Journal of Differential Equations

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1 J. Differential Equations 5 ) Contents lists available at ScienceDirect Journal of Differential Equations Global solutions for micro macro models of polymeric fluids Zhen Lei a,,yunwang b a School of athematical Sciences; LNS and Shanghai Key Laboratory for Contemporary Applied athematics, Fudan University, Shanghai 433, China b The Institute of athematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong article info abstract Article history: Received 5 August Available online January SC: 35Q35 76D3 Keywords: FENE dumbbell model Co-rotational Smoluchowski equation Global existence We provide a new proof for the global well-posedness of systems coupling fluids and polymers in two space dimensions. Compared to the well-known existing method based on the losing a priori estimates, our method is more direct and much simpler. The corotational FENE dumbbell model and the coupling Smoluchowski and Navier Stokes equations are studied as examples to illustrate our main ideas. Elsevier Inc. All rights reserved.. Introduction The dynamics of many polymeric fluids are described by two-scale micro macro models. The systems usually consist of a macroscopic momentum equation and a microscopic Fokker Planck type equation. The fluid is described by the macroscopic equation sometimes the incompressible Navier Stokes equations), with an induced elastic stress. The stress is the micro macro interaction. The particles in the system are represented by a probability distribution ψt, x, R) or ψt, x,m) that depends on time t, macroscopic physical location x and particle configuration R or m. The Lagrangian transport of the particles is modeled using a Taylor expansion of the velocity field, which accounts for a drift term that depends on the spatial gradient of velocity. The system attempts to describe the behavior of this complex mixture of polymers and fluids. For more physical and mechanical backgrounds, see [3,]. * Corresponding author. addresses: leizhn@yahoo.com, zlei@fudan.edu.cn Z. Lei), wangyun66@hotmail.com Y. Wang). -396/$ see front matter Elsevier Inc. All rights reserved. doi:.6/j.jde...5

2 384 Z. Lei, Y. Wang / J. Differential Equations 5 ) The FENE Finite Extensible Nonlinear Elastic) dumbbell model is one of the typical and extensively studied micro macro models. In this model a polymer is idealized as an elastic dumbbell consisting of two beads joined by a spring which can be modeled by a vector R. athematically, this system reads t u + u )u + p = ν u + div τ, x R, div u =, x R, [ t ψ + u )ψ = div R W u) Rψ + β R ψ + R Uψ ], x, R) R, R ), R Uψ + β R ψ) n =, on, R ), t = : ut, x) = u x), ψt, x, R) = ψ x, R)..) In the above system, u = ut, x) denotes the velocity field of the fluid, p = pt, x) denotes the pressure, ψt, x, R) is the distribution function for the internal configuration, ν > is the viscosity of the fluid and β is related to the temperature of the system. oreover, the spring potential U and the induced elastic stress τ is given by UR) = k ln R / R ), τ =,R ) R i R j U)ψt, x, R) dr..) Here k > is a constant. The boundary condition insures the conservation of the polymer density. Assume that W u) = u u)t, which corresponds to the co-rotational case. For simplicity of writing, assume that β = and R = and denote, R ) by. In what follows, without special claim, represents x, and div represents div x. The Smoluchowski equation coupled with the incompressible Navier Stokes equations is another extensively studied micro macro model. The Smoluchowski equation describes the temporal evolution of the probability distribution function ψ for directions of rod-like particles in a suspension. athematically, the system reads t u + u )u + p = ν u + div τ, x R, div u =, x R, ) t ψ + u )ψ + div g Gu,ψ)ψ g ψ =, x,m) R, t = : ut, x) = u x), ψt, x,m) = ψ x,m)..3) Here is a d-dimensional smooth compact Riemannian manifold without boundary of and dm is the Riemannian volume element of, u = ut, x) and p = pt, x) denote the velocity field and the pressure of fluid respectively, ψ = ψt, x,m) is the distribution function, Gu,ψ)= g U + W stands for a meanfield potential resulting from the excluded volume effects due to steric forces between molecules with W = c αm) j u i. esides, the added stress tensor τ and the potential U are given by Ut, x,m) = Km, q)ψt, x, q) dq, τ t, x) = γ ) m)ψt, x,m) dm + γ ) m,m )ψt, x,m )ψt, x,m ) dm dm..4)

3 Z. Lei, Y. Wang / J. Differential Equations 5 ) Here the kernel K is a smooth symmetric function defined on. γ ) and γ ) are smooth, time independent and x independent. At present there have been extensive and systematical studies on the existence and regularity theories of those D micro macro models of polymeric fluids [,8,7,9,,,3]. For example, the first global well-posedness result for FENE.) was derived by Lin, Zhang and Zhang [] for k > 6. asmoudi [3] extends it to the case of k > by a crucial observation on the linear operator. Very recently the global existence of weak solutions to the FENE dumbbell model of polymeric flows for a very general class of potentials was also obtained by asmoudi [4]. The global well-posedness of nonlinear Fokker Planck system coupled with Navier Stokes equations.3) in D has been proven by Constantin and asmoudi in []. When the nonlinear Fokker Planck equation is driven by a time averaged Navier Stokes system in D, global well-posedness has been obtained by Constantin, Fefferman, Titi and Zarnescu [9]. ost proofs of the above global well-posedness theorems are based on an important analytic technique called losing a priori estimate in the spirit of ahouri and Chemin [] and Chemin and asmoudi [7]. In [7], we studied the blow-up criteria of a macroscopic viscoelastic Oldroyd- system avoiding using the losing a priori estimates. The main purpose of this paper is to extend the method in [7] to micro macro models and provide a new proof for global well-posedness of the co-rotational FENE dumbbell model.) and the coupling Smoluchowski and incompressible Navier Stokes equations.3). Compared to the proofs of the theorems in [3,], which are based on the technique of losing a priori estimates, ours are direct and much simpler. For the co-rotational FENE model.), we have Theorem.. Assume that u H s R )s > ), ψ H s R ; L r L ) for some r such that r )k > with ψ dr =, a.e. in x. Then there exists a unique global solution u,ψ) of the FENE model.) in C[, ); H s ) C[, ); H s R ; L )). oreover, u L loc, ; H s+ ) and ψ L loc, ; H s R ; L,r )). For the definition of L r and L,r, please refer to Section. Similarly, for the coupling Smoluchowski and Navier Stokes system.3), we have Theorem.. Take u W +ɛ,r R ) L R ) and ψ W,r R, H s )), for some r >, ɛ >,s> d + and ψ. ψ dm L R ) L R ).Then.3) has a global solution in u L loc, ; W,r ) L loc, ; W,r ) and ψ L loc, ; W,r R ; H s ))). oreover, for T > T >, we have u L T, T ); W ɛ,r ). We end this introduction by mentioning some other results on micro macro models. Global existence of weak solutions can be found in [4,] and local existence of strong solutions are studied in [,3,5]. For macroscopic models, we refer the readers to [4,5,8,9,6,6] as references. The paper is organized as follows. In Section, we give some preliminaries. Then we give some a priori estimates for the FENE model.) in Section 3 and the coupling Smoluchowski and Navier Stokes equations.3) in Section 4. The a priori estimates obtained in Sections 3 and 4 are enough to get the global existence of systems.) and.3) and prove Theorems. and. [3,].. Definitions and useful lemmas We will use the Littlewood Paley decomposition in the following sections. Define C to be the ring { C = ξ R : 3 4 ξ 8 }, 3 and define D to be the ball { D = ξ R : ξ 4 }. 3

4 386 Z. Lei, Y. Wang / J. Differential Equations 5 ) Let χ and ϕ be two smooth nonnegative radial functions supported respectively in D and C, such that χξ) + q ϕ q ξ ) = forξ R, and ϕ q ξ ) = forξ R \{}. q Z Let us denote by F the Fourier transform on R and denote h = F ϕ, h = F χ. The frequency localization operator is defined by q u = F [ ϕ q ξ ) Fu) ] = q ) h q y ux y) dy, R and S q u = F [ χ q ξ ) Fu) ] = q R h ) q y ux y) dy. Hence, for s < /p or s = /p and r =, the homogeneous esov space Ḃ s p,r closure of compactly supported smooth functions under the norm Ḃs, p,r is defined as the u Ḃs p,r = ) qs q u L p q Z lr Z), when p = r =, s = k +α where k N and α, ), thenḃ s p,r turns to be the homogeneous Hölder space Ċ k+α. Another kind of space to be used is L p t, t ; Ċ r ), which is the space of distributions u such that For the FENE model, let u L p t,t ;Ċ r ) sup qr q u L p t,t ;L ) <. q Z R) = e UR) e UR) dr = c R ) k, where the constant c isgivensuchthat dr =. In fact, u,ψ)=, ) defines a stationary solution of.). For r, denote L r and L,r the weighted spaces L r = {ψ: ψ rl r = { L,r = ψ L r : ψ r L =,r ψ We will need to use the following well-known inequalities. r } dr <, ) ψ r } R dr <.

5 Z. Lei, Y. Wang / J. Differential Equations 5 ) Lemma. ernstein inequalities). See [5].) For s R, p r and q Z,onehas q u Lr R d ) d p r )q q u L p R d ), c qs q u L p s q u L p qs q u L p, s S q u L p qs u L p, ce Cqt q u L e t q u L e cqt q u L. Here C and c are positive constants independent of s, p and q. We will also need the following lemma, whose proof can be found in [7]. Lemma.. Assume that β>. Then there exists a positive constant C > such that T gs, ) L ds t + T t gs, ) L ds + sup q T t q gs, ) L ds ln e + T t )) gs, ) Ċ β ds. 3. Proof of Theorem. Since local existence of smooth solutions has been derived by N. asmoudi [3], here we only focus on the a priori estimate which is sufficient for proving Theorem.. As explained in [] or [3], to get the global existence we just need to control the L -norm of u, i.e., u L.Definetheflow associated with u by Φt, x), which means Φ satisfies the ODEs, { t Φt, x) = u t,φt, x) ), Φt =, x) = x. 3.) Step I: Uniform estimates for ψ and τ with respect to t. Due to the special structure of the equation about ψt, x, R), we have the following bounded estimates of ψ. For r >, multiplying the third equation of.) by r ψ r ψ ψ, and integrating over, thenweget t ψ r dr + u ψ r 4r ) dr = r ) r ψ R dr. Therefore, ψ r ) t,φt, x), R dr r ψ x, R) dr. Since the flow is incompressible, then ψ L t,x L r ) ψ L x L r ), ψ L t L xl r )) ψ L x L r ). 3.)

6 388 Z. Lei, Y. Wang / J. Differential Equations 5 ) To estimate τ, we need a lemma. Lemma 3.. For any p such that pk >, it holds that ψ dr R ψ p+ ψ p ) p+ = ψ L p+. Proof. y the Hölder s inequality, ψ dr R ψ R ) kp/p+) ψ p/p+) dr R ) +/p k ) p p+ ψ p+ ψ p dr ) p+. Since pk >, the result follows. Noting that τ t, x) = R i R j U)ψt, x, R) dr, one has τ t, x) R U ψt, x, R) dr k R R ψt, x, R) dr ψt, x, R) dr R ψt, x, ) L r, where in the last step we used Lemma 3.. Hence, we have τ L,T ;L ) ψ L x L r ), τ L,T ;L ) ψ L x L r ). 3.3) Step II: A priori estimates for u. We need a useful lemma whose proof was established by Chemin and asmoudi [7] see also [7]). Lemma 3. Chemin asmoudi). Let v be a solution of the Navier Stokes equations with initial data v L R ), and an external force f L, T ; Ċ ) L, T ; Ḣ ):

7 Z. Lei, Y. Wang / J. Differential Equations 5 ) t v v + v )v + p = f, in R, T ), div v =, in R, T ), 3.4) vt =, x) = v x), in R. Then we have the following a priori estimates, v L,T ;L ) + v L,T ;L ) v L + f L,T ;Ḣ ), and v L,T ;Ċ ) { sup C q v L }) exp c q T + ) v L + f L,T ;Ḣ ) v L q,t ;L ) + sup q T q q f s) ) L ds. Furthermore, if f L, T ; Ċ ),then ɛ >, there exists t ɛ), T ) such that v L t,t ;Ċ ) ɛ. Particularly for our problem, since we have shown that τ L, T ; L ) L, T ; L ), applying Lemma 3., we know that and u L, T ; L ) L, T ; Ḣ ) L, T ; Ċ ) ɛ >, t ɛ), T ), such that u L t,t ;Ċ ) ɛ. 3.5) Step III: Hölder estimates for u. For t < T, choose some α satisfying < α < min {s, }, define N r q t, x) = q ψt, x, R) At) = sup us, ) Ċ +α, s<t Dt) = sup s<t q Z r dr = q ψ r Lr t, x), t) = sup τ s, ) Ċα, s<t sup αq Nq s, ) L. Here q is the frequency operator with respect to x. efore the detailed estimates, we will introduce an inequality for later use, which can be considered as an extension of Hölder inequality. Lemma 3.3. For any u L 4 R ) Ċ +α R ), there holds that u u Ċ +α u L 4 u Ċ +α, with some constant C independent of u.

8 38 Z. Lei, Y. Wang / J. Differential Equations 5 ) Proof. For any q Z, using ony s para-product decomposition [], we have q u u) L +α)q p q 5 + q S p u p u) L +α)q p q 3 p r I + I. q p u r u) L +α)q I can be estimated as I p q 5 p q 5 p q 5 u L 4 u Ċ +α, here the first inequality is due to Lemma.. While I can be estimated as I = p q 3 p r p q 3 p r p q 3 S p u p u L 4 +α)q S p u L 4 p u L +α)q u L 4 +α)p p u L +α)q p) u Ċ +α u L 4, q p u r u) L +α)q p u L r u L 4 +α)q r +α)p p u L u L 4 +α)q p) here in the second inequality we also used Lemma.. These above estimates complete the proof of Lemma 3.3. First, applying q to the first equation of the FENE system, then we obtain t q u ν q u + q p = q τ u u), hence t q u = e νt q u + e νt s) P q τ u u) ) ds, where P is the Helmoltz Weyl projection operator.

9 Z. Lei, Y. Wang / J. Differential Equations 5 ) q+α) q u L t) t e cνqt q u L q+α) + C e cνq t s) q+α) q τ u u) L ds t u Ċ +α + C e cq t s) q+α) q τ L + q u u) L ) ds. 3.6) Applying Lemma., we obtain t t e cq t s) q+α) q τ L ds e cq t s) q+α) q q τ L ds t e cq t s) q αq q τ L ds t). On the other hand, applying Lemma. again, we obtain t q+α) e cq t s) q u u) L ds t e cq t s) q+α) q u u) L ds t e cq t s) 3 q u u Ċ +α ds t t t e cq t s) 3 q u L 4 u Ċ +α ds u 4 L 4 u 4 Ċ +α ds ) 4 t u L u L u 4 Ċ +α ds ) 4 e cq t s) q ds, ) 3 4 where we used Lemma 3.3 and interpolation inequality for the third and last inequality respectively. Taking the supreme of both sides of 3.6) with respect to q, onegets ut) 4 Ċ +α u Ċ +α + t) ) 4 + C t us) L us) L us) 4 Ċ +α ds ).

10 38 Z. Lei, Y. Wang / J. Differential Equations 5 ) y the Gronwall s inequality and the estimates in Step II, we have At) u Ċ +α + t) ) + t) ). 3.7) As a result of the relationship between τ and ψ and Lemma 3., t) = sup s<t sup s<t Combining the above inequality and 3.7), sup αq q τ s, ) L q sup αq q q ψ R U dr sup sup αq q ψ L x L r )s) s<t q = CDt). L At) + Dt) ). 3.8) Step IV: Hölder estimates of ψ. The remaining part is to estimate the Ċ α -norm of ψ. Take the operator q to the third equation, then t q ψ + u q ψ = div R W u) R q ψ ) )) q ψ + div R R ultiply 3.9) by r qψ r q ψ and integrate over, + div R [ q, W u) ] Rψ ) +[u, q ]ψ. 3.9) t N r q + u Nr q 4r ) + r [ div R q, W u) ] Rψ ) J + J. ) r/ q ψ R dr q ψ r q ψ rdr+ [u, q ]ψ q ψ r rdr y the Young s inequality and the Hölder inequality, Hence J C [ q, W u) ] Rψ L r Nq r + r r J [u, q ]ψ L r N r q. ) r/ q ψ R dr, t Nq r + u Nr q [ q, W u) ] Rψ L r Nq r + C[u, q ]ψ L r Nq r,

11 Z. Lei, Y. Wang / J. Differential Equations 5 ) which implies that αqr N q r L t) αqr t + C αqr [ q, W u) ] Rψ L x L r ) N q r L s) ds t [u, q ]ψ L x L r ) N q r L s) ds. 3.) Note that [ q, W u) ] Rψ [ = hy) )] W u)x) W u) x q y Rψ x )) q y dy R αq W u) Ċα R hy) Rψ x q y ) dy. Then we get that [ q, W u) ] Rψ L x L r ) u Ċ α ψ L x L r ) αq u Ċα αq. And by the ony s para-product formula, [u, q ]ψ p q 3 p q + q q 5 [ p u, q ] q ψ + [ q u, q ]S q ψ q q 5 [S q u, q ] q ψ with each term having the following estimate: Since [ p u, q ] q ψs, x, R) [ = hy) p us, x) p u s, x )] q y q ψ s, x q y, ) R dy, R and q q ψx) = ) h q 3q y ψx y) dy = ) h q y ψx y) dy R R then

12 384 Z. Lei, Y. Wang / J. Differential Equations 5 ) αq [ p u, q ] q ψ L x L r ) s) p q 3 p q αq q p u L s) q ψ L x L r )s) p q 3 p q αq q p u L s) q ψ L x L r )s) p q 3 p q αp p u L s) αq p) ψ L x L r )s) p q 3 As) ψ L x L r )s) As). Similarly, αq [S q u, q ] q ψ L x L r ) s) q q 5 αq q S q u L s) q ψ L x L r )s) q q 5 q q 5 u L s) αq q ψ L x L r )s) u L s) Ds), αq [ q u, q ]S q ψ L x L r ) s) q q 5 αq q q u L s) S q ψ L x L r )s) q q 5 As) ψ L x L r )s) As). Therefore, taking the supreme of 3.) with respect to q, by the Young s inequality and 3.8), t Dt) r C [ As) r + Ds) r] ds + C t u L s)ds) r ds t C [ + Ds) r] + C u L Ds) r ds. Then the Gronwall s inequality implies that Dt) D) + ) e C t u L +) ds.

13 Z. Lei, Y. Wang / J. Differential Equations 5 ) Hence according to Lemma., e + Dt) D) + ) C exp { C C D) + ) exp {C t t u L + ) ds + ) { [ )]} C D) + exp C + ɛ ln e + Dt) ) ) Cɛ, C D) + e + Dt) t t } u L ds + ɛ ln e + t t )At) ))} where C is some positive constant depending on the solution u on [, t ]. Choosing ɛ = C, DT ) [ CC D) + )]. Then by 3.8), AT ) is bounded, which implies that u L is bounded on [, T ] since u L u L + u Ċ +α ). 4. Proof of Theorem. The proof of Theorem. is similar to that of Theorem., we just give the sketch. As above, to get the global existence we only need to control u L,see[]. Step I: Uniform estimates for ψ and τ. Integrate the third equation of.3) on, then t ψt, x,m) dm + u ψt, x,m) dm =. y the maximum principle for evolutionary equation, ψ is always nonnegative. Hence ψ L x L x L )) t) = ψ L x L x L )), 4.) τ L,T ;L ) ψ L 4 x L )) + ψ L x L ))), 4.) τ L,T ;L ) ψ L x L )) + ). 4.3) Since s > d + and is a smooth compact manifold without boundary, ψ L t,x H s )) ψ L x L )). 4.4) Step II: A priori estimates for u. Applying Lemma 3. and the estimates 4.), 4.3), we get that u L, T ; L ) L, T ; Ḣ ) L, T ; Ċ ) and ɛ >, there exists t ɛ), T ), such that u L t,t ;Ċ ) ɛ.

14 386 Z. Lei, Y. Wang / J. Differential Equations 5 ) Step III: Hölder estimates for u. Denote H = g + I) s, N q t, x) = At) = sup us, ) Ċ +α, s<t Dt) = sup s<t q Z H q ψt, x,m) dm, t) = sup τ s, ) Ċα, s<t sup αq Nq s, ) L. As in Section 3, we have the estimates At) u Ċ +α + t) ) + t) ), and by 4.4), q Z, Nq t, ) L Hψt,, ) L x L )) ψ L x L )), αq q τ s, x) L αq Hm H + αq αq p q 5 + αq H γ ) p q 5 + αq m γ ) q Hm ψs, x,m )H m ψs, x,m ) ) dm dm L m)h q ψs, x,m) dm Hm Hm γ ) Hm Hm γ ) p q 3 p r + C αq Nq s, ) L L m,m )S p Hψm ) p Hψm ) dm dm L Hm Hm γ ) Ds) Hψ L x L ))s) + CDs) Ds) m,m ) p Hψm )S p Hψm ) dm dm L p Hψm ) r Hψm ) dm dm L which implies that t) Dt). Therefore, At) + Dt) ). 4.5)

15 Z. Lei, Y. Wang / J. Differential Equations 5 ) Step IV: Hölder estimates for ψ. Take the operator H and q to the third equation, multiply by q Hψ and integrate over, then t q Hψ dm + u q Hψ dm + g q Hψ dm ) = [u, q ]Hψ q Hψ dm q H div g Gu,ψ)ψ q Hψ dm = [u, q ]Hψ q Hψ dm j u i + [ j u i, q ]H c αψ ) g q Hψ dm + [ + q H g U, H ] Hψ g q Hψ dm. H div g c α q ψ ) H q ψ dm q g U Hψ) g q Hψ dm y the Young s inequality, we have q Hψ L x L )) t) t [u, q ]Hψ q Hψ dm ds + L t ju i H div g c α q ψ ) q Hψ dm ds L t t t [ j u i, q ]H cαψ ) L x L )) ds [ q H g U, H ] Hψ ) L x L )) ds t q g U Hψ) L x L )) ds J + J + J 3 + J 4 + J 5 ) ds. 4.6) As in Section 3, applying 4.5), for every q Z, αq J s) As) Hψ L x L )) Ds) + C u L Ds) Ds) + ) u L s) + ). J, J 4 and J 5 are estimated as in [9],

16 388 Z. Lei, Y. Wang / J. Differential Equations 5 ) H div g c α q ψ ) q Hψ dm div g c α H q ψ ) q Hψ dm + [ H divg cα, H ] H q ψ ) q Hψ dm divg cα) q Hψ dm + [ H divg cα, H ] H q ψ ) q Hψ dm q Hψ L ), which deduces that Using ony s decomposition, αq J s) αq u L s) N q s) u L s) Ds). αq q g U Hψ) L x L )) αq S p g U p Hψ L x L )) + + p q 5 p q 3 p r p g U r Hψ L x L )) g U L x L )) sup αp p Hψ L x L )) p p q 5 + C sup αp p g U L x L )) Hψ L x L )). p Combining the relationship between U and ψ, p g U S p Hψ L x L )) αq J 4 s) αq Hψ L x L )) s) N q s) Ds). Similarly, αq J 5 s) Hψ L s) x L sup αp N )) p s) Ds). p Since [ j u i, q ]H cαψ ) = R [ hy) j u i x) )] j u i x q y H cαψ ) x ) q y,m dy, αq J 3 s) αq αq u Ċ α s) H c αψ ) L x L )) s) As) [ c α Hψ L x L )) + [ H c α, H ] Hψ ] L x L )) As) Hψ L x L )) As).

17 Z. Lei, Y. Wang / J. Differential Equations 5 ) Therefore, taking the supreme of 4.6) αq with respect to t and q, Dt) t u L + ) + Ds) ) ds. The remaining proof is just the same as that in Section 3. We omit the details. Acknowledgments The work was in part supported by NSFC grants Nos. 89 and 9384), FANEDD, Shanghai Rising Star Program QA43) and SGST 9DZ79. Part of the work was done when Zhen Lei was visiting the Institute of athematical Sciences of CUHK. Zhen Lei would like to thank the hospitality of Professor Zhouping Xin and the Institute. References [] J.. ony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. 4) 4 ) 98) [] H. ahouri, J.-Y. Chemin, Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides Transport equations for non-lipschitz vector fields and fluid mechanics), Arch. Ration. ech. Anal ) 59 8 in French). [3] R.. ird, C.F. Curtis, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, vol., Kinetic Theory, Weiley Interscience, New York, 987. [4] J.W. arrett, C. Schwab, E. Süli, Existence of global weak solutions for some polymeric flow models, ath. odels ethods Appl. Sci. 5 6) 5) [5] J.Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Ser. ath. Appl., vol. 4, Clarendon Press/Oxford University Press, New York, 998. [6] Y. Chen, P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in and 3 space dimensions, Comm. Partial Differential Equations 3 ) 6) [7] J.Y. Chemin, N. asmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIA J. ath. Anal. 33 ) ) 84. [8] P. Constantin, Nonlinear Fokker Planck Navier Stokes systems, Commun. ath. Sci. 3 4) 5) [9] P. Constantin, C. Fefferman, E. Titi, A. Zarnescu, Regularity for coupled two-dimensional nonlinear Fokker Planck and Navier Stokes system, Comm. ath. Phys. 7 7) [] P. Constantin, N. asmoudi, Global well-posedness for a Smoluchowski equation coupled with Navier Stokes Equations in D, Comm. ath. Phys. 78 8) []. Doi, S.F. Edwards, The Theory of Polymeric Dynamics, Oxford University Press, Oxford, UK, 986. [] W.N. E, T.J. Li, P.W. Zhang, Well-posedness for the dumbbell model of polymeric fluids, Comm. ath. Phys. 48 ) 4) [3]. Jourdain, T. Lelièvre, C. Le ris, Existence of solutions for a micro macro model of polymeric fluid: the FENE model, J. Funct. Anal. 9 ) 4) [4] Z. Lei, Global existence of classical solutions for some Oldroyd- model via the incompressible limit, Chinese Ann. ath. Ser. 7 5) 6) [5] Z. Lei, On D viscoelasticity with small strain, Arch. Ration. ech. Anal. 98 ) ) [6] Z. Lei, C. Liu, Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. ech. Anal. 88 3) 8) [7] Z. Lei, N. asmoudi, Y. Zhou, Remarks on the blowup criteria for Oldroyd models, J. Differential Equations 48 ) ) [8] Z. Lei, Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit, SIA J. ath. Anal. 37 3) 5) [9] F.H. Lin, C. Liu, P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. ath. 58 ) 5) [] F.H. Lin, C. Liu, P. Zhang, On a micro macro model for polymeric fluids near equilibrium, Comm. Pure Appl. ath. 6 6) 7) [] P.L. Lions, N. asmoudi, Global existence of weak solutions to micro macro models, C. R. ath. Acad. Sci. Paris 345 ) 7) 5. [] F.H. Lin, P. Zhang, Z.F. Zhang, On the global existence of smooth solutions to the -d FENE dumbbell model, Comm. ath. Phys. 77 ) 8)

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