Nonlinear Fokker-Planck Navier-Stokes Systems

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1 Nonlinear Fokker-Planck Navier-Stokes Systems Peter Constantin Department of athematics, The University of Chicago, Chicago, Il September 27, 2005 Abstract We consier Navier-Stokes equations couple to nonlinear Fokker- Planck equations escribing the probability istribution of particles interacting with fluis. We escribe relations etermining the coefficients of the stresses ae in the flui by the particles. These relations link the ae stresses to the kinematic effect of the flui s velocity on particles an to the inter-particle interaction potential. In equations of type I, where the ae stresses epen linearly on the particle istribution ensity, energy balance requires a response potential. In equations of type II, where the ae stresses epen quaratically on the particle istribution, energy balance can be achieve without a ynamic response potential. In unforce energetically balance equations, all the steay solutions have flui at rest an particle istributions obeying an uncouple Onsager equation. Systems of equations of type II have global smooth solutions if inertia is neglecte. Key Wors: Fokker-Planck equations, Navier-Stokes equations, Smoluchowski equations, microscopic inclusions. AS classification numbers:35q30, 82C31,76A05. 1 Introuction Fluis with complex rheological properties are of great scientific interest an have a rich phenomenology. Their mathematical escription remains 1

2 challenging. With the exception of ([20]) most of the progress is rather recent. Global existence of weak solutions, with mollyfie velocities an linear Fokker-Planck equations with aitional bounary conitions has been obtaine in ([1]). Global existence for shear flow Hookean umbell moels was prove in ([11]). The local existence of various systems has been obtaine ([5], [12], [17]). Global existence for small ata for linear Fokker-Planck couple with Navier-Stokes equations was obtaine in [14]. The recent work ([19]) is the only global regularity result for large ata that I am aware of. It applies to the case of couple linear Fokker-Planck an Stokes system. The stuy of time asymptotics is also in its early evelopment stage. High intensity asymptotics for uncouple Smoluchowski equations have been stuie in ([2], [6], [15], [16]). The long time effects of shear in Doi-Smoluchowski equations have been investigate in ([7], [8]). The long time asymptotics of couple systems using entropy methos has been stuie in ([13]). In this paper we consier a complex flui with microscopic inclusions. The flui is governe by the incompressible Navier-Stokes equation, an the microscopic insertions influence the flui through an ae macroscropic stress. The flui s velocity u(x, t) obeys thus the three imensional Navier- Stokes equations { x u = 0, t u + u x u ν x u + x p = iv x σ + F Derivatives with respect to coorinates x R 3 will be inicate by a subscript x. F (x, t) are given smooth boy forces, ν > 0 is the kinematic viscosity. The expression iv x σ, (1) iv x σ = x σ = 3 j=1 σ ij x j, represents the forces ue to the presence of microscopic insertions: the tensor σ ij (x, t) is the aitional stress in the flui. The insertions are objects parameterize by a microscopic variable m which belongs to a connecte smooth Riemannian manifol of imension. In the case of ro-like particles, is the unit sphere in three imensions, an m R 3 represents the irector of the ros. ore complicate particles require more egrees of freeom for the configuration space. For instance, articulate ros with several articulations, require a phase space which is a prouct of spheres. 2

3 We will use local coorinates m = m(φ) with φ = (ϕ 1,..., ϕ ) R. In the paper we use the fact that a smooth Riemannian metric g ij exists on. As it is customary g ij will enote the inverse of g ij ([9]). Compactness of will also be assume, for the sake of simplicity, although a non-compact manifol coul be allowe as well. (In the case of non-compact some of the assumptions on the coefficients woul have to be augmente an some arguments woul have to be moifie, mostly technical changes). Derivatives with respect to the microscopic variable are esignate by the subscript g: g h = ( ϕ1 h,... ϕ h). When f, h are scalars an V is a (0, 1) tensor on we efine iv g (V f) = 1 g α,β=1 ϕα ( gg αβ V β f ) an recall that the volume element on is locally the Laplace-Beltrami operator is m = gφ, g h = iv g g h an hiv g (V f)m = f(v g h)m hols for smooth functions where V g h = g αβ V β h ϕ α. We will attempt to istinguish between Greek alphabet inices α, β,... running from 1 to an relate to the microscopic variables m, an Roman alphabet inices i, j,... relate to the spatial variable x an running from 1 to 3. One of the funamental assumptions of the current literature on the subject is that the ae stresses σ(x, t) o not epen explicitly on the microscopic variable m. Consequently, the velocity u = u(x, t) oes not epen on the variable m. The microscopic insertions at time t an macroscopic place x are escribe by the probability f(x, m, t)m. The suspension stress tensor is then given by an expansion σ(x, t) = σ (1) (x, t) + σ (2) (x, t) (2) 3

4 where an σ (2) ij (x, t) = σ (1) ij (x, t) = γ (1) ij (m)f(x, m, t)m (3) γ (2) ij (m, n)f(x, m, t)f(x, n, t)mn. (4) This, an more general expansions for σ are encountere in the polymer literature ([4]). The structure coefficients in the expansion, γ (1) ij, γ(2) ij are smooth, time inepenent, x inepenent, an o not epen on f. In this work we will use moels in which the particles interact through potentials that epen linearly an nonlocally on the particle ensity istribution f ([18]). Because of this, an expansion (2) with only two terms is sufficient. One coul consier interaction potentials that epen nonlinearly on f an with them, higher nonlinear epenence of σ on f, as well as explicit epenence of σ on x u. In the ilute cases, when the interaction of particles is moelle by an f-inepenent (zeroth power) expression, as in the umbell, FENE an polynomial force moels, then the customary expression for σ is linear in f, an is given for instance, for ro-like particles by ( σ (1) kt ij (x, t) = m i m j δ ) ij f(x, m, t)m (5) 4π 3 where kt is an energy scale associate to the microscopic suspension. oelling the ae stresses is a complicate task. The purpose of this work is twofol. First, we iscuss moels from an energy principle point of view, an we reveal necessary relationships ictate by energy requirements. We istinguish between the two types of relationship between σ an f. We refer to the case in which σ (2) = 0, so that σ epens linearly on f as equations of type I, an to the case in which σ (2) 0 as equations of type II. The secon purpose of the article is to prove a global existence result for equations of type II. The evolution of the ensity f is governe by a nonlinear Fokker-Planck equation t f + u x f + iv g (Gf) = ɛ g f (6) Here ɛ 0 is an inverse time scale associate to iffusion of the microscopic particles. The tensor G is mae of two parts G = g U + W, (7) 4

5 The (0, 1) tensor fiel W is obtaine from the macroscopic graient of velocity in a linear smooth fashion, given locally as ( 3 ) W (x, m, t) = (W α (x, m, t)) α=1,..., = c ij α (m) u i (x, t) (8) x j i,j=1 α=1,...,. The smooth coefficients c ij α (m) o not epen on the solution, time or x an, like the coefficients γ (1) ij, γ(2) ij are a constitutive part of the moel. In the case of ro-like particles W is the projection of x u(x, t)m onto the tangent plane to at m: W (x, m, t) = ( x u(x, t))m (( x u(x, t))m m)m. (9) This etermines the coefficients c ij α (m) for this case. The fiel W represents the rotation an stretching of microscopic insertions ue to macroscopic flow; in jargon, this is terme a macro-micro interaction. In the same vein, σ is a micro-macro interaction. There are two types of potentials. The nonlocal microscopic interaction potential is ([18]) (Kf) (x, m, t) = K(m, q)f(x, q, t)q. (10) Here K(m, q) is a smooth, time inepenent, x inepenent, symmetric function K : R. This is yet another constitutive parameter in the system; the potential Kf represents a micro-micro interaction. The moel for the total potential U epens on equation type. In orer to have an energy principle, even in the absence of interparticle interactions (K = 0), it turns out that the coefficients γ (1) ij are etermine by the coefficients c ij α. Therefore, if the macro-micro effect W is impose by kinematic physical consierations, (as is the case in (9)), then a linear constitutive form σ (1) of the ae stresses is ictate by energetics requirements alone, irrespective of particle interaction potentials. In orer to restore an energy balance for equations of type I, in the presence of interparticle interactions, the total potential U is given by U(x, m, t) = 1 τ ((Kf)(x, m, t) + δv (x, m, t)) (11) where τ is a time scale associate with the microscopic interactions an the term δv obeys the equation t δv + u x δv + W g δv = W g (Kf). (12) 5

6 This potential is a macro-micro term, an is evolutionary: It arises in response to the micro-micro interactions, an plays an absolutely crucial role in the energetics of type I equations. Because of (11) an (12) it follows that t U + u x U + W g U = 1 τ D t(kf). (13) This shows that the potential U evolves incorporating both macroscopic an microscopic effects. We enote a 1 ɛτ = a 1 the nonimensional ratio of microscopic time scales an enote by V V = Kf + δv (14) the nonimensional total potential, so that U ɛ = av. In the case of type II equations one can satisfy the energetic requirements in the presence of interparticle interactions by relating the coefficients γ (2) to K. 2 Energetics For Type I Equations We start with a concrete example: ro-like particles with = with σ = σ (1) given in (5). Then the system has a Lyapunov functional, the energy E = 1 u(x, t) 2 x + F (15) 2 with free energy F given by F = kt 20π { f log f a (δv + 12 fkf )} mx. (16) Theorem 1 Let (u, f, δv ) be a smooth solution of the system (1, 6, 12) with constitutive relations (5, 7, 9, 10, 11) an F = 0. Then E(t) = ν x u(x, t) 2 x kt ɛ f g (log f av ) 2 mx (17) t 20π 6

7 Proof. The ientity iv g W (x, m, t) = 5( x u(x, t))m m (18) follows from the efinition (9). Using the efinition (5), it follows that 20π f(x, m, t)iv g W (x, m, t)mx = u(x, t) x σ(x, t)x (19) kt ultiplying the equation (1) by u, integrating by parts an using the ientity (19) we obtain the Navier-Stokes energy balance u 2 x + ν x u(x, t) 2 x = kt (fiv g W )mx (20) 2t 20π Let us enote now an E(t) = D t = t + u x ( ɛf log f 1 τ fδv 1 ) 2τ fkf mx. (21) Note that F = kt 20πɛ E (22) Using the fact that D t fmx = 0, the fact that x u = 0, the relation (11), the fact that K is a symmetric operator an that D t commutes with K we have that { t E = (D t f) (ɛ log f U) fd t U + 1 } τ fd tkf mx an thus, in view of (13) t E = {(D t f) (ɛ log f U) + fw g U} mx. The equation (6) has the structure D t f = iv g (f( g (ɛ log f U) W )). 7

8 Using it, we have t E = + f g (ɛ log f U) 2 mx+ {iv g (UfW ) ɛiv g (fw ) log f} mx Therefore t E = f g (ɛ log f U) 2 mx ɛ (fiv g W )mx (23) ultiplying by kt an using U 20πɛ ɛ = av we obtain t F = kt ɛ f g (log f av ) 2 mx kt (fiv g W )mx 20π 20π (24) Aing to (20) we finish the proof. Note that when F = 0 the only steay solutions have no flow, (u = 0 or any constant, by Galilean invariance), an solve Onsager s equation ([18], [2]) log f = av log Z (25) with Z a constant serving as normalizing factor. Let us use the calculation above as a guie to unerstan the energetics for more general type I equations. Assume thus that σ = σ (1) with σ (1) given in (3). Then the Navier-Stokes energy balance is, after one integration by parts, u 2 x + ν x u 2 x = 2t ( 3 ) u i (x, t)γ (1) ij x (m) f(x, m, t)xm. (26) j i,j=1 This balance occurs no matter what is the equation for f. Let us assume first that the equation for f is the simplest possible, with only a macro-micro interaction W, D t f = ɛ g f iv g (W f) 8

9 with W given by (8). Then the evolution of the entropy is f log fmx = ɛ g f 2 f 1 mx (iv g W )fmx t Because iv g W = 3 ij=1 iv g (c ij ) u i x j (27) we see that the system has a Lyapunov functional of the form kinetic energy + λ entropy if an only if there exist constants λ an µ so that γ (1) ij = λiv g c ij + µδ ij (28) hols. This happens in the case of rigi ros where (9) prouces the balance (18) which is of the form above. Thus, energy requirements impose that the coefficients of (8) etermine those of (3). Once the relation (28) has been establishe then the general type I calculation is very similar to the the concrete example an gives: Theorem 2 Type I equations (1, 6, 12), with constitutive equations (2, 3, 7, 8, 10, 11), with F = 0, σ (2) = 0 an (28), have a Lyapunov functional E = 1 { u 2 x + λ f log f a (δv + 12 ) } 2 Kf f mx. (29) If u, f, δv is a smooth solution then E t = ν x u 2 x λɛ f g (log f av ) 2 mx (30) hols. If the smooth solution is time inepenent, then u = 0, δv = δv t=0 an f solves the Onsager equation with an appropriate normalizing constant Z. log f = av + log(z 1 ) (31) 9

10 Proof. We form the integral E as in (21): E(t) = ( ɛf log f 1 τ fδv 1 2τ fkf ) mx. (32) The calculation leaing to (23) can be repeate verbatim an leas to t E = f g (ɛ log f U) 2 mx ɛ (fiv g W )mx. (33) Using (27), (28 ) an the fact that ( x u) is traceless we have λfiv g W = f 3 ij=1 γ (1) ij (m) u i x j. (34) We obtain (30) by multiplying (33) by λɛ 1 an aing to (26). 3 Energetics For Type II Equations In this case the energy balance in the Navier-Stokes equations is moifie because of the presence of the terms in σ (2). The balance (26) becomes u 2 x + ν x u 2 x = 2t ( 3 ) u i (x, t)γ (1) ij x (m) f(x, m, t)xm j ( 3 i,j=1 i,j=1 u i x j (x, t)γ (2) ij (m, n) ) f(x, m, t)f(x, n, t)xmn (35) This balance hols in a type II equation, irrespective of the form of the equation obeye by f. If this equation is of the form D t f = ɛiv g (f g (log f akf)) iv g (W f) (36) (i.e. δv = 0), then one can maintain energy balance if (28) hols, an if there exists a constant η so that γ (2) ij (m, n) = aλcij α (m)g αβ (m) β K(m, n) + ηδ ij (37) 10

11 Theorem 3 Type II equations (1, 36) with constitutive equations (2, 3, 4, 8, 10), with F = 0, (28, 37) have a Lyapunov functional E(t) = 1 u 2 x + λ {f log f a } 2 2 (Kf)f xm. (38) If (u, f) is a smooth solution then E t = ν x u 2 x λɛ f g (log f akf) 2 mx. (39) If the smooth solution is time inepenent, then u = 0 an f solves the Onsager equation log f = akf + log(z 1 ) (40) with an appropriate normalizing constant Z. Proof. Using (36) we have {f log f a } t 2 (Kf)f mx = ɛ + f g (log f akf) 2 mx {aw g (Kf) iv g W } fmx (41) The conition (37) implies that λaw g (Kf) = 3 ij=1 u i x j γ (2) ij (m, n)f(x, n, t)n (42) ultiplying (41) by λ, aing to (35) an using (42) an (34) finishes the proof. Remark. We note that, for type II equations, knowlege of the micromicro potential K an of the nature of the macro-micro effect (8) i.e., the coefficients c ij α, couple with energetic balance, etermine the coefficients γ (1) ij an γ (2) ij. 11

12 4 A Global Regularity Result For Type II Equations We consier a nonlinear Fokker-Planck system t f + u x f + iv g (W f) + 1 τ iv g(f g (Kf)) = ɛ g f (43) where W is given in (8) an K in (10). We use the time scales τ an ɛ 1 separately, so that we can state a theorem that allows for the limit cases ɛ = 0, τ =. The velocity is relate to f via the Stokes equations: ν x u + x p = iv x σ + F, x u = 0. (44) The ae stresses are given by the type II relation (2, 3, 4). perioic bounary conitions for the Stokes equations. We take Theorem 4 Let ɛ 0, τ (0, ]. If the initial istribution f(x, m, 0) is smooth, positive an normalize, f 0 (x, m)m = 1, an the initial velocity is smooth, then the system (43), (44), with constitutive relations (8), (10), (2, 3, 4) has global smooth solutions. This theorem was prove in ([19]) for ɛ > 0, τ =, = using the nonlinear issipative structure of the Fokker-Planck equation. Our proof oes not use this issipation. Proof. We consier the L 2 () selfajoint pseuoifferential operator ([10]) R = ( g + I) s 2 (45) with s > + 1. We will use the following properties of R: 2 [R, x ] = 0, (46) R g : L 1 () L 2 () is boune, (47) R g : L 2 () L () is boune, (48) 12

13 [ g c, R 1 ] : H s () L 2 () is boune, (49) for any smooth function c : R, an R : L 2 () H s () is boune. (50) We ifferentiate (43) with respect to x, apply R, multiply by R x f an integrate on. Let us enote by N(x, t) 2 = R x f(x, m, t) 2 m (51) the square of the L 2 norm of R x f on the unit sphere. The following lemma is the main tool for regularity: Lemma 1 Let u(x, t) be a smooth, ivergence-free function an let f solve (43). There exists an absolute constant c > 0 (epening only on imensions of space, the coefficients c ij α an, but not on u, f, ɛ, τ) so that hols pointwise in (x, t). ( t + u x ) N c( x u + 1 τ )N + c x x u (52) Proof of Lemma 1. The normalization f(x, m, t)m = 1 an smoothness of f(x, t) are easy to prove, an we will omit the proof. The evolution equation of N is with 1 2 ( t + u x ) N 2 = D + I + II + III + IV (53) D = ɛ g R x f 2 m (54) I = u j x k ( R f x j ) ( R f ) m (55) x k 13

14 an II = III = 2 α=1 IV = 1 τ u i ( x ) x j 2 α=1 u i x j (Riv g (c ij α f))( x Rf)m, (56) (Riv g (c ij α x f))(r x f)m, (57) Riv g ( x {f g (Kf)})R x fm. (58) Now we start estimating terms. D 0 will be iscare. Clearly In orer to boun II we use (47) to boun so that we have I c x u N 2. (59) R g (c ij α f) L 2 () c f L 1 () = c II c x x u N. (60) In orer to boun III we nee to use the commutator carefully. We start by writing Riv g (c ij α x f) = Riv g (c ij α R 1 R x f) = The secon term obeys iv g (c ij α R x f) + [ Riv g c ij α, R 1] R x f. [ Riv g c ij α, R 1] R x f L 2 () cn because, in view of (49) an (50) one has that [ Rivg c ij α, R 1] : L 2 () L 2 () is boune. The first term nees to be integrate against R x f an integration by parts gives (iv g (c ij α R x f))r x fm = 1 (iv g c ij α ) R x f 2 m. 2 We obtain thus III c x u N 2 (61) 14

15 The term IV is split in two, IV = A + B A = 1 Riv g ({( x f) g (Kf)})R x fm (62) τ an B = 1 τ Riv g ({f g (K x f)})r x fm. (63) The (0, 1) tensor Φ(x, m, t) = ( g Kf)(x, m, t) is smooth in m for fixe x, t an Φ(x,, t) W s, () c s hols for any s, with c s epening only on the kernel K. We write the term A A = 1 Riv τ g ({( x f)φ})r x fm = 1 R 1 (R τ x f){φ g R 2 x f))m = 1 iv 2τ g {Φ} R x f 2 m + 1 (R τ x f) [R 1, Φ g ] R(R x f)m In view of (49), (50), the operator [ R 1, Φ g ] R : L 2 () L 2 () is boune with norm boune by an a priori constant. It follows that A c τ N 2 (x, t) hols. The term B is easier to boun, because (K x f)(x, m, t) = R 1 K(m, n)r x f(x, n, t)n an thus ( g K x f)(x,, t) L () cn(x, t). Using (47) it follows that B c τ N 2 (x, t) 15

16 an consequently IV c τ N 2 (x, t). (64) Putting together the inequalities (59), (60), (61) an (64) we finish the proof of the lemma. We return to the proof of the theorem. The Stokes system (44) is elliptic, an it is well known that u L (x) C σ L (x) { 1 + log+ x σ L q (x)} + F L q (x) (65) hols for q > 3. Also x x u L q (x) C { } x σ L q (x) + F L q (x) hols for any 1 < q <. Now it follows from (2, 3, 4) that (66) σ(x, t) c (67) hols with a constant that epens only on the coefficients γ (1) ij, γ(2) ij. Differentiating in x (3) an (4) it follows that x σ(x, t) cn(x, t) (68) hols with a constant c that epens only on the smooth coefficients γ (1) ij, γ(2) ij. Inee, in the case of σ (2) for example ( x σ (2) (x, t) = R 1 m γ (2) (m, n) ) (R m x f(x, m, t)) f(x, n, t)mn+ + ( R 1 n γ (2) (m, n) ) (R n x f(x, n, t)) f(x, m, t)mn where we use R m an R n to enote R acting in the m variables or n variables. The inequality (68) follows from the smoothness of the coefficients γ an the positivity an normalization of f. Now the proof follows along the classical lines: we prove easily a local existence an smoothness result of the system. We euce from (52, 65, 66, 67, 68) that, for q > 3, the norm y(t) = N(, t) L q (x) obeys y t c(1 + log + y)y + Cy + with constant c, C,. This proves that this norm is boune apriori by a finite function of time, for all time. Once this is achieve, higher erivatives are controlle inuctively. We omit further etails. 16

17 5 Conclusions Type I nonlinear Fokker-Planck equations (6) couple with Navier-Stokes equations (1) are equations in which the ae stresses σ epen in a linear fashion (3) on the ensity of particles. Such systems are energetically balance if (28) hols an if the ynamical response potential δv obeying (12) is inclue in the system. Type II nonlinear Fokker-Planck equations (43) couple with Navier-Stokes equations are equations in which the ae stresses (2, 3, 4) epen quaratically on the ensity of particles. Such systems are energetically balance if both (28) an (37) hol. Energetically balance means in both situations that the natural total energy of the system is issipate in the absence of external or bounary forcing. Steay solutions of unforce energetically balance systems have necessarily the flui at rest, an solve the steay uncouple Onsager equation for the particle istribution. The energy balance is sufficient to etermine the coefficients of the stresses ae by the microscopic insertions in the flui. This provies a guiing principle for moelling that is inepenent of closure strategies. Type II equations have global smooth solutions if inertia is neglecte, so that the flui s equation is Stokes equation (44). In the couple system (43, 44) the particle istribution ensity f = f(x, m, t) obeys a transport equation D t f + iv g (W f) 0 if we neglect both particle iffusion an interparticle potential interaction. The fiel W is proportional to x u (8), the flui s physical space graient. The Stokes equation with perioic bounary conitions imposes a balance x u = Hσ with H a singular integral operator of Caleron-Zygmun type. Therefore the equation for f appears close to D t f + iv g ((H(σ)f)) 0. The constitutive relations for σ are of the type σ = γ (1) (f) + γ (2) (f f). Substituting, we see that the equation for f has the appearance of a nonlocal invisci Burgers equation. Why oes it not shock, then? The important facts for global regularity are: the smoothing properties in microscopic variables of the transformation f σ, an the tensorial nature of (8) where x u acts on particles multiplicatively. The presence of inertia is ifficult to hanle but some two imensional an filtere three-imensional type II equations have global smooth solutions ([3]). Acknowlegment Research partially supporte by NSF-DS grant 17

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