Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D.

Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D. P. Constantin Department of Mathematics, The University of Chicago 5734 S. University Avenue, Chicago, Il 6637 email const@math.uchicago.edu and Nader Masmoudi Courant Institute, New York University 251 Mercer St, New York NY 112 email:masmoudi@cims.nyu.edu May 11, 27 Abstract We prove global existence for a nonlinear Smoluchowski equation a nonlinear Fokker- Planck equation coupled with Navier-Stokes equations in 2d. The proof uses a deteriorating regularity estimate in the spirit of [5] see also [1] Key words Nonlinear Fokker-Planck equations, Navier-Stokes equations, Smoluchowski equation, micro-macro interactions. AMS subject classification 35Q3, 82C31, 76A5. 1 Introduction Systems coupling fluids and particles are of great interest in many branches of applied physics and chemistry. The equations attempt to describe the behavior of complex mixtures of particles and fluids, and as such, they present numerous challenges, simultaneously at three levels: at the level of their derivation, the level of their numerical simulation and that of their mathematical treatment. In this paper we concentrate solely on one aspect of the mathematical treatment, the regularity of solutions. The particles in the system are described by a probability distribution ft, x, m that depends on time t, macroscopic variable x R n, and particle configuration m M. Here M is a smooth compact Riemannian manifold without boundary. The particles are transported by a fluid, agitated by thermal noise, and interact among themselves. This is reflected in a kinetic equation for the evolution of the probability distribution of the particles [2, 8]. The interaction between particles a micro-micro interaction is modeled in a mean-field fashion by a potential that represents the tendency of particles to favor certain coherent configurations. The interaction between particles occurs only when the concentration of particles is sufficiently high. Mathematically, this term is responsible for the nonlinearity of the Smoluchowski Fokker-Planck equation, and physically, it is responsible 1

for nematic phase transitions. Because the particles are considerably small, and for smooth flows, the Lagrangian transport of the particles is modeled using a Taylor expansion of the velocity field. This gives rise to a drift term in the Smoluchowski equation that depends on the spatial gradient of velocity. It is a macro-micro term, and it causes mathematical difficulties in the regularity theory. The fluid is described by the incompressible Navier-Stokes equations. The microscopic particles add stresses to the fluid. This is the micro-macro interaction and it is the most puzzling and important physical aspect of the problem. Indeed, while a macro-micro interaction can be derived, in principle, by assuming that the macroscopic entities vary little on the scale of the microscopic ones, the scaling up of the effect of microscopic quantities to the macroscopic level is more mysterious. A principle based on an energy dissipation balance, and that recovers familiar results in simple cases was proposed in [6], where the regularity of nonlinear Fokker-Planck systems coupled with Stokes equations in 3D was also proved. The linear Fokker-Planck system coupled with Stokes equations was considered in [22]. The nonlinear Fokker-Planck equation driven by a time averaged Navier-Stokes system in 2D was studied in [7]. An approximate closure of the linear Fokker-Planck equation reduces the description to closed viscoelastic equations for the added stresses themselves. This leads to well-known non- Newtonian fluid models that have been studied extensively. For regularity results we refer to Lions and Masmoudi [19] where the existence of global weak solutions was proved for an Oldroyd-type model. In Guillopé and Saut [13] and [14], the existence of local strong solution was proved. Also, Fernández-Cara, Guillén and Ortega [11], [1] and [12] proved local well posedness in Sobolev spaces. We also mention Lin, Liu and Zhang [16] where a formulation based on the ormation tensor is used to study the Oldroyd-B model. An other model for the polymers is the FENE dumbbell model. From mathematical point of view, this model was studied by several authors. In particular W. E, Li and Zhang [9], Jourdain, Lelievre and Le Bris [15] and Zhang and Zhang [23] proved local well-posedness. Moreover, Lin, Liu and Zhang [17] proved global existence near equilibrium. After the completion of the present work, we learned that Lin, Zhang, and Zhang [18] proved a result similar to our result for the co-rotational FENE model see also [21]. Existence of global weak solutions was also proved in [2]. 1.1 The model Consider the system v t + v v ν v + p = τ in Ω, T f t + v f + div ggv, ff g f = in Ω, T divv = in Ω, T, 1 where τ ij = M γ1 ij mft, x, mdm+ M M γ2 ij m 1, m 2 ft, x, m 1 ft, x, m 2 dm. We denote Gv, f = g U + W where W = c ij α j v i and U = Kf is a potential given by Ut, x, m = Km, qft, x, q dq 2 M with a kernel K which is a smooth, time and space independent symmetric function K : M M R. We also take Ω = R 2. 2

1.2 Statement of the result Theorem 1.1 Take v W 1+ε,r L 2 R 2 and f W 1,r H s, for some r > 2 and ε > and f, M f L 1 L. Then 1 has a global solution in v L loc W 1,r L 2 loc W 2,r and f L loc W 1,r H s. Moreover, for T > T >, we have v L T, T ; W 2 ε,r. 1.3 Preliminaries We ine C to be the ring of center, of small radius 1/2 and great radius 2. There exist two nonnegative radial functions χ and ϕ belonging respectively to DB, 1 and to DC so that χξ + q ϕ2 q ξ = 1, 3 p q 2 Supp ϕ2 q Supp ϕ2 p =. 4 For instance, one can take χ DB, 1 such that χ 1 on B, 1/2 and take ϕξ = χξ/2 χξ. Then, we are able to ine the Littlewood-Paley decomposition. Let us denote by F the Fourier transform on R d. Let h, h, q, S q q Z be ined as follows: where h = F 1 ϕ and h = F 1 χ, q u = F 1 ϕ2 q ξfu = 2 qd h2 q yux ydy, S q u = F 1 χ2 q ξfu = 2 qd h2 q yux ydy. We use the para-product decomposition of Bony [3] T u v = q Z uv = T u v + T v u + Ru, v S q 1 u q v and Ru, v = q q 1 We ine the inhomogeneous and homogeneous Besov spaces by q u q v. Definition 1.2 Let s be a real number, p and r two real numbers greater than 1. Then we ine the following norm u eb s = S u L p + 2 qs l q u L p p,r q N r N and the following semi-norm Definition 1.3 u B s p,r = 2 qs q u L p q Z l r Z. Let s be a real number, p and r two real numbers greater than 1. We denote by B p,r s the space of tempered distributions u such that u eb s is finite. p,r 3

If s < d/p or s = d/p and r = 1 we ine the homogeneous Besov space B s p,r as the closure of compactly supported smooth functions for the norm B s p,r. We refer to [4] for the proof of the following results and for the multiplication law in Besov spaces. Lemma 1.4 q u L b 2 d 1 a 1 b q q u L a for b a 1 The following corollary is straightforward. e t q u L b C2 ct22q q u L b Corollary 1.5 If b a 1, then, we have the following continuous embeddings Ba,r s B s d 1 a 1 b b,r. Definition 1.6 Let p be in [1, ] and r in R; the space L p T Cr is the space of distributions u such that u el = sup 2 qr p,t ;C r q u L p T q L <. We will use the following theorem from [5] Theorem 1.7 Let v be the solution in L 2 T H1 of the two dimensional Navier-Stokes system v + v v ν v = p + f NS ν t divv = v t= = v. with an initial data in L 2 and an external force f in L 1 T C 1 L 2 T H 1 ; then, for any ε, a T in the interval ], T [ exists such that v el 1 [T,T ] C ε. 2 A deteriorating regularity estimate The main part of this section is the proof of a deteriorating regularity estimate for transport equations in the spirit of [1] and [5]. After this proof, we will apply this estimate in order to prove Theorem 1.1. We also denote H = g + I s/2 with s > d/2 + 1. Theorem 2.1 Let σ and β be two elements of ], 1[ such that σ + β < 1. A constant C exists that satisfies the following properties. Let T and be two positive numbers and v a smooth divergence free vector field so that σ v el 1 T C β and σ + v L e1 1 β. 5 T C Consider two smooth functions f and v so that f is the solution of { t f + v f + div g Gv, ff g f = f t= = f. 6 4

where Then we have, if 3C, M σ f 3 f B σ p, H s + 3C M σ+1 v 7 M σ v M σ f Φ q, t, t = sup 2 qσ Φq,t q vt L p or 8 t [,T ],q = sup = t [,T ],q 2 qσ Φ q,t q ft L p H s with 9 t S q 1 vt L + 1dt, Φ q, t = Φ q, t,. 1 We will use the notation f q = q f. Applying the operator q to the transport equation 6, we get { t f q + S q 1 v f q + div g GS q 1 v, S q 1 ff q g f q + R q v, f = f q t= = q f. 11 where R q is a rest term. We denote Nq 2 t, x = Applying H to 11 and taking the L 2 norm on M, we get M Hf q 2 dm 12 1 2 tnq 2 + S q 1 v Nq 2 + V S q 1 v, S q 1 U, f q + g Hf q 2 + Hf q HR q v, fdm = 13 M where V v, h, f = j v i M Hdiv g c ij α fhfdm. + Hdiv g g hfhfdm. 14 M Hence, arguing as in [7], we have V S q 1 v, S q 1 U, f q C S q 1 v + HS q 1 f L 2 MN 2 q. We will use now the following lemma, postponing its proof: Lemma 2.2 R q v, f satisfies 2 qσ Φq,t HR q vt, ft L p L 2 Ce C v L e1 C T M σ+1 v + 1 + S q vt L + Taking the L p norm of N q, we get q q N q vt L M σ f. 15 N q t L p N q L p + C After multiplication by 2 qσ Φ q,t, we get HR q vt, ft L p L 2 + 1 + S q vt L N q t L pdt. 5

2 qσ Φ q,t N q t L p 2 qσ N q L p + + 2 Φ q,t,t 2 qσ Φ q,t S q vt L N q L pdt 2 Φ q,t,t 2 qσ Φ q,t HR q vt, ft L p L 2 dt. Then, using the inequality 15 and taking the sup over q, we get M σ f f B σ p, H s + e C v e L 1 T C M σ+1 v + M σ f 1 + 2 S q vt L + q q N sup t [,T ],q 2 Φ q,t,t 16 q vt L dt. 17 As v el 1 T C is smaller than σ β, we have e C v e L 1 T C e Cσ β. Moreover, by inition of Φ q, t, t, it is obvious that Then, we obtain that 2 Φ q,t,t S q vt L + 1dt 1 log 2 M σ f f B σ p, H s + C M σ+1 v + C v el 1 T C M σ f + C M σ f f B σ p, H s + C M σ+1 v + 2C M σ f. This proves the theorem of course if we prove the estimate 15 of the lemma. First of all, let 6

us decompose the operator R q. We have Indeed, R q v, f = R 1 qv, f = R 2 qv, f = R 3 qv, f = R 4 qv, f = R 5 qv, f = R 6 qv, f = + 6 Rqv, l f l=1 q T j f v j, [ q, T v j j ]f, with q j Rv j, f + q 1 v j j q+1 f q q 2 v j j q 1 f q i, i, i, i, div g c ij α q T f j v i + div g q T f g U, div g c ij α [ q, T j v i]f + div g[ q, T gu]f div g c ij α R j v i, f + q 1 j v i q+1 f q q 2 i v j q 1 f q div g R g U, f + q 1 g U q+1 f q q 2 g U q 1 f q q v f = q Then, we use that T v j j f q = Hence, In the same way, we have = T j f v j + T v j j f + Rv j, j f 2 Rqv, l f + l=1 q q 1 T v j j q f + q Rv j, j f, S q 1v j j q f q = S q 1 v j j f q + q q 1 S q 1v j S q 1 v j j q f q = S q 1 v j j f q + q 1 v j j q+1 f q q 2 v j j q 1 f q q v f = q div g Gv, ff = 3 Rqv, l f + S q 1 v f q. l=1 6 Rqv, l f + div g GS q 1 v, S q 1 ff q. l=4 7

Let us estimate the six terms appearing above. We have Let us begin with R 1 qv, f. By inition of the paraproduct, we have R 1 qv, f = q q S q 1 j f q v j. As, if q q > 2 then the above term is equal to, we deduce that HRqvt, 1 ft L p L 2 C HS q 1 f L L 2 q vt L p. Using the fact that, if q q 2, then HS q 1 f L L 2 C2 q Hft L L 2 C2 q, we infer that HRqvt, 1 ft L p L 2 C2 q q vt L p C q vt L p. Hence 2 qσ Φ q,t HR 1 qvt, ft L p L 2 CM σ+1 But, it is obvious that S q vt L dt Using the fact that q q 2, we get So it turns out that S q vt L dt v S q vt L dt 2 R t Sq vt L dt + R t S q vt L dt. S q S q vt L dt. S q vt L dt C q q v el 1 T C. 18 2 qσ Φ q,t HR 1 qvt, ft L p L 2 2 C v e L 1 T C M σ+1 v. 19 Now let us look at R 2 qv, f. By inition of the paraproduct, we have R 2 qv, f = = [S q 1v j j q, q ]f q [S q 1v j, q ] j q f. q The terms of the above sum are equal to except if q q 2. Moreover, by inition of the operators q, we have [S q 1v j, q ] j q fx = 2 qd R d h2 q x ys q 1v j x S q 1v j y j q fydy. So we infer that H[S q 1v j, q ] j q fx L 2 M 2 q S q 1v L 2 qd 2 q h2 q H j q f L 2 M x. 8

Hence, H[S q 1v j, q ] j q fx L p L 2 M 2 q S q 1v L H j q f L p L 2 M. Then, we have, using Inequality 18, 2 qσ Φq,t H[S q 1v j, q ] j q f L p L 2 M CM σ f 2 C v L e1 C 1 T S q 1 S q vt L + S q vt L. So, we get Hence, 2 qσ Φq,t HRqvt, 2 ft L p L 2 CM σ f2c v L e1 C 1 S q vt L + q vt L. 2 For R 3 q, we have HR 3 qv, f L p L 2 C q q 1 q q 2 C 2 qσ Φ q,t HR 3 qv, f L p L 2 C q q 2 Then, we see that the sum converges since q q 2 2 q q v L p H q f L L 2 2 q q q v L p Hf L L 2. 2 1+σq q Φ q, t+φ q, t M σ+1 v Hf L L 2. Φ q, t Φ q,t v el 1 T C q q σ β q q 21 and 1 + σ σ β = 1 + β >. Hence, we get 2 qσ Φ q,t HR 3 qv, f L p L 2 CM σ+1 v Hf L L 2. The estimate for Rqv, 4 f = Rq 4,1 v, f + Rq 4,2 f is the same as the estimate for Rqv, 1 f. Indeed, we have HRq 4,1 vt, ft L p L 2 C g HS q 1f L L 2 q vt L p C q vt L p where we used that g HS q 1f L L 2 C. Hence, we conclude as for R 1 qv, f. Besides, HR 4,2 q ft L p L 2 C C 9 g HS q 1f L L 2 q g U L p q ft L p

Hence, we conclude as for R 1 qv, f and get 2 qσ Φq,t HRq 4,2 ft L p L 2 2 C v L e1 C T M σ f. 22 We write Rqv, 5 f = Rq 5,1 v, f + Rq 5,2 f. The estimate for Rqv, 5 f is similar to the one for Rqv, 2 f with the only difference that we have to use the regularity of v. We have [ q, T j v i]f = [S q 1 j v i, q ] j q f. q The terms of the above sum are equal to except if q q 2. Moreover, by inition of the operators q, we have [S q 1 j v i, q ] q fx = 2 qd R d h2 q x ys q 1 j v i x S q 1 j v i y q fydy. So we infer that HRq 5,1 v, f L 2 M 2 q 2 S q 1v 2 qd 2 q h2 q Hence, Hence, g H q f L 2 M x. HRq 5,1 v, f L p L 2 M 2 q 2 S q 1v L p g H q f L L 2 M. Then, we have, using Inequality 18, 2 qσ Φ q,t HR 5,1 q v, f L p L 2 M C q q 1 2 qσ Φ q,t HR 5,1 q v, f L p L 2 M C 2 σ 1q q Φ q, t+φ q, t M σ+1 v g H q f L L 2. q q+1 2 βq q M σ+1 v g Hf L L 2 and the sum is uniformly bounded since σ 1 + v el 1 T C similar way for HRq 5,2 f L p L 2 M and get β. Then, we argue in a HRq 5,2 f L p L 2 M 2 q S q 1f L p L 2 M g H q f L L 2 M. and we conclude as above with M σ+1 v replaced by M σf. Finally, the estimate for Rqv, 6 f is exactly the same as the one for Rqv, 3 f since, we also have that g Hf L L 2 C. 3 Global existence Now, we turn to the proof of our main theorem. First, we notice that the local existence in with v L loc [, T ; W 1,r L 2 loc [, T ; W 2,r and f L loc [, T ; W 1,r H s can be easily deduced from standard arguments. Moreover, from regularity estimates for the heat equation, we have for all < T < T, v L loc T, T ; W 2 ε,r. 1

We want to prove that we can extend the solution beyond the time T. It is enough to prove that v L, T R 2. The local existence result tells that, for any T in ], T [, the solution v, f of 1 belongs to the space L loc [T, T [; W 2 ε,r W 1,r H s for any ε >. Sobolev type embeddings of Corollary 1.5 imply that v, τ L loc [T, T [; 2 ε 2 1 r B 1 p p, 1 2 1 r B 1 p p, Choosing ε < 1 2/r and p = in the above assertion implies that v, τ L loc C 1+σ C σ H s where σ = 1 ε 2/r >. So we can apply Theorem 1.7 and we can choose T such that, with the notations of Theorem 2.1, we have v el 1 [T,T ] C minσ β, 1 σ β 3 The deteriorating regularity estimate of Theorem 2.1 applied with σ and between T and T tells exactly that f satisfies M σ f 3 ft C σ H s + 3C M σ+1 v. 23 Now, we have to estimate v. The two dimensional Navier-Stokes equation can be written as t v ν v = P v v + P Dτ where P denotes the Leray projector on the divergence free vector field. Exactly along the same lines as in the proof of Theorem2.1, we have 2 qσ+1 Φq,t P v v P S q v q v L CM σ+1 v S q vt L + 2 q q q vt L. q q. Moreover, it is obvious that 2 qσ 1 2 Φq,t P S q v q v L C vt 1 H 2 M σ+1 v. So it turns out that 2 qσ+1 Φq,t q P v v L CM σ+1 v S q vt L + q q 24 Using well known estimates on the heat equation see for instance [4] and Inequalities 23 and 24, we get that with M σ+1 v vt C σ+1 + Hölder inequality implies immediately that C + 2 3q 2 Fq T, T F q T, T = sup e cν22q t t vt 1 t [T,T ] T H 2 dt. F q T, T C ν 3 4 2 3q 2 v L 4 T,T ]H 1 2. 2 q q vt L +2 3q 2 vt H 1 2. M σ+1 v + C ν M σ τ 11

Moreover, it is easy to see that So, we infer that M σ τ M σ f. M σ+1 v vt C σ+1 + 3C ν τ C σ + C + C ν + C v ν 3 4 L 4 T,T ] H 1 2 M σ+1 v Now it is enough to choose T such that the quantity C + C ν + C v ν 3 4 L 4 T,T ] H 1 2 is small enough. Then as σ is greater than, the solution v, τ of the system 1 is such that v, τ belongs to L [T, T ] R 2 ; this concludes the proof of Theorem 1.1. 4 Acknowledgments The work of P.C. is partially supported by NSF-DMS grant 54213. The work of N. M. is partially supported by NSF-DMS grant 43983. References [1] H. Bahouri and J.-Y. Chemin. Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides. Arch. Rational Mech. Anal., 1272:159 181, 1994. [2] R. B. Bird, C. Curtiss, R. Amstrong, and O. Hassager. Dynamics of polymeric liquids, Kinetic Theory Vol. 2,. Wiley, New York, 1987. [3] J.-M. Bony. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 4, 142:29 246, 1981. [4] J.-Y. Chemin. Théorèmes d unicité pour le système de Navier-Stokes tridimensionnel. Journal d Analyse Mathématique, 77?:27 5, 1999. [5] J.-Y. Chemin and N. Masmoudi. About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal., 331:84 112 electronic, 21. [6] P. Constantin, Nonlinear Fokker-Planck Navier-Stokes systems, Commun. Math. Sci. 3 4, 531-544 25. [7] P. Constantin, C. Fefferman, E. Titi and A. Zarnescu Regularity for coupled twodimensional nonlinear Fokker-Planck and Navier-Stokes systems Comm. Math. Phys to appear [8] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1986. [9] W. E, T. Li, and P. Zhang. Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys., 2482:49 427, 24. [1] E. Fernández-Cara, F. Guillén, and R. R. Ortega. Some theoretical results for viscoplastic and dilatant fluids with variable density. Nonlinear Anal., 286:179 11, 1997. 12

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