Smoluchowski Navier-Stokes Systems

Transcription

1 Smoluchowski Navier-Stokes Systems Peter Constantin Mathematics, U. of Chicago CSCAMM, April 18, 2007

2 Outline: 1. Navier-Stokes 2. Onsager and Smoluchowski 3. Coupled System

3 Fluid: Navier Stokes Equation t u + u u + p = ν u + σ u = 0 The tensor σ ij (x, t) : added stress. Sufficient for regularity, if σ smooth T 0 u 2 L (dx) dt < Necessary for applications T 0 u L (dx) dt <

4 2D, Bounded stress Theorem 1 Let σ L (dtdx). Let u 0 L 2 (dx). There exists a unique weak solution of the forced 2D NS eqns, with Moreover, u L (dt)(l 2 (dx)) L 2 (dt)(w 1,2 (dx)) T q 1 0 u q L q (dx) dt <, q 2 T 0 u p L (dx) dt <, p < 2.

5 Open questions T 0 u 2 L (dx) dt < T 0 u L (dx) <? T 0 u 2 L (dx) dt <? T 0 u L (dx) dt <? Partial regularity?

6 Navier-Stokes with nearly singular forces t u + u x u ν x u + x p = div x σ, x u = 0 Theorem 2 Let u be a solution of the 2D Navier-Stokes system with divergence-free initial data u 0 W 1,2 (R 2 ) W 1,r (R 2 ). Let T > 0 and let the forces σ obey with r > 2. Then σ L 1 (0, T ; L (R 2 )) L 2 (0, T ; L 2 (R 2 )) σ L 1 (0, T ; L r (R 2 )) L 2 (0, T ; L 2 (R 2 )) T 0 u(t) L dt K log (B)

7 and also 1 M M q=1 T 0 q u(t) L dt K with K depending on T, norms of σ and the initial velocity, but not on gradients of σ nor M, and B depending on norms of the spatial gradients of σ. u q q (u) Littlewood-Paley decomposition. F( q (u))(k) 0 k 2 q

8 Particles in Equilibrium f(p)dp: probability density of director p. Mean field interaction potential Free energy: (Kf)(p) = E = M M K(p, q)f(q)dq (f log f 1 2 fkf)dp

9 δe δf = 0 log f(m) = Onsager equation M K(m, p)f(p)dp log Z Z a normalizing constant. f = Z 1 e Kf Nonlinear, nonlocal.

10 General Onsager Equation Conjecture: The prolate nematic state is the generic stable limit. Partition function Z(f, b) = M ebkf dm with b concentration parameter. Define, for φ : M R, [φ](f, b) = (Z(f, b)) 1 M φ(m)ebkf dm. K(m, p) = j=1 µ j φ j (m)φ j (p)

11 φ j real, complete, orthonormal in L 2 (M), Kφ j = µ j φ j Expand f: Onsager s equation v j (f) = M f(p)φ j(p)dp. f = Z 1 e bkf is equivalent to the system v j (f) = [φ j ](f, b). Onsager solution is a critical point of the free energy F(v, b) = log Z(v, b) b j=1 µ j v 2 j 2

12 Differentiation: For any function φ(p) [φ] v i = bµ i {[φφ i ] [φ][φ i ]} Therefore the Hessian 2 F v i v j is H ij = b 2 µ i µ j [ξ i ξ j ] bµ i δ ij with ξ j = φ j [φ j ]. For b small the isotropic state v = 0 is stable. lim [φ](v, b) = φ(p(v)) b Morse lemma.

13 Smoluchowski (Nonlinear Fokker-Planck) Equation t f + u x f + div g (Gf) = 1 τ gf The (0, 1) tensor field W is: Example, rod-like particles: G = 1 τ gkf + W, ( W (x, m, t) = 3i,j=1 = c ij α (m) u i x (x, t) j ) α=1,...,d. W (x, m, t) = ( x u(x, t))m (( x u(x, t))m m)m. Macro-Micro Effect: from first principles, in principle...

14 Energetics σ ij (x) = ɛ M ( divg c ij + c ij g Kf(x, m) ) f(x, m)dm Micro-Macro Effect: from Energetics! f = Z 1 e Kf σ = 0 K = 0, W = ( x u)m m(( x u)m m) σ = ɛ (3n n 1)dm

15 γ (2) ij = ɛdiv g c ij + λδ ij (m, n) = ɛc ij α (m)g αβ (m) β K(m, n) + µδ ij γ (1) ij σ(x, t) = σ (1) (x, t) + σ (2) (x, t) +... where σ (1) ij (x, t) = M γ(1) ij (m)f(x, m, t)dm σ (2) ij (x, t) = M M γ (2) ij (m, n)f(x, m, t)f(x, n, t)dmdn.

16 Theorem 3 Liapunov functional If (u, f) is a smooth solution then E(t) = 1 2 u 2 dx+ +ɛ { f log f 1 2 (Kf)f } dxdm. ɛ τ de M dt = ν x u 2 dx f g (log f Kf) 2 dmdx. If the smooth solution is time independent, then u = 0 and f solves the Onsager equation f = Z 1 e Kf.

17 Time dependent Stokes and Nonlinear Fokker-Planck in 3D t f + u x f + div g (W f) + 1 τ div g(f g (Kf)) = ɛ g f t u ν x u + x p = div x σ + F, x u = 0. Theorem 4 Assume u 0 is divergence-free and belongs to W 2,r (T 3 ), r > 3, assume that f 0 is positive, normalized, and f 0 L (dx; C(M)) x f 0 L r (dx; H s (M)), s 2 d + 1. Then the solution exists for all time and u L p [(0,T );W 2,r (dx)] <, x f L [(0,T );L r (dx;h s (M))] < for any p > r 3 2r, T > 0 τ, ɛ 0.

18 Global existence, NSE and Nonlinear Fokker-Planck 2D Theorem 5 (C-Masmoudi) Let u 0 ( W α,r L 2) (R 2 ) be divergencefree, and f 0 W 1,r (H s (M)), with r > 2, α > 1, s d 2 +1 and f 0 0, M f 0dm (L 1 L )(R 2 ). Then the coupled NS and nonlinear Fokker- Planck system in 2D has a global solution u L loc (W 1,r ) L 2 loc (W 2,r ) and f L loc (W 1,r (H s )). Moreover, for T > T 0 > 0, we have u L ((T 0, T ); W 2 0,r ). No a priori bound. sup k λ α 1 k k t 0 xs k 1 (u(s)) L ds k (u)(t) L p

19 Basic Fokker-Planck Lemma t f + u x f ( x u)div g (cf) + div g (f g (Kf)) t N + u x N c x u N + c x x u pointwise at (x, t), with N(x, t) = M (I g ) 2 s x f(x, m, t) 2 dm s > d

20 Growth d n c(gn + G) dt n(t) = N(, t) L r (dx) L 2 (dx), g(t) = u(, t) L G(t) = x x u(, t) L r (dx) L 2 (dx). n(t) { n(0) + t 0 G(s)ds } e c t 0 g(s)ds

21 Bounds t 0 n = I t 0 g K log I t 0 G KI log I x u = e t x u(0) + t 0 e(t s) H {(u u) σ} ds

22 Linear piece t 0 e(t s) H(D)(σ(s))ds H(D)σ L C σ L log B t 0 e(t s) φ(s)ds L C φ L log B together: one log, not two

23 Nonlinear piece U(t) = t 0 e(t s) H(D)((u u)(s))ds. Idea = Chemin and Masmoudi: take time integral first q 3 q (U) = C(t) + I(t)

24 C(t) = q 3 t 0 e(t s) H(D) q p (u(s)) p (u(s)) p p 2 ds T 0 C(t) L dt ce 1

25 q (( p u(s)) ( p (u(s))) L c2 2q q (( p u(s)) ( p (u(s))) L 1 c2 (2q 2p) p u(s) L 2 p u(s) L 2 p q 2, p p 2, p = p + j, j [ 2, 2] C(t) L 2 p=1 j= 2 c t 0 p u(s) L 2 p+j u(s) L 2 { } p+2 q=3 e 22(q 1) (t s) 2 2q 2 2(q p) ds T0 C(t) L dt c 2 p=1 T0 p+2 j= 2 p ( u(s)) L 2 p+j ( u(s)) L 2 q=3 22(q p) ds c 2 T0 p=1 j= 2 p ( u(s)) L 2 p+j ( u(s)) L 2ds.

26 I(t) = q 3 t 0 e(t s) H(D) q p (u) p (u) p p 3 ds T 0 I(t) L dt ce 1 log ( c(1 + T )R1 ɛ ) + ɛ

27 I(t) = I 1 (t) + I 2 (t) I 1 (t) = q 3 t0 e (t s) H(D) q ( p p +3 p(u(s)) p (u(s)) ) ds I 2 (t) = q 3 t0 e (t s) H(D) q ( p p+3 p(u(s)) p (u(s)) ) ds with I 1 (t) = 2 t j= 2 q 3 0 e(t s) H(D) q (J q (s))ds J q (s) = q+j (u(s)) S q+j 3 (u(s)). q M: q (J q (s)) L c S q+j 3 (u(s)) L q+j (u(s)) L c [ u(s) L 2 + M + 2 u(s) L 2] q+j (u(s)) L

28 ( S q+j (u(s)) L c u(s) L 2 + ) q + j u(s) L 2. Bernstein s inequality: q+j u(s) L c q+j u(s) L 2 q (J q (s)) L c ( u(s) L 2 + M + 2 u(s) L 2) q+j (u(s)) L 2 q M: q (J q (s)) L c ( u(s) L 2 + u(s) L 2) 2 q q+j (u(s)) L 2 with A = c 2 M T j= 2 q=3 T 0 I 1(t) L dt A + B 0 q(j q (s)) L ( T s ) 22q e (t s)22(q 1) dt ds

29 and B = c 2 j= 2 q=m T 0 q(j q (s)) L ( T s 22q e (t s)22(q 1) dt ) ds A: A c ( T 0 u(s) L 2 + ( Mq=3 M + 2 u(s) L 2) q+j u(s) L 2) ds c ( T 0 u(s) L 2 + ( M + 2 u(s) L 2) M u(s) L 2) ds. We have therefore A cme 1 B: B c T 0 ( u(s) L 2 + u(s) L 2) q M ( 2 q ) q+j (u(s)) L 2 ds

30 and therefore where B c2 M T 0 u(s) 2 H 2 ds = c2 M E 2 E 2 = T 0 u(t) 2 H 2 dt. B c2 M (1 + T ) { σ 2 L 2 (0,T ;H 1 ) + u(0) 2 H 1 } = c2 M (1 + T )R 1 ( ) c(1 + T )R1 M = log ɛ

31 Coming attractions: A priori bound in 2D Stochastic system: existence and bounds Traveling waves, standing waves (with Berestycki, Ryzhik, Tsogka)

32 References P. Constantin, Nonlinear Fokker-Planck Navier-Stokes Systems, CMS, (2005) P. Constantin, C. Fefferman, E. Titi, A. Zarnescu, Regularity of coupled two-dimensional Nonlinear Fokker-Planck and Navier-Stokes systems, CMP (2007), to appear. P. Constantin, N. Masmoudi, Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D, preprint 2007.

33 More References P. Constantin, I. Kevrekidis, E. S. Titi, Remarks on a Smoluchowski Equations, Discrete and Continuous Dyn. Syst, (2004) P. Constantin. I Kevrekidis, E. S. Titi Asymptotic States of a Smoluchowski Equation, Arch. Rational Mech. Analysis, (2004) P. Constantin, E.S. Titi, J. Vukadinovic, Dissipativity and Gevrey Regularity of a Smoluchowski Equation, Indiana U. Math J., (2004) P. Constantin, J. Vukadinovic, Note on the number of steady states for a 2D Smoluchowski equation, Nonlinearity (2004).

34 References for Stochastic representation G. Iyer, Thesis, U of C P. Constantin, G. Iyer, A stochastic representation of the 3D incompressible NSE, Comm. Pure Appl. Math., (2007) to appear. P. Constantin, G. Iyer, Stochastic Lagrangian transport and generalized relative entropies, CMS, (2006). P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: local theory. Journal of AMS 14 (2001),

35 P. Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations, Comm. Math. Phys. 216 (2001), K. Ohkitani, P. Constantin, Numerical study of the Eulerian-Lagrangian formulation of the NSE, Phys. fluids 15 (2003)