ϵ0 > 0 such that (25) holds for δ = 1/γ and ϵ = ϵ0 . Then we have d ∥u(t)∥2H˙ k 6 C(ϵ0 )(e + ′ max ∥u(t′ )∥H˙ k ) dt t ∈[T ∗ −ϵ0 ,T1 )
(32)
for all t ∈ [T ∗ − ϵ0 , T1 ). Integrating from T ∗ − ϵ0 to t ∈ (T ∗ − ϵ0 , T1 ), we arrive (e + ∥u(t)∥H˙ k )2 6 C1 (ϵ0 ) + C(ϵ0 )ϵ0 (e +
max
t′ ∈[T ∗ −ϵ0 ,T1 )
∥u(t′ )∥H˙ k )
(33)
and consequently (e +
max
t′ ∈[T ∗ −ϵ0 ,T1 )
∥u(t′ )∥H˙ k ) 6 C2 (ϵ0 ) < ∞.
(34)
As C2 (ϵ0 ) is independent of T1 , setting T1 ↗ T ∗ we have max
t′ ∈[T ∗ −ϵ0 ,T ∗ )
∥u(t′ )∥H˙ k < ∞
(35)
which contradicts the assumption that T ∗ is a blow-up time. Remark 1. It is clear that we can replace ∥p∥W m,∞ by ∥u∥W m,∞ . Acknowledgment The authors would like to thank the anonymous referees for the valuable comments and suggestions that greatly improved the quality of the article. The authors are also grateful to Prof. Jishan Fan for pointing out an error in the earlier version of the paper. Part of this research was carried out when CVT was visiting the University of Alberta, whose hospitality is gratefully acknowledged. XY is partially supported by NSERC Discovery grant RES0020476. References [1] Jean Leray, On the motion of a viscous liquid filling space, Acta Math. 63 (1934) 193–248. [2] E.B. Fabes, B.F. Jones, N.M. Riviere, The initial value problem for the Navier–Stokes equations with data in Lp , Arch. Ration. Mech. Anal. 45 (1972) 222–248.
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[3] T. Ohyama, Interior regularity of weak solutions to the Navier–Stokes equation, Proc. Japan Acad. 36 (1960) 273–277. [4] Jame Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 9 (1962) 187–191. [5] L. Escauriaza, G. Seregin, V. Sverak, L3,∞ -solutions of Navier–Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 (2) (2003) 3–44; Russian Math. Surveys 58 (2) (2003) 211–250. [6] Stefano Bosia, Vittorino Pata, James C. Robinson, A weak-Lp Prodi-Serrin type regularity criterion for the Navier–Stokes equations, J. Math. Fluid Mech. 16 (2014). [7] Chi Hin Chan, Alexis Vasseur, Log improvement of the Prodi-Serrin criteria for Navier–Stokes equations, Methods Appl. Anal. 14 (2) (2007) 197–212. [8] Jishan Fan, Song Jiang, Gen Nakamura, Yong Zhou, Logarithmically improved regularity criteria for the Navier–Stokes and MHD equations, J. Math. Fluid Mech. 13 (4) (2011) 557–571. [9] Yi Zhou, Zhen Lei, Logarithmically improved criteria for Euler and Navier–Stokes equations, Commun. Pure Appl. Anal. 12 (6) (2013) 2715–2719. [10] Yong Zhou, A new regularity criterion for the Navier–Stokes equations in terms of the direction of vorticity, Monatsh. Math. 144 (3) (2005) 251–257. [11] Yong Zhou, Sadek Gala, Logarithmically improved regularity criteria for the Navier–Stokes equations in multiplier spaces, J. Math. Anal. Appl. 356 (2) (2009) 498–501. [12] Luigi C. Berselli, Giovanni P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations, Proc. Amer. Math. Soc. 130 (2002) 3585–3595. [13] Dongho Chae, Jihoon Lee, Regularity criterion in terms of pressure for the Navier–Stokes equations, Nonlinear Anal. TMA 46 (5) (2001) 727–735. [14] Samia Benbernou, A note on the regularity criterion in terms of pressure for the Navier–Stokes equations, Appl. Math. Lett. 22 (9) (2009) 1438–1443. [15] Qionglei Chen, Zhifei Zhang, Regularity criterion via the pressure on weak solutions to the 3D Navier–Stokes equations, Proc. Amer. Math. Soc. 135 (2007) 1829–1837. [16] Jishan Fan, Song Jiang, Guoxi Ni, On regularity criteria for the n-dimensional Navier–Stokes equations in terms of the pressure, J. Differential Equations 244 (11) (2008) 2963–2979. [17] Zhengguang Guo, Sadek Gala, Remarks on logarithmical regularity criteria for the Navier–Stokes equations, J. Math. Phys. 52 (6) (2011) 63503. [18] Zhengguang Guo, Sadek Gala, A note on the regularity criteria for the Navier–Stokes equations, Appl. Math. Lett. 25 (3) (2012) 305–309. [19] Xiaowei He, Sadek Gala, Regularity criterion for weak solutions to the Navier–Stokes equations in terms of the pressure −1 (R3 ))}, Nonlinear Anal. RWA 12 (6) (2011) 3602–3607. in the class L2 (0, T ; B˙ ∞,∞ [20] Michael Struwe, On a Serrin-type regularity criterion for the Navier–Stokes equations in terms of the pressure, J. Math. Fluid Mech. 9 (2) (2007) 235–242. [21] Xingwei Zhang, Yan Jia, Bo-Qing Dong, On the pressure regularity criterion of the 3D Navier–Stokes equations, J. Math. Anal. Appl. 393 (2) (2012) 413–420. ˇ ak, Navier–Stokes equations with lower bounds on the pressure, Arch. Ration. Mech. Anal. 163 (1) [22] G. Seregin, V. Sver´ (2002) 65–86. [23] Sadek Gala, Multiplier spaces, Muckenhoupt weights and pseudo-differential operators, J. Math. Anal. Appl. 324 (2006) 1262–1273. [24] Rupert L. Frank, Robert Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (12) (2008) 3407–3430. [25] Hichem Hajaiej, Xinwei Yu, Zhichun Zhai, Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms, J. Math. Anal. Appl. 396 (2) (2012) 569–577. [26] Yoshikazu Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier–Stokes system, J. Differential Equations 62 (2) (1986) 186–212. [27] Tosio Kato, Strong Lp -solutions of the Navier–Stokes equation in Rm , with applications to weak solutions, Math. Z. 187 (1984) 471–480. [28] Ning Ju, Geometric constraints for global regularity of 2D quasi-geostrophic flows, J. Differential Equations 226 (2006) 54–79.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Logarithmic improvement of regularity criteria for the Navier–Stokes equations in terms of pressure Chuong V. Tran a , Xinwei Yu b,∗ a
School of Mathematics and Statistics, University of St. Andrews, St Andrews KY16 9SS, United Kingdom Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada b
article
abstract
info
Article history: Received 15 December 2015 Received in revised form 9 February 2016 Accepted 9 February 2016 Available online 17 February 2016
In this article we prove a logarithmic improvement of regularity criteria in the multiplier spaces for the Cauchy problem of the incompressible Navier–Stokes equations in terms of pressure. This improves the main result in Benbernou (2009). © 2016 Elsevier Ltd. All rights reserved.
Keywords: Navier–Stokes Global regularity Pressure Prodi–Serrin Multiplier spaces Logarithmic improvement
1. Introduction At the center stage of mathematical fluid mechanics are the incompressible Navier–Stokes equations ut + u · ∇u = −∇p + ν△u, div u = 0,
(x, t) ∈ Ω × (0, ∞)
(x, t) ∈ Ω × (0, ∞)
u(x, 0) = u0 (x),
x∈Ω
(1) (2) (3)
with appropriate boundary conditions. Here Ω ⊆ Rd is a domain with certain regularity, u : Ω → Rd is the velocity field, p : Ω → R is the pressure, and ν > 0 is the (dimensionless) viscosity. The system (1)–(3) on one hand describes the motion of viscous Newtonian fluids, while on the other hand serves as the starting point of mathematical modeling of many other types of fluids, such as non-Newtonian fluids, magnetic fluids, electric fluids, and ferro-fluids. In this article we focus on the Cauchy problem of (1)–(3), where Ω = Rd . ∗ Corresponding author. E-mail addresses: [email protected] (C.V. Tran), [email protected] (X. Yu).
http://dx.doi.org/10.1016/j.aml.2016.02.003 0893-9659/© 2016 Elsevier Ltd. All rights reserved.
C.V. Tran, X. Yu / Applied Mathematics Letters 58 (2016) 62–68
63
As (1)–(3) serve as the foundation of the modern quantitative theory of incompressible fluids, it is important to have complete mathematical understanding of these equations. However the achievement of this goal is still out of the question. In particular, there is still no satisfactory answer to the question of well-posedness of the Cauchy problem of (1)–(3). The first systematic study of this well-posedness problem (for the case d = 3) was carried out by Jean Leray in [1], where it is shown that for arbitrary T ∈ (0, ∞] there is at least one function u(x, t) satisfying the following: i. ii. iii. iv.
u ∈ L∞ (0, T ; L2 (Rd )) ∩ L2 (0, T ; H 1 (Rd )); u satisfies (1) and (2) in the sense of distributions; u takes the initial value in the L2 sense: limt↘0 ∥u(·, t) − u0 (·)∥L2 = 0; u satisfies the energy inequality ∥u(·, t)∥2L2 + 2ν
t
∥∇u(·, τ )∥2L2 dτ 6 ∥u0 ∥2L2
(4)
0
for all 0 6 t 6 T . Such a function u(x, t) is called a Leray–Hopf weak solution for (1)–(3) in Rd × [0, T ). It is easy to show that if a Leray-Hopf weak solution is smooth, then it satisfies (1)–(3) in the classical sense. One could further show that such a smooth Leray-Hopf solution must be unique. Therefore the wellposedness problem would be settled if all Leray-Hopf solutions could be shown to be smooth. However such a general result has not been established up to now. On the other hand, various additional assumptions guaranteeing the smoothness of Leray-Hopf solutions have been discovered. For example, it has been shown that if a Leray-Hopf solution u(x, t) further satisfies u ∈ Lr (0, T ; Ls (Rd ))
with
2 d + 6 1, d < s 6 ∞, r s
(5)
then u(x, t) is smooth and is thus a classical solution, see e.g. [2–4]. The borderline case u ∈ L∞ (0, T ; L3 ) is much more complicated and requires a totally different approach. It was settled much later by Escauriaza, Seregin, and Sverak in [5]. Many generalizations and refinements of (5) have been proved, see e.g. [6–11]. If we formally take the divergence of (1) we obtain the following relation between u and p: − △p = div(div(u ⊗ u))
(6)
where u ⊗ u is a d × d matrix with i-j entry ui uj . Thus intuitively we have p ∼ u2 . Transforming (5) via this relation, we expect that p ∈ Lr (0, T ; Ls (Rd ))
with
2 d d + 6 2,
(7)
should guarantee the smoothness of u. This is indeed the case and was confirmed in [12,13]. Many efforts have been made to refine (7), see e.g. [14–21]. It is worth mentioning that the relation (6) has also played crucial roles in the proofs of other regularity criteria not of the Prodi-Serrin type. For example, 2 in [22] it is used to show that Leray-Hopf weak solutions are regular as long as either | u | +2p is bounded above or p is bounded below. Among generalizations of (7), it is shown in [14] that u is smooth as long as p ∈ L2/(2−r) (0, T ; X˙ r (Rd )d ) for 0 < r 6 1 where X˙ r (Rd ) is the multiplier space. Multiplier spaces are defined for 0 6 r < d/2 and functions f ∈ L2loc (Rd ) through the norm ∥f ∥X˙ r :=
sup ∥g∥H˙ r 61
∥f g∥L2 < ∞,
(8)
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where H˙ r (Rd ) is the completion of the space D(Rd ) with respect to the norm ∥u∥H˙ r = ∥(−△)r/2 u∥L2 , see e.g. [23] for properties of such spaces. Among its properties the following are most relevant in the present context. • Ld/r ⊂ X˙ r for 0 6 r < d/2, • This inclusion is strict. For example by the Hardy inequality for fractional Laplacians (see e.g. [24,25]) −r we have | x | ∈ X˙ r (Rd ). Thus the criterion p ∈ L2/(2−r) (0, T ; X˙ r (Rd )d ) refines (7). In this article we will present the following logarithmic improvement of this criterion. Theorem 1. Let u0 ∈ L2 (Rd ) ∩ Lq (Rd ) for some q > d, and ∇ · u0 = 0. Let u(t, x) be a Leray–Hopf solution of NSE in [0, T ). If the pressure p satisfies
2/(2−r)
∥p∥X˙
T
r
log(e + ∥p∥W m,∞ )
0
dt < ∞
(9)
for some m ∈ N ∪ {0} and r ∈ (0, 1], then u(t, x) is smooth up to T and could be extended beyond T . 2. Proof of theorem Without loss of generality, we take ν = 1 in (1) to simplify the presentation. We apply the following result from [26,27] to guarantee short-time smoothness of the solution and thus relieving us from worrying about the legitimacy of the various integral and differential manipulations below. Theorem 2. Let u0 ∈ Ls (Rd ), s > d. Then there exist T > 0 and a unique classical solution u ∈ BC(0, T ; Ls (Rd )). Moreover, let (0, T∗ ) be the maximal interval such that the solution u stays in C(0; T∗ ; Ls (Rd )), s > d. Then for any t ∈ (0, T∗ ), ∥u(·, t)∥Ls >
C (T∗ − t)
(10)
s−3 2s
where the constant C is independent of T∗ and s. We also recall that (6) implies p=
d
Ri Rj (ui uj )
(11)
i,j=1
where Rj , j = 1, . . . , d are the Riesz transforms. As a consequence of the standard theory of singular integrals, the following holds: For any s ∈ (1, ∞) and m ∈ N, α ∈ (0, 1), ∥p∥Ls 6 C∥u∥2L2s ,
∥p∥C m,α 6 C
max
i,j=1,2,...,d
∥ui uj ∥C m,α
(12)
where the constant C depends on s, m, α but not on p or u. Proof of Theorem 1. Assume the contrary. Let T ∗ 6 T be the first “blow-up” time. By Theorem 2 we must have lim supt↗T ∗ ∥u(·, t)∥Ls = ∞ for all s > d. In the following we will prove in two steps that under such assumption ∥u(·, t)∥H k stays bounded up to T ∗ for some k > d2 +m, thus reaching contradiction as H k ↩→ Ls . Note that again by Theorem 2 we can assume u to be smooth in (0, T ∗ ) and freely manipulate all functions in integration and differentiation.
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(1) Ls estimate. Pick any s > max{4, d}. We multiply (1) by |u|s−2 u· and integrate in Rd to obtain ∥u∥s−1 Ls
d ∥u∥Ls = − dt =
|u|
s−2
|u|s−2 u · △udx
u · ∇pdx +
Rd
Rd
pu · ∇(|u|s−2 )dx + |u|s−2 u · △udx Rd s−2 = (s − 2) p|u| (ˆ u · ∇|u|)dx + |u|s−2 u · △udx Rd
Rd
(13)
Rd
u (if u = 0, just define u ˆ = 0 too). where u ˆ := |u| Recalling the identity
u · △u = ∇ · (|u|∇|u|) − |∇u|2 ,
(14)
we easily derive
|u|s−2 u · △udx = −
Rd
4(s − 2) ∥∇|u|s/2 ∥2L2 − ∥|∇u||u|s/2−1 ∥2L2 , s2
(15)
and reach the following estimate d ∥u∥sLs + ∥|∇u||u|s/2−1 ∥2L2 + ∥∇|u|s/2 ∥2L2 . dt
p|u|s−2 (ˆ u · ∇|u|)dx.
(16)
Rd
From here on we will use A . B to denote A 6 cB for some constant c whose value does not depend on u or p. Since s−2 s−2 s−2 6 2 2 ∇|u| dx, |p||u| p|u| (ˆ u · ∇|u|)dx (17) |u| d d R
R
application of Young’s inequality turns (16) into d ∥u∥sLs + ∥|∇u||u|s/2−1 ∥2L2 + ∥∇|u|s/2 ∥2L2 . dt
|p|2 |u|s−2 dx.
(18)
Rd
Now let w := |u|s/2 . From (18) it follows that d 4 + s2 ∥u∥sLs + ∥∇w∥2L2 . dt s2
|p|2 |w|2(1−2/s) dx.
(19)
Rd
We have Rd
|p|2 |w|2(1−2/s) dx 6 ∥pw∥L2 ∥p∥Ls/2 ∥w1−4/s ∥L2s/(s−4) 1−4/s
6 ∥p∥X˙ r ∥w∥H˙ r ∥u∥2Ls ∥w∥L2 = ∥p∥X˙ r ∥w∥H˙ r ∥w∥L2 (2−r)
6 ∥p∥X˙ r ∥w∥L2 2/(2−r)
6 C∥p∥X˙
r
∥w∥rH˙ 1
1 ∥w∥2L2 + ∥w∥2H˙ 1 2
where we have applied H¨ older’s inequality and the definition of X˙ r norm (8).
(20)
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From (20) we conclude 2/(2−r)
2 ∥p∥X˙ d 2−r r ∥u∥Ls . ∥p∥X log(e + ∥p∥W m,∞ )∥u∥Ls . ˙ r ∥u∥Ls = dt log(e + ∥p∥W m,∞ )
(21)
Now let k > d2 + m be arbitrary. By Sobolev embedding theorems we have ∥u∥C m,α . ∥u∥H k for some α > 0. Application of (12) now gives ∥p∥W m,∞ 6 ∥p∥C m,α .
max
i,j=1,2,...,d
∥ui uj ∥C m,α . ∥u∥2C m,α . ∥u∥2H k .
(22)
Consequently (21) yields 2/(2−r)
∥p∥X˙ d r ∥u∥Ls . log(e + ∥u∥H k )∥u∥Ls . dt log(e + ∥p∥W m,∞ )
(23)
Let ϵ > 0 be small and T ∗ − ϵ < t < T ∗ . Integrating (23) from T ∗ − ϵ to t we see that ∥u∥Ls (t) 6 ∥u0 ∥Ls exp
T ∗ −ϵ
2/(2−r)
T∗
∥p∥X˙
r
log(e + ∥p∥W m,∞ )
dt max log(e + ∥u∥H k ) .
(24)
[0,T ]
Thanks to the integrability assumption (9), for any δ > 0, we can take ϵ > 0 small enough to have ∥u(t)∥Ls 6 C(ϵ)(e +
max
t′ ∈[T ∗ −ϵ,T1 )
∥u(t′ )∥H k )δ
(25)
for all T1 ∈ (T ∗ − ϵ, T ∗ ) and t ∈ [T ∗ − ϵ, T1 ). Note that by our assumption C(ϵ) −→ ∞ as δ ↘ 0. In the following we will see that it is possible to take a fixed positive value of δ and thus exclude this possibility. 2d (2) H k estimate. Let 2 < q < d−2 and p > s be such that 1q + p1 = 12 . Let k > d2 + m be a large natural number to be specified later. Let Λ := (−△)1/2 . We multiply (1) by Λ2k u and integrate: d 1 2 2 2k ∥u∥H˙ k + ∥u∥H˙ k+1 . u · ∇u · Λ udx dt 2 d R k−1 k+1 = Λ ∇ · (u ⊗ u) · (Λ u)dx d R
. ∥u∥Lp ∥Λk u∥Lq ∥Λk+1 u∥L2
(26)
Here we have used the product estimate ∥Λs (f g)∥Lp 6 ∥f ∥Lp1 ∥g∥H s,p2 + ∥f ∥H s,p3 ∥g∥Lp4 ,
(27)
which holds for p, p2 , p3 ∈ (1, +∞) and 1/p = 1/p1 + 1/p2 = 1/p3 + 1/p4 , from Lemma 2.1 of [28]. Now we interpolate ∥u∥Lp . ∥u∥θLs ∥Λk+1 u∥1−θ L2 , k
∥Λ u∥Lq .
∥u∥µLs ∥Λk+1 u∥1−µ L2
(28) (29)
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1−d/2+d/q 1 where θ := k+1−d/2+d/p k+1−d/2+d/s , µ := k+1−d/2+d/s . Note that since q > 2, µ < k+1 when k is large enough. From now on we fix k > d2 + m to be such a large natural number. Substituting (28) and (29) into (26) yields
d 1 k+1 . ∥u∥2H˙ k + ∥u∥2H˙ k+1 . ∥u∥θ+µ u∥3−θ−µ Ls ∥Λ L2 dt 2
(30)
As s > d we have θ + µ > 1. Application of Young’s inequality now gives d ∥u∥2H˙ k . ∥u∥γLs dt
(31)
for some positive number γ. Now we fix ϵ0 > 0 such that (25) holds for δ = 1/γ and ϵ = ϵ0 . Then we have d ∥u(t)∥2H˙ k 6 C(ϵ0 )(e + ′ max ∥u(t′ )∥H˙ k ) dt t ∈[T ∗ −ϵ0 ,T1 )
(32)
for all t ∈ [T ∗ − ϵ0 , T1 ). Integrating from T ∗ − ϵ0 to t ∈ (T ∗ − ϵ0 , T1 ), we arrive (e + ∥u(t)∥H˙ k )2 6 C1 (ϵ0 ) + C(ϵ0 )ϵ0 (e +
max
t′ ∈[T ∗ −ϵ0 ,T1 )
∥u(t′ )∥H˙ k )
(33)
and consequently (e +
max
t′ ∈[T ∗ −ϵ0 ,T1 )
∥u(t′ )∥H˙ k ) 6 C2 (ϵ0 ) < ∞.
(34)
As C2 (ϵ0 ) is independent of T1 , setting T1 ↗ T ∗ we have max
t′ ∈[T ∗ −ϵ0 ,T ∗ )
∥u(t′ )∥H˙ k < ∞
(35)
which contradicts the assumption that T ∗ is a blow-up time. Remark 1. It is clear that we can replace ∥p∥W m,∞ by ∥u∥W m,∞ . Acknowledgment The authors would like to thank the anonymous referees for the valuable comments and suggestions that greatly improved the quality of the article. The authors are also grateful to Prof. Jishan Fan for pointing out an error in the earlier version of the paper. Part of this research was carried out when CVT was visiting the University of Alberta, whose hospitality is gratefully acknowledged. XY is partially supported by NSERC Discovery grant RES0020476. References [1] Jean Leray, On the motion of a viscous liquid filling space, Acta Math. 63 (1934) 193–248. [2] E.B. Fabes, B.F. Jones, N.M. Riviere, The initial value problem for the Navier–Stokes equations with data in Lp , Arch. Ration. Mech. Anal. 45 (1972) 222–248.
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