Miami, Florida, USA. Engineering, University of California, Irvine, California, USA. Science, Rehovot, Israel

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1 This article was downloaded by:[weizmann Institute Science] On: July 008 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, 7-41 Mortimer Street, London W1T JH, UK Online Publication Date: 01 March 008 Communications in Partial Differential Equations Publication details, including instructions for authors and subscription information: Regularity Criterion for Solutions of Three-Dimensional Turbulent Channel Flows Chongsheng Cao a ; Junlin Qin b ; Edriss S. Titi cd a Department of Mathematics, Florida International University, University Park, Miami, Florida, USA b Department of Mathematics, Xian Jiaotong University, Xian, Shaanxi, China c Department of Mathematics and Department of Mechanical and Aerospace Engineering, University of California, Irvine, California, USA d Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel To cite this Article: Cao, Chongsheng, Qin, Junlin and Titi, Edriss S. (008) 'Regularity Criterion for Solutions of Three-Dimensional Turbulent Channel Flows', Communications in Partial Differential Equations, :, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Communications in Partial Differential Equations, : , 008 Copyright Taylor & Francis Group, LLC ISSN print/15-41 online DOI: / Regularity Criterion for Solutions of Three-Dimensional Turbulent Channel Flows CHONGSHENG CAO 1, JUNLIN QIN, AND EDRISS S. TITI 4 1 Department of Mathematics, Florida International University, University Park, Miami, Florida, USA Department of Mathematics, Xian Jiaotong University, Xian, Shaanxi, China Department of Mathematics and Department of Mechanical and Aerospace Engineering, University of California, Irvine, California, USA 4 Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel In this paper we consider the three-dimensional Navier Stokes equations in infinite channel. We provide a regularity criterion for solutions of the three-dimensional Navier Stokes equations in terms of the vertical component of the velocity field. Keywords Regularity criterion; Three-dimensional Navier Stokes equations; Turbulent channel flows. Mathematics Subject Classification 5Q5; 65M Introduction Turbulence stands out as a prototype of multi-scale phenomenon that occurs in nature. It involves wide ranges of spatial and temporal scales which makes it very difficult to study analytically and prohibitively expensive to simulate computationally. Turbulent channel flows are considered to be the simplest flows confined within physical boundaries that can be simulated numerically and that demonstrates many of the common features of turbulence. In this paper we consider three-dimensional finite energy turbulent flows of viscous incompressible homogeneous fluids in the infinite channel = L, subject to the no-slip Dirichlet boundary conditions. These flows are governed by the Received September 1, 006; Accepted March 1, 007 Address correspondence to Edriss S. Titi, Department of Mathematics, University of California, Irvine, CA , USA; etiti@math.uci.edu 419

3 40 Cao et al. three-dimensional Navier Stokes system of equations: t p = f in (1) u = 0 in () u = 0 on () lim ut x = 0 (4) x u0x= u 0 x in (5) Here, u = 1 represents the unknown velocity vector field, and p is the unknown pressure scalar; where >0, the constant kinematic viscosity, f, the body forcing term, and u 0, the initial velocity, are given. Mathematically, it is well-known that the three-dimensional system (1) (5) has global (for all time and all initial data) weak solutions (see, e.g., Constantin and Foias, 1988; Galdi, 1994; Ladyzhenskaya, 1969; Sohr, 001; Temam, 198, 001 and references therein). The question of well-posedness, in the sense of Hadamard, and in particular the question of uniqueness, of these weak solutions is still an open problem. On the other hand, it is also well-established (see, e.g., Constantin and Foias, 1988; Galdi, 1994; Ladyzhenskaya, 1969; Sohr, 001; Temam, 198, 001 and references therein) that the system (1) (5) possesses a unique strong (regular) solution, which depends continuously on the initial data, for a short interval of time 0T, where T depends on the size of initial datum, u 0,onf, and L. Moreover, it is also well-known that the existence (for all time) and uniqueness of strong (regular) solutions is guaranteed under suitable additional assumptions (see, e.g., Berselli, 00; Berselli and Galdi, 00; Cao and Titi, 007; Chae and Lee, 001; He, 00; Kato, 1984; Kukavica and Ziane, 006; Moise, 006; Pokorný, 00; Prodi, 1959; Serrin, 196; Zhou, 00 and references therein). In particular, some of these recent results involve conditions on only one component of the velocity field of the D NSE in the whole space or under periodic boundary conditions (see, e.g., He, 00; Kukavica and Ziane, 006; Pokorný, 00; Zhou, 00). In this paper, we study this type of sufficient conditions for the global regularity of the D NSE in the infinite channel, subject to no-slip Dirichlet boundary condition on the physical boundary of the channel (see also, Moise, 006). Using the geophysical terminology, our condition is formulated in terms of the third component of the baroclinic mode ũ (see (16), below, for the definition of the barotropic mode (vertically averaged mode), ū, and the baroclinic mode (the fluctuation about the barotropic mode)). Specifically, our results states that if ũ satisfies ũ L 0 L (6) then the strong (regular) unique solution of the D Naiver Stokes equation (1) (5) exists for all time. Let us observe that our condition (6) seems to be slightly tighter than the former ones (cf. e.g., He, 00; Kukavica and Ziane, 006; Pokorný, 00; Zhou, 00). However, unlike the previous works we study here the D Navier Stokes in a domain with physical boundaries under the no-slip Dirichlet boundary conditions. Furthermore, we emphasize that the techniques developed here, which are inspired

4 Regularity of D Navier Stokes Equations 41 by ideas presented in Cao and Titi (007), are totally different than the previous ones. Let us denote by L q L q, and H m H m the usual L q -Lebesgue and Sobolev spaces, respectively (Adams, 1975). We denote by Let q q dx 1 dx dx for every L q q = q q dx 1 dx for every L q = v C 0 v = 0 Since we are interested in flows of finite energy in the infinite channel, we consider the spaces H and V, defined to be the closures of the set in L under L - topology, and in H 1 under H 1 -topology, respectively. Denote by PL H, the orthogonal projection, and let A = P be the Stokes operator subject to the homogeneous Dirichlet boundary condition (). It is well known that the Navier Stokes equations (NSE) (1) (5) are equivalent to the functional differential equation (see, e.g., Constantin and Foias, 1988; Sohr, 001; Temam, 198, 001) (7) du dt Au B u = f (8) u0 = u 0 (9) where B u = P the nonlinear (bilinear) term. We say u is a Leray Hopf weak solution to the system (8) (9) if u satisfies (see, e.g., Constantin and Foias, 1988; Temam, 198, 001) (1) u C0 T H-weak L 0 T V and t u L 1 0 T V where V is the dual space of V, () the weak formulation: ut x xdx ut 0 x xdx t = s x x s x s x xdx ds t 0 t t 0 fs x xdx ds for every and almost every t, t 0 0T. () the energy inequality: t t t t 0 s ds t 0 t 0 for all t 0T, and for almost every t 0 in the interval 0t. fs x us xdx ds (10)

5 4 Cao et al. Moreover, a weak solution is called strong solution of (8) (9) on 0T if, in addition, it satisfies u C0 T V L 0 T H For convenience, we recall the following Gagiliardo Nirenberg, Ladyzhenskaya, and Sobolev inequalities (cf. e.g., Adams, 1975; Constantin and Foias, 1988; Galdi, 1994; Ladyzhenskaya, 1985, 1969; Sohr, 001) in : L r C r /r L for every H 1 and in : r r L C 6 L H 1 H 1 r < (11) (1) for every u H 1 6 Here C r and C are scale invariant constants. We also recall the Pioncaré inequality: v C 0 L v v V (1) Av C 0 L v v A (14) where C 0 is a scale invariant constant. Also, we recall the integral version of Minkowsky inequality for the L r spaces, r 1. Let 1 m 1 and m be two measurable sets, where m 1 and m are two positive integers. Suppose that is measurable over 1. Then, [ 1 ) r ] 1/r d d. Global Existence of the Strong Solution /r r d d (15) 1 In this section we will show the global existence of the strong solutions to the three-dimensional Navier Stokes system (1) (5) under assumption (6). We will denote by x 1 x = 1 L x L 1 x x dx and = (16) Following the geophysical fluid dynamics terminology we will call the barotropic mode and the baroclinic mode. From now on, we will denote by h = and x 1 x h =. First, let us x1 x prove the following lemma. Lemma 1. Suppose that x 1 x H 1 H 1, and L. Then, dx 1 dx dx C 1/ h 1/ 1/ h 1/

6 Regularity of D Navier Stokes Equations 4 Proof. Notice that dx 1 dx dx = [ L dx )]dx 1 dx We will estimate the above term by applying the same method used to establish Proposition. in Cao and Titi (00). First, by Cauchy Schwarz inequality, we obtain L L L dx dx dx Thus, by the above and Hölder inequality, we reach dx 1 dx dx By using Minkowsky inequality (15), we get [ [ L L dx dx ]dx 1 dx 4 [ L [ dx 1 dx dx ) dx 1 dx ] dx] L ) ] 1 L dx dx 1 dx 4 dx 1 dx dx Thanks to (11) with r = 4, for every fixed x we have 4 dx 1 dx 4 C 1 L 1 H 1 As a result of the above and the Cauchy Schwarz inequality, we obtain L Therefore, 4 dx 1 dx [ L dx C L H 1 dx L L C dx L C h dx H 1 L ) ] 1 4 dx dx 1 dx C h 1/ 1/ (17) By using (11) with r = 4, we have [ 4 dx 1 dx ] 1 4 C 1/ L 1/ H 1

7 44 Cao et al. Thus, by the above and (17), we get dx 1 dx dx C 1/ h 1/ 1/ h 1/ Theorem. Let f L 0 L u 0 V. Let u = 1 be a weak solution of the system (8) (9) in 0. Suppose that for T>0 ũ L 0 T L ; that is, ũ satisfies Then u is the strong solution of the system (8) (9) on 0T. sup ũ t < (18) 0 t T Proof. Let u 0 V. Following, for instance, the Galerkin method one can show that there exists a unique strong solution u for the system (8) (9), with the initial datum u 0, for a short interval of time (see, e.g., Constantin and Foias, 1988; Ladyzhenskaya, 1969; Sohr, 001; Temam, 198, 001. Suppose that 0T is the maximal interval of existence of this strong solution u. It is also well known (see, e.g., the above references) that there exists a Leray-Hopf weak solution for the system (8) (9), with the same initial datum u 0, which exists globally in time, i.e. for all time t 0. Most importantly, following the work of Sather and Serrin (196) one can show that all the Leray Hopf weak solutions coincide with the unique strong solution, u, on the interval 0T. To conclude our proof we need to show that T<T. Suppose, arguing by contradiction, that T T, and that (18) holds. If we show that the lim sup t T t H 1 < then 0T is not a maximal interval of existence, which leads to a contradiction. For the rest of this proof we consider the strong solution, u, in the the interval 0T. From the energy inequality of (10), which is satisfied by all the Leray-Hopf weak solutions (see, for example, Constantin and Foias, 1988; Ladyzhenskaya, 1969; Sohr, 001; Temam, 198, 001, for details), we have for all t 0T, where t C L4 F t In particular, since t<t T we have 0 s ds C L F t e t L 0 (19) 0 (0) F =f L 0L (1) t t s ds K 1 () 0

8 Regularity of D Navier Stokes Equations 45 where K 1 = C F L 4 T 0 () Taking the inner product of the equation (8) with h u in H, and notice that P xi = xi P for i = 1, we get 1 d h u dt h u hu z = f B u h udx 1 dx dx By integration by parts we get B u h udx 1 dx dx = {( ) ( 1 = = = x 1 kl=1 { x kl=1 jl=1 { ũ x kl=1 ū [ jl=1 x k x x l k x l l=1 ) ( 1 x 1 x jl=1 [( ) ( 1 x 1 x ( k k u x l x x k l x x j j x l x l l=1 [( ) ( 1 x 1 x ( k ũ k ũ u x l x x k l x [ jl=1 x j ũ x j j x l x l k j k dx x j x l x 1 dx dx l )[( 1 x j j x l x l ) 1 x 1 x ) ( ) 1 x x 1 x l=1 x l x l ( ) ( 1 x x 1 ) u k u u k x l x k l x l x l x ( 1 ) } dx x 1 x x l x 1 dx dx l ( ) ( 1 ) 1 x 1 x ( ) j ( )] j j x l x l x l x x l l=1 x x 1 ) u k ũ u u k x l x k x l x l x x l=1 l ( 1 ) } ũ dx x 1 x x l x 1 dx dx l x x 1 } dx 1 dx dx ] ) ] 1 x x 1 ) ] 1 x x 1 [( 1 x 1 x ) ]] x l Then, from the above and the Cauchy Schwarz inequality we obtain 1 d h u dt C h u hu z F hu C ū h dx 1 dx dx ũ h u dx 1 dx dx C ũ h dx 1 dx dx

9 46 Cao et al. where F is given in (1). By applying Lemma 1 we obtain ū h dx 1 dx dx Cū 1/ ū h ū 1/ 1/ h 1/ h Cū h Cū 1/ h ū 1/ 1/ h 1/ h C h C 1/ h u 1/ 1/ h / (4) By Hölder inequality, we reach ũ h u dx 1 dx dx Cũ h u 4 Cũ h u 1/ h u h / (5) And also by Hölder inequality, we obtain ũ h dx 1 dx dx C x 1 x ũ x 1 x h x 1 x dx 1 dx { 1/ C x 1 x 1/ x x 1 x ũ x 1 x h x 1 x }dx 1 dx { } 1/8 { C x 1 x 4 dx 1 dx x x 1 x { } 1/4 ũ x 1 x 4 dx 1 dx h C 1/4 h u 1/4 x 1/4 h u x By (4) (6) and Young s inequality we get 1/4 4 dx 1 dx } 1/8 ũ 1/ h ũ 1/ h (6) d h u dt h CF C ( ) C 1/ ũ 4 x ũ 4 h u where F is given in (1). Thanks to Gronwall inequality, we obtain, for all t 0T, t h ut h s ds K 0

10 Regularity of D Navier Stokes Equations 47 where K = e CK 1 CTK 1 max 0 s T ũ s 4 0 F H 1 CK 1 (7) Recall that V = u Au dx 1 dx dx. It is well know that V is equivalent to (see, e.g., Constantin and Foias, 1988). Taking the inner product of the equation (8) with Au in H, weget 1 d V dt Au = f B u Au dx 1 dx dx FAu C 6 Au FAu C h u / x Here, we used (cf., e.g., Galdi, 1994, p. ) 6 C x 1 By Young s inequality we obtain d V dt 1/ x 1/ Au CF C h u 8/ x 1/ 1/ x 1/ Au / 4/ V By Gronwall inequality and (7), we obtain, for all t 0T, where Therefore, t V t 0 Aus ds K K = e CK4/ K / 1 0 F H 1 (8) lim sup t t T V K which leads to a contradiction that 0T is the maximal interval of existence, and this completes the proof. Acknowledgments E.S.T. would like to thank the Bernoulli Center of the École Polytechnique Fédéral de Lausanne for the kind hospitality where this work was completed. This work was supported in part by the NSF grant no. DMS , the BSF grant no , the ISF grant no. 10/06, and by the MAOF Fellowship of the Israeli Council of Higher Education.

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