Attractors for a deconvolution model of turbulence

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1 Aracors for a deconvoluion model of urbulence Roger Lewandowski and Yves Preaux April 0, 2008 Absrac We consider a deconvoluion model for 3D periodic flows. We show he exisence of a global aracor for he model. MCS Classificaion : 76D05, 35Q30, 76F65, 76D03 hal , version - 0 Apr 2008 Key-words : Navier-Sokes equaions, Large eddy simulaion, Deconvoluion models. Inroducion This noe is concerned by he deconvoluion model of order N inroduced in [6] (model (2.7) below) for 3D periodic flows. This model akes inspiraion in he class of he so called α-models (see in [2] and [4] and references inside) and also in he class of ADM models (see in [7]). We are ineresed by he quesion of he exisence of a global aracor for his model. The quesion of aracors has already been considered for he alpha model (see []), corresponding o he case N = 0. We prove in his work he exisence of an aracor for each N (see Theorem 3.). In order o make he paper self conained, we describe carrefully how is consruced he deconvoluion model. Nex, we recall basic noions on he aracors, noions ha can be founded in he book of R. Temam (see [8]). Finally we prove he exisence of he aracor. The quesion of is dimension is under progress. 2 The Deconvoluion model 2. Funcion Spaces for s IR, le us define he space funcion (2.) H s = { w = k ŵeik x, w = 0, ŵ(0) = 0, k k 2s ŵ(k, ) 2 < }. IRMAR, UMR 6625, Universié Rennes, Campus Beaulieu, Rennes cedex FRANCE; Roger.Lewandowski@univ-rennes.fr, hp://perso.univ-rennes.fr/roger.lewandowski/ Lycée du Puy de Lôme, rue du Puy de Lôme, Bres, Yves.Preaux@free.fr

2 We define he H s norms by (2.2) w 2 s = k k 2s ŵ(k, ) 2, where of course w 2 0 = w 2. I can be shown ha when s is an ineger, w 2 s = s w 2 (see [3]). We denoe by P L The Helmholz-Leray orhogonal projecion of (L 2 ) 3 ono H 0 and by A he Sokes operaor defined by A = P L on D(A) = H 0 (H 2 ) 3. We noe ha in he space-periodic case Aw = w for all w D(A). The operaor A is a sef-adjoin posiive definie compac operaor from H s ono H s, for s = and s = 2 (see [5]). We denoe λ he smalles eigenvalue of A. We inroduce he rilinear form b defined by (2.3) b(u,v,w) = u i i v j w j dx. i,j wherever he inegrals make sense. Noe ha b(u, w, w) = 0 when u = The Filer and he deconvoluion process Le w H 0 and w H be he unique soluion o he following Sokes problem wih periodic boundary condiions: (2.4) δ 2 w + w + r = w in R 3, w = 0, w = 0. We denoe he filering operaion by G so ha w = Gw. Wriing w(x, ) = k ŵ(k, )e ik x, i is easily seen ha r = 0 and w(x, ) = ŵ(k, ) 2e ik x. + δ 2 k k Then wriing w = G(w), we see ha in he corresponding spaces of he ype H s, he ransfer funcion of G, denoed by Ĝ, is he funcion Ĝ(k) =, and we +δ 2 k 2 also can wrie on he H s ype spaces (2.5) δ 2 w + w = w in R 3, w = 0, w = 0. The procedure of deconvoluion by he Van Cier approximaion is described in [6]. This yields he operaor D N w = N n=0 (I G)n w. Definiion 2. The runcaion operaor H N : H s H s is defined by H N w := D N w = (D N G)w. Noe ha, for any s 0 we have he following proprieies (see [6]) : (2.6) H N w s w s, H N w s+2 C(δ, N) w s. 2

3 2.3 The model Le u 0 H 0, f H. For δ > 0, le he averaging be defined by (2.4). The problem we consider is he following: for a fixed T > 0, find (w, q) w L 2 ([0, T],H ) L ([0, T],H 0 ), w L 2 ([0, T],H ) q L 2 ([0, T], L 2 per,0 (2.7) ), w + (H N (w) )w ν w + q = H N (f) in D ([0, T] IR 3 ), w(x, 0) = H N (u 0 ) = w 0. where L 2 per,0 denoes he scalar fields in L2 loc (IR3 ), 2π-periodic wih zero mean value. We prove in [6] he following resul. Theorem 2. Problem (2.7) admis a unique soluion (w, q), w L ([0, T],H ) L 2 ([0, T],H 2 ), and he following energy equaliy holds: (2.8) 2 w() 2 + ν w 2 dxd = 2 H N(u 0 ) 2 + H N (f).w dxd. 3 Main resul 0 3. Recall of basic noions abou aracors We denoe by w(, ) = S()(w 0 ) he (unique) soluion of sysem (2.7) a ime. We recall he definiions of a global aracor and an absorbing se (see in [8]). Definiion 3. We say ha A H 0 is a global aracor for he dynamical sysem (2.7) if and only if (P) A is compac in he space H 0, (P2) IR, S()(A) A, (P3) For every bounded subse B H 0, ρ(s()(b), A) goes o zero when goes o infiniy, where ρ(s()(b), A) = sup v B inf u A u v. 0 Definiion 3.2. A se A H 0 is an absorbing se if and only if for every bounded subse B H 0 here exiss > 0 such ha for all one has S()(B) A. 2. We say ha he semi group S() is uniformly compac if and only if for every bounded subse B H 0 here exiss 2 = 2 (B) such ha S()(B) is compac We denoe by ω(a) he se ω(a) = S()(A). s 0 s Proposiion 3. Assume ha here exiss an absorbing bounded se A and ha he semi group S() is uniformly compac, hen A = ω(a) is he global aracor for he dynamical sysem defined by S(). see he proof in [8]). 3

4 3.2 Exisence of a global aracor We are now in order o sae and prove he main resul of his noe. Theorem 3. The sysem (2.7) has a global aracor. Proof. Thanks o Proposiion 3., i remains o prove ha sysem (2.7) has an absorbing se and ha S() is uniformly compac, in he sense of definiion 3.2. Boh hings are derived from basic esimaes ha we deail in he following. Absorbing se in H 0 : We ake he inner produc of he firs quaion of sysem (2.7) wih w o obain (3.) d 2 d w 2 + b(h N (w),w,w) + ν w 2 = (H N(f),w). Observing ha b(h N (w),w,w) = 0 due o H N (w) = 0, applying Young inequaliy, Poincare inequaliy w λ 2 w and using (2.6) here remains (3.2) d d w 2 + νλ w 2 νλ f 2. So, noing ρ 0 = νλ f and applying Gronwall lema we obain (3.3) w 2 w 0 2 e νλ + ρ 2 0 ( e νλ ). Considering w 0 included in a ball B(0, R) and choosing ρ 0 > ρ 0, he previous inequaliy implies ha, for > T 0, (3.4) w() 2 < ρ 2 0, wih T 0 = ln νλ ρ 0 R 2 2 ρ 0 2. Since each bounded se of H 0 is included in a ball B(0, R), one deduces ha B(0, ρ 0 ) is an absorbing se in H 0. More, as an alernaive of (3.2) we may obain (3.5) d d w 2 + ν w 2 νλ f 2. Inegraing beween and + r, ) we observe han, for u 0 B(0, R), ρ 0 > ρ 0 and > T 0 (wih T 0 = νλ ln R2 ρ 02 ρ 2 : 0 (3.6) w(s) 2 ds r f 2 + ρ 0 ν 2 λ ν. 2 Absorbing se in H : We ake now he inner produc of he firs equaion of sysem (2.7) wih Aw o obain d (3.7) 2 d w 2 + b(h N(w),w, Aw) + ν Aw 2 = (H N (f), Aw), 4

5 leading o (3.8) d 2 d w 2 + +ν Aw 2 ν H N(f) 2 + ν 4 Aw 2 + b(h N (w),w, Aw), The rilinear form b saisfies he folowing inequaliy (see in [6]) : (3.9) b(u,v,w) c u /4 u 3/4 v /4 Av 3/4 w. Therefore, one has (3.0) b(h N (w),w, Aw) c H N (w) /4 H N (w) 3/4 w /4 Aw 7/4. Using (2.6) we have H N (w) H N (w) 2 C(δ, N) w and using (2.6) : (3.) b(h N (w),w, Aw) C (δ, N) w w /4 Aw 7/4. By Young inequaliy we obain (3.2) b(h N (w),w, Aw) ν 4 Aw 2 + C (δ, N) 2 hus (3.3) w 8 w 2, d d w 2 + ν Aw 2 2 ν H N(f) 2 + C (δ, N) w 8 w 2 We now use a Gronwall ype proposiion (see he proof in [8]): Proposiion 3.2 Assume ha y, g and h are posiive, localy inegrable funcions on ] 0, + [, and ha for 0, dy d gy + h, y(s)ds k, g(s)ds, k 2, h(s)ds k 3, where r, k, k 2, k 3 are four posiive consans, hen ( ) k y( + r) r + k 3 e k 2, 0. We can now finish he proof. Thanks o (3.4) and (3.6), using his lemma wih y = w 2, g = C (δ, N) w 8 and h = 2 H ν N(f) 2, we obain, ( ) (3.4) w() 2 k r + k 3 e k 2, T 0 + r, wih k = r ν 2 λ f 2 + ν ρ 02, k 2 = C (δ, N)ρ 8 0, k 3 = 2r ν f 2. Thus, afer a ime T = T ( w 0, f,ν), w is included in a ball or radius R = R ( f, ν, δ, N). One deduces ha here exiss an absorbing se in H. Le B be a bounded se in H. Esimae (3.4) implies ha S()B is a bounded T 0 +r se in H wich is compacly imbeded in H 0, so S() is uniformly compac. Esimae (3.4) also implies he exisence of an absorbing bonded se since k, k 2 and k 3 are independan of w 0. Thanks o (3.), his achieves he proof of he heorem. 5

6 References [] V. V. Chepyzhov, E. S. Tii, and M. I. Vishik, On he convergence of he leray-alpha model o he rajecory aracor of he 3D Navier-Sokes sysem, Maemaicheskii Sbornik, 2 (2007), pp [2] A. Cheskidov, D. D. Holm, E. Olson, and E. S. Tii, On a leray-α model of urbulence, Royal Sociey London, Proceedings, Series A, Mahemaical, Physical and Engineering Sciences, 46 (2005), pp [3] C. Doering and J. Gibbon, Applied analysis of he Navier-Sokes equaions, Cambridge Universiy Press, 995. [4] C. Foias, D. D. Holm, and E. S. Tii, The Navier-Sokes-alpha model of fluid urbulence, Physica D, 52 (200), pp [5] C. Foias, O. Manley, R. Rosa, and R. Temam, Navier-Sokes Equaions and Turbulence, Cambridge Universiy Press, 200. [6] W. Layon and R. Lewandowski, A high accuracy leray-deconvoluion model of urbulence and is limiing behavior, Analysis and Applicaions, 6 (2008), pp. 27. [7] S. Solz, N. A. Adams, and L. Kleiser, An approximae deconvoluion model for large-eddy simulaion wih applicaion o incompressible wall-bounded flows, Physics of fluids, 3 (200), pp [8] R. Temam, Infinie Dimensional Dynamical Sysems in Mechanics and Physics, Springer Verlag,