XY is partially supported by a grant from NSERC.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 [S. Benbernou, A note on the regularity criterion in terms of pressure for the Navier-Stokes equations, Applied Mathematics Letters 22 (2009) 1438–1443].PostprintPeer reviewe

Tran, Chuong Van

Yu, Xinwei

Applied Mathematics Letters

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 [S. Benbernou, A note on the regularity criterion in terms of pressure for the Navier-Stokes equations, Applied Mathematics Letters 22 (2009) 1438–1443]

University of St. Andrews - Pure

Logarithmic improvement of regularity criteria for the Navier-Stokes equations in terms of pressure

XY is partially supported by a grant from NSERC.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 [S. Benbernou, A note on the regularity criterion in terms of pressure for the Navier-Stokes equations, Applied Mathematics Letters 22 (2009) 1438–1443].Peer reviewe

St Andrews Research Repository

Accepted ManuscriptLogarithmic improvement of regularity criteria for the Navier–Stokesequations in terms of pressureChuong V. Tran, Xinwei YuPII: S0893-9659(16)30047-7DOI: http://dx.doi.org/10.1016/j.aml.2016.02.003Reference: AML 4944To appear in: Applied Mathematics LettersReceived date: 15 December 2015Revised date: 9 February 2016Accepted date: 9 February 2016Please cite this article as: C.V. Tran, X. Yu, Logarithmic improvement of regularity criteria forthe Navier–Stokes equations in terms of pressure, Appl. Math. Lett. (2016),http://dx.doi.org/10.1016/j.aml.2016.02.003This is a PDF file of an unedited manuscript that has been accepted for publication. As aservice to our customers we are providing this early version of the manuscript. The manuscriptwill undergo copyediting, typesetting, and review of the resulting proof before it is published inits final form. Please note that during the production process errors may be discovered whichcould affect the content, and all legal disclaimers that apply to the journal pertain.LOGARITHMIC IMPROVEMENT OF REGULARITY CRITERIA FOR THENAVIER-STOKES EQUATIONS IN TERMS OF PRESSURECHUONG V. TRAN, XINWEI YUAbstract. In this article we prove a logarithmic improvement of regularity criteria in the mul-tiplier spaces for the Cauchy problem of the incompressible Navier-Stokes equations in terms ofpressure. This improves the main result in [S. Benbernou, A note on the regularity criterionin terms of pressure for the Navier-Stokes equations, Applied Mathematics Letters 22 (2009)1438–1443].1. IntroductionAt the center stage of mathematical fluid mechanics are the incompressible Navier-Stokes equa-tionsut + u · ∇u = −∇p+ ν△u, (x, t) ∈ Ω× (0,∞) (1)div u = 0, (x, t) ∈ Ω× (0,∞) (2)u(x, 0) = u0(x), x ∈ Ω (3)with appropriate boundary conditions. Here Ω ⊆ Rd is a domain with certain regularity, u : Ω 7→ Rdis the velocity field, p : Ω 7→ R is the pressure, and ν > 0 is the (dimensionless) viscosity. Thesystem (1)–(3) on one hand describes the motion of viscous Newtonian fluids, while on the otherhand serve as the starting point of mathematical modeling of many other types of fluids, such asnon-Newtonian fluids, magnetic fluids, electric fluids, and ferro-fluids. In this article we focus onthe Cauchy problem of (1)–(3), where Ω = Rd.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 theachievement of this goal is still out of the question. In particular, there is still no satisfactoryanswer 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 outby Jean Leray in [19], where it is shown that for arbitrary T ∈ (0,∞] there is at least one functionu(x, t) satisfying the following:i. u ∈ L∞(0, T ;L2(Rd)) ∩ L2(0, T ;H1(Rd));ii. u satisfies (1) and (2) in the sense of distributions;iii. u takes the initial value in the L2 sense: limtց0 ‖u(·, t)− u0(·)‖L2 = 0;Date: February 9, 2016.2000 Mathematics Subject Classification. 35Q35, 76B03, 76W05.Key words and phrases. Navier-Stokes, Global regularity, Pressure, Prodi-Serrin, Multiplier spaces, Logarithmicimprovement.1*ManuscriptClick here to view linked References2 CHUONG V. TRAN, XINWEI YUiv. u satisfies the energy inequality‖u(·, t)‖2L2 + 2ν∫ t0‖∇u(·, τ)‖2L2dτ 6 ‖u0‖2L2 (4)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) inthe classical sense. On could further show that such a smooth Leray-Hopf solution must be theunique solution to (1)–(3). Therefore the well-posedness problem would be settled if all Leray-Hopfsolutions could be shown to be smooth. However such an general result has not been establishedup to now. On the other hand, various additional assumptions guaranteeing the smoothness ofLeray-Hopf solutions have been discovered. For example, it has been shown that if a Leray-Hopfsolution u(x, t) further satisfiesu ∈ Lr(0, T ;Ls(Rd)) with 2r+ds6 1, d < s 6∞, (5)then u(x, t) is smooth and is thus a classical solution, see e.g. [8], [20], [22]. The borderline caseu ∈ L∞(0, T ;L3) is much more complicated and requires a totally different approach. It was settledmuch later by Escauriaza, Seregin, and Sverak in [7]. Many generalizations and refinements of (5)have been proved, see e.g. [3], [5], [9], [25], [26], [27].If we formally take 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 uiuj . Thus intuitively we have p ∼ u2. Transforming(5) via this relation, we expect thatp ∈ Lr(0, T ;Ls(Rd)) with 2r+ds6 2, d2< s 6∞ (7)should guarantee the smoothness of u. This is indeed the case and was confirmed in [2], [4].Many efforts have been made to refine (7), see e.g. [1], [6], [10], [14], [15], [17], [23], [24]. It isworth mentioning that the relation (6) has also played crucial roles in the proofs of other regularitycriteria not of the Prodi-Serrin type. For example, in [21] it is used to show that Leray-Hopf weaksolutions are regular as long as either |u|2 + 2p is bounded above or p is bounded below. Amonggeneralizations of (7), in [1] it is shown that u is smooth as long as p ∈ L2/(2−r)(0, T ; X˙r(Rd)d) for0 < r 6 1 where X˙r(Rd) is the multiplier space. Multiplier spaces are defined for 0 6 r < d/2 andfunctions f ∈ L2loc(Rd) through the norm‖f‖X˙r := sup‖g‖H˙r61‖fg‖L2 <∞, (8)where H˙r(Rd) is the completion of the spaceD(Rd) with respect to the norm ‖u‖H˙r = ‖(−△)r/2u‖L2,see e.g. [12] for properties of such spaces. Among its properties we would emphasize the followingtwo.• Ld/r ⊂ X˙r for 0 6 r < d/2,• This inclusion is strict. For example by the Hardy inequality for fractional Laplacians (seee.g. [11], [16]) we have |x|−r ∈ X˙r(Rd).Thus the above criterion refines (7).In this article we will present the following logarithmic improvement of this criterion.LOGARITHMIC IMPROVEMENT OF REGULARITY CRITERIA FOR THE NAVIER-STOKES EQUATIONS IN TERMS OF PRESSURE3Theorem 1. Let u0 ∈ L2(Rd)∩Lq(Rd) for some q > d, and ∇·u0 = 0. Let u(t, x) be a Leray-Hopfsolution of NSE in [0, T ). If the pressure p satisfies∫ T0‖p‖2/(2−r)X˙rlog(e + ‖p‖Wm,∞)dt <∞ (9)for some m ∈ N ∪ {0} and r ∈ (0, 1], then u(t, x) is smooth up to T and could be extended beyondT .2. Proof of TheoremWithout loss of generality, we take ν = 1 in (1) to simplify the presentation. We apply thefollowing result from [13], [18] to guarantee short-time smoothness of the solution and thus relievingus from worrying about the legitimacy of the various integral and differential manipulations below.Theorem 2. Let u0 ∈ Ls(Rd), s > d. Then there exists T > 0 and a unique classical solutionu ∈ BC(0, T ;Ls(Rd)). Moreover, let (0, T∗) be the maximal interval such that the solution u staysin C(0;T∗;Ls(Rd)), s > d. Then for any t ∈ (0, T∗),‖u(·, t)‖Ls > C(T∗ − t) s−32s(10)where the constant C is independent of T∗ and s.We also recall that (6) impliesp =d∑i,j=1RiRj(uiuj) (11)where Rj , j = 1, . . . , d are the Riesz transforms. As a consequence of the standard theory of singularintegrals, the following holds: For any s ∈ (1,∞) and m ∈ N, α ∈ (0, 1),‖p‖Ls 6 C‖u‖2L2s, ‖p‖Cm,α 6 C maxi,j=1,2,...,d‖uiuj‖Cm,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 havelim suptրT∗ ‖u(·, t)‖Ls = ∞ for all s > d. In the following we will prove in two steps that undersuch assumption ‖u(·, t)‖Hk stays bounded up to T ∗ for k > d2 +m, thus reaching contradictionas Hk →֒ Ls. Note that again by Theorem 2 we can assume u to be smooth in (0, T ∗) and freelymanipulate all functions in integration and differentiation.(1) Ls estimate. Pick any s > max{4, d}. We multiply (1) by |u|s−2u· and integrate in Rd toobtain‖u‖s−1Lsddt‖u‖Ls = −∫Rd|u|s−2u · ∇pdx+∫Rd|u|s−2u · △udx=∫Rdpu · ∇(|u|s−2)dx+∫Rd|u|s−2u · △udx= (s− 2)∫Rdp|u|s−2(uˆ · ∇|u|)dx+∫Rd|u|s−2u · △udx. (13)where uˆ := u|u| (if u = 0, just define uˆ = 0 too).4 CHUONG V. TRAN, XINWEI YURecalling the identityu · △u = ∇ · (|u|∇|u|)− |∇u|2, (14)we easily derive∫Rd|u|s−2u · △udx = −4(s− 2)s2‖∇|u|s/2‖2L2 − ‖|∇u||u|s/2−1‖2L2, (15)and reach the following estimateddt‖u‖sLs + ‖|∇u||u|s/2−1‖2L2 + ‖∇|u|s/2‖2L2 .∫Rdp|u|s−2(uˆ · ∇|u|)dx. (16)From here on we will use A . B to denote A 6 cB for some constant c whose value doesnot depend on u.Since ∣∣∣∣∫Rdp|u|s−2(uˆ · ∇|u|)dx∣∣∣∣ 6 ∫Rd(|p||u| s−22) ∣∣∣|u| s−22 ∇|u|∣∣∣dx, (17)application of Young’s inequality turns (16) intoddt‖u‖sLs + ‖|∇u||u|s/2−1‖2L2 + ‖∇|u|s/2‖2L2 .∫Rd|p|2|u|s−2dx. (18)Now let w := |u|s/2. From (18) it follows thatddt‖u‖sLs +4 + s2s2‖∇w‖2L2 .∫Rd|p|2|w|2(1−2/s)dx. (19)We have ∫Rd|p|2|w|2(1−2/s)dx 6 ‖pw‖L2‖p‖Ls/2‖w1−4/s‖L2s/(s−4)6 ‖p‖X˙r‖w‖H˙r‖u‖2Ls‖w‖1−4/sL2= ‖p‖X˙r‖w‖H˙r‖w‖L26 ‖p‖X˙r‖w‖(2−r)L2 ‖w‖rH˙16 C‖p‖2/(2−r)X˙r‖w‖2L2 +12‖w‖2H˙1(20)where we have applied Holder’s inequality and the definition of X˙r norm (8).From (20) we concludeddt‖u‖Ls . ‖p‖22−rX˙r‖u‖Ls =‖p‖2/(2−r)X˙rlog(e + ‖p‖Wm,∞) log(e+ ‖p‖Wm,∞)‖u‖Ls . (21)Now take k > d2 +m. By Sobolev embedding theorems we have ‖u‖Cm,α . ‖u‖Hk for someα > 0. Application of (12) now gives‖p‖Wm,∞ 6 ‖p‖Cm,α . maxi,j=1,2,...,d‖uiuj‖Cm,α . ‖u‖2Cm,α . ‖u‖2Hk . (22)Consequently (21) yieldsddt‖u‖Ls .‖p‖2/(2−r)X˙rlog(e+ ‖p‖Wm,∞) log(e+ ‖u‖Hk)‖u‖Ls. (23)LOGARITHMIC IMPROVEMENT OF REGULARITY CRITERIA FOR THE NAVIER-STOKES EQUATIONS IN TERMS OF PRESSURE5Let ǫ > 0 be small and T ∗ − ǫ < t < T ∗. Integrating (23) from T ∗ − ǫ to t we see that‖u‖Ls(t) 6 ‖u0‖Ls exp∫ T∗T∗−ǫ‖p‖2/(2−r)X˙rlog(e + ‖p‖Wm,∞)dtmax[0,T ]log(e+ ‖u‖Hk) . (24)Thanks to the integrability assumption (9), for any δ > 0, we can take ǫ > 0 small enoughto have‖u(t)‖Lp 6 C(ǫ)(e + maxt′∈[T∗−ǫ,T1)‖u(t′)‖Hk)δ (25)for all T1 ∈ (T ∗ − ǫ, T ∗) and t ∈ [T ∗ − ǫ, T1). Note that by our assumption C(ǫ) −→ ∞ asδ ց 0. In the following we will see that it is possible to take a fixed positive value of δ andthus exclude this possibility.(2) Hk estimate. Fix a natural number k > d2 +m. Let Λ := (−△)1/2. We multiply (1) byΛ2ku and integrate:ddt12‖u‖2H˙k+ ‖u‖2H˙k+1.∣∣∣∣∫Rdu · ∇u · Λ2kudx∣∣∣∣=∣∣∣∣∫RdΛk−1∇ · (u ⊗ u) · (Λk+1u)dx∣∣∣∣6 C‖u‖L∞‖u‖H˙k‖u‖H˙k+1= C‖u‖L∞‖u‖αH˙k‖u‖1−αH˙k ‖u‖H˙k+1. (26)Here we have used the calculus inequality ‖Dm(uv)‖L2 . ‖u‖L∞‖Dmv‖L2+‖v‖L∞‖Dmu‖L2.The parameter α ∈ [0, 1] will be determined in a short while.Now we interpolate‖u‖L∞ . ‖u‖θLs‖Λk+1u‖1−θL2 , (27)‖u‖H˙k . ‖u‖µLs‖Λk+1u‖1−µL2 (28)where θ := k+1−d/2k+1−d/2+d/s , µ :=1k+1−d/2+d/s . Application of (27) and (28) to (26) ( (28) to‖u‖αH˙konly) yields‖u‖L∞‖u‖H˙k‖u‖H˙k+1 . ‖u‖γLs‖u‖1−αH˙k ‖u‖[1−θ+α(1−µ)]+1H˙k+1(29)where γ is some positive number which is finite for all values of α.As d < s we observe that θ + µ > 1. Thus there is α ∈ (0, 1) such that1− α+ [1− θ + α(1 − µ)] + 1 = 2 + (1− θ − αµ) < 2. (30)Take this α in (26). We obtain∣∣∣∣∫ u · ∇u · Λ2ku∣∣∣∣ . ‖u‖γLs‖u‖κH˙k‖u‖κ′H˙k+1 . ‖u‖2γ/(2−κ′)Ls ‖u‖2κ/(2−κ′)H˙k + 12‖u‖2H˙k+1 (31)We set δ0 :=(2− 2κ2−κ′). As κ + κ′ < 2, δ0 > 0. Fix ǫ0 > 0 such that (25) holds for(2γ2−κ′)−1δ02 and ǫ0. Then we haveddt‖u(t)‖2H˙k6 C(ǫ0)(e+ maxt′∈[T∗−ǫ0,T1)‖u(t′)‖H˙k)δ0/2‖u(t)‖2−δ0H˙k (32)6 CHUONG V. TRAN, XINWEI YUfor all t ∈ [T ∗ − ǫ0, T1). Integrating from T ∗ − ǫ0 to t ∈ (T ∗ − ǫ0, T1), we arrive(e+ ‖u(t)‖H˙k)2 6 C1(ǫ0) + C(ǫ0)ǫ0(e + maxt′∈[T∗−ǫ0,T1)‖u(t′)‖H˙k)2−δ02 (33)and consequently(e+ maxt′∈[T∗−ǫ0,T1)‖u(t′)‖H˙k)δ0/2 6 C2(ǫ0) <∞. (34)As both C2(ǫ0) and δ0 are independent of T1, setting T1 ր T ∗ we havemaxt′∈[T∗−ǫ0,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‖Wm,∞ by ‖u‖Wm,∞.Acknowledgment.The authors would like to thank the anonymous referees for the valuable com-ments and suggestions. Part of this research was carried out when CVT was visiting the Universityof Alberta, whose hospitality is gratefully acknowledged. XY is partially supported by a grant fromNSERC.References[1] Samia Benbernou. A note on the regularity criterion in terms of pressure for the Navier-Stokes equations. AppliedMathematics Letters, 22(9):1438–1443, September 2009.[2] Luigi C. Berselli and Giovanni P. Galdi. 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On the pressure regularity criterion of the 3D Navier-Stokesequations. Journal of Mathematical Analysis and Applications, 393(2):413–420, September 2012.[25] Yi Zhou and Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communicationson Pure and Applied Analysis, 12(6):2715–2719, November 2013.[26] Yong Zhou. A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity.Monatsh. Math., 144(3):251–257, 2005.[27] Yong Zhou and Sadek Gala. Logarithmically improved regularity criteria for the Navier-Stokes equations inmultiplier spaces. Journal of Mathematical Analysis and Applications, 356(2):498–501, August 2009.Chuong V. Tran: School of Mathematics and Statistics, University of St. Andrews, St AndrewsKY16 9SS, United KingdomE-mail address: chuong@mcs.st-and.ac.ukXinwei Yu: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton,AB, T6G 2G1, CanadaE-mail address: xinwei2@ualberta.ca

Logarithmic improvement of regularity criteria for the Navier-Stokes equations in terms of pressure

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