Existence of global weak solutions to implicitly constituted kinetic models of incompressible homogeneous dilute polymers

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1 1 / 31 Existence of global weak solutions to implicitly constituted kinetic models of incompressible homogeneous dilute polymers Endre Süli Mathematical Institute, University of Oxford joint work with J.W. Barrett, Imperial College London, and Miroslav Buĺıček and Josef Málek, Charles University Prague To Eitan Tadmor, on the occasion of his 60th birthday.

2 2 / 31 Viscoelastic fluids Gareth McKinley s Non-Newtonian Fluid Dynamics Group, MIT Jonathan Rothstein s Non-Newtonian Fluids Dynamics Lab, University of Massachusetts

3 3 / 31 Statement of the model Ω R d, d = 2,3: bounded open Lipschitz domain, T : length of the time interval of interest, and Q := Ω (0,T ): the associated space-time domain.

4 3 / 31 Statement of the model Ω R d, d = 2,3: bounded open Lipschitz domain, T : length of the time interval of interest, and Q := Ω (0,T ): the associated space-time domain. Consider the following system of nonlinear PDEs: ρ(u t + div(u u)) divt = ρf in Q, divu = 0 in Q, (1) u(,0) = u 0 ( ) in Ω, and the boundary condition u = 0 on Ω (0,T ).

5 4 / 31 We assume that the Cauchy stress T is decomposed as T = pi + S v + S e, where p : Q R is the pressure;

6 4 / 31 We assume that the Cauchy stress T is decomposed as T = pi + S v + S e, where p : Q R is the pressure; S v : Q R d d sym is the viscous part of the deviatoric stress;

7 4 / 31 We assume that the Cauchy stress T is decomposed as T = pi + S v + S e, where p : Q R is the pressure; S v : Q R d d sym is the viscous part of the deviatoric stress; S e : Q R d d sym is the elastic part of the deviatoric stress.

8 4 / 31 We assume that the Cauchy stress T is decomposed as T = pi + S v + S e, where p : Q R is the pressure; S v : Q R d d sym is the viscous part of the deviatoric stress; S v and D(u) := 1 2 ( u + ( u)t ) are assumed to be related via a maximal monotone graph described by the implicit relation: where G : R d d sym R d d sym R d d sym G(S v,d(u)) = 0, (2) is a continuous mapping. S e : Q R d d sym is the elastic part of the deviatoric stress.

9 5 / 31 Examples of G(S v,d(u)) = 0 Newtonian (Navier Stokes) fluids: S v = 2µ D(u), with µ > 0;

10 5 / 31 Examples of G(S v,d(u)) = 0 Newtonian (Navier Stokes) fluids: S v = 2µ D(u), with µ > 0; Power-law fluids: S v = 2µ D(u) r 2 D(u), 1 r < ;

11 5 / 31 Examples of G(S v,d(u)) = 0 Newtonian (Navier Stokes) fluids: S v = 2µ D(u), with µ > 0; Power-law fluids: S v = 2µ D(u) r 2 D(u), 1 r < ; Generalized power-law fluids: S v = 2 µ( D(u) 2 )D(u);

12 5 / 31 Examples of G(S v,d(u)) = 0 Newtonian (Navier Stokes) fluids: S v = 2µ D(u), with µ > 0; Power-law fluids: S v = 2µ D(u) r 2 D(u), 1 r < ; Generalized power-law fluids: S v = 2 µ( D(u) 2 )D(u); Stress power-law fluids and generalizations: D(u) = α( S v 2 )S v ;

13 5 / 31 Examples of G(S v,d(u)) = 0 Newtonian (Navier Stokes) fluids: S v = 2µ D(u), with µ > 0; Power-law fluids: S v = 2µ D(u) r 2 D(u), 1 r < ; Generalized power-law fluids: S v = 2 µ( D(u) 2 )D(u); Stress power-law fluids and generalizations: D(u) = α( S v 2 )S v ; Fluids with the viscosity depending on the shear rate and shear stress S v = 2ˆµ( D(u) 2, S v 2 )D(u);

14 5 / 31 Examples of G(S v,d(u)) = 0 Newtonian (Navier Stokes) fluids: S v = 2µ D(u), with µ > 0; Power-law fluids: S v = 2µ D(u) r 2 D(u), 1 r < ; Generalized power-law fluids: S v = 2 µ( D(u) 2 )D(u); Stress power-law fluids and generalizations: D(u) = α( S v 2 )S v ; Fluids with the viscosity depending on the shear rate and shear stress S v = 2ˆµ( D(u) 2, S v 2 )D(u); Activated fluids, such as Bingham and Herschel Bulkley fluids: S v τ D(u) = 0 and S v > τ S v = τ D(u) D(u) + 2ν( D(u) 2 )D(u).

15 5 / 31 Examples of G(S v,d(u)) = 0 Newtonian (Navier Stokes) fluids: S v = 2µ D(u), with µ > 0; Power-law fluids: S v = 2µ D(u) r 2 D(u), 1 r < ; Generalized power-law fluids: S v = 2 µ( D(u) 2 )D(u); Stress power-law fluids and generalizations: D(u) = α( S v 2 )S v ; Fluids with the viscosity depending on the shear rate and shear stress S v = 2ˆµ( D(u) 2, S v 2 )D(u); Activated fluids, such as Bingham and Herschel Bulkley fluids: S v τ D(u) = 0 and S v > τ S v = τ D(u) D(u) + 2ν( D(u) 2 )D(u). i.e. 2ν( D(u) 2 )(τ + ( S v τ ) + )D(u) = ( S v τ ) + S v, τ > 0.

16 Examples of S v (= τ) vs. D(u)(= γ) 6 / 31

17 7 / 31 We identify the implicit relation (2) with a graph A R d d sym R d d sym, i.e., G(S,D) = 0 (D,S) A. We assume that, for some r (1, ), A is a maximal monotone r-graph: (A1) A includes the origin; i.e., (0,0) A; (A2) A is a monotone graph; i.e., (S 1 S 2 ) (D 1 D 2 ) 0 for all (D 1,S 1 ),(D 2,S 2 ) A; (A3) A is a maximal monotone graph; i.e., for any (D,S) R d d sym R d d sym, if ( S S) ( D D) 0 for all ( D, S) A, then (D,S) A; (A4) A is an r-graph; i.e., there exist positive constants C 1, C 2 such that S D C 1 ( D r + S r ) C 2 for all (D,S) A.

18 8 / 31 K. R. Rajagopal. On implicit constitutive theories. Appl. Math., K. R. Rajagopal. On implicit constitutive theories for fluids. J. Fluid Mech., K. R. Rajagopal and A. R. Srinivasa. On the thermodynamics of fluids defined by implicit constitutive relations. Z. Angew. Math. Phys., M. Buĺıček, P. Gwiazda, J. Málek, and A. Swierczewska-Gwiazda. Adv. Calc. Var., M. Buĺıček, J. Málek, and K. R. Rajagopal. Indiana Univ. Math. J., M. Buĺıček, J. Málek, and K. R. Rajagopal. SIAM J. Math. Anal., M. Buĺıček, P. Gwiazda, J. Málek, and A. Swierczewska-Gwiazda. SIAM J. Math. Anal., 2012.

19 9 / 31 Definition of S e : kinetic theory of polymers Large number of internal degrees of freedom statistical physics.

20 9 / 31 Definition of S e : kinetic theory of polymers Large number of internal degrees of freedom statistical physics. R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager: Dynamics of Polymeric Liquids, Vol. II: Kinetic Theory. Wiley, P.G. de Gennes: Scaling Concepts in Polymer Physics. CUP, H.C. Öttinger: Stochastic Processes in Polymeric Fluids. Springer, M. Doi: Introduction to Polymer Physics. OUP, T. Kawakatsu: Statistical Physics of Polymers. Springer, 2004.

21 10 / 31 Definition of S e : kinetic theory of polymers Let D i R d, i = 1,...,K, be bounded open balls centred at 0.

22 10 / 31 Definition of S e : kinetic theory of polymers Let D i R d, i = 1,...,K, be bounded open balls centred at 0. Consider the Maxwellian M(q) := M 1 (q 1 ) M K (q K ), with q i D i, where M i (q i ) := e U i( 1 2 q i 2 ) D i e U i( 1 2 p i 2) dp i, i = 1,...,K.

23 11 / 31 S e is defined by the Kramers expression: S e (x,t) := k B T K i=1 D M(q) qi ψ(x,q,t) q i dq, where q = (q T 1,...,qT K )T D 1 D K =: D and ψ := ψ/m is the normalized probability density function, that is the solution of a Fokker Planck equation.

24 12 / 31 Fokker Planck equation The function ψ = ψ/m satisfies the Fokker Planck equation: (M ψ) t + div(m ψu) + div q (M ψ( u)q) = (M ψ) + div q A(M q ψ) (3) in O (0,T ), with O := Ω D, subject to the boundary conditions: M ψ n = 0 on Ω D (0,T ), (M ψ( u)q i A i (M q ψ)) n i = 0 on Ω D i (0,T ), for all i = 1,...,K, and the initial condition ψ(x,q,0) = ψ 0 (x,q) in O. A R K K symm: Rouse matrix (symmetric, positive definite).

25 13 / 31 J.W. Barrett & E. Süli (M3AS, 21 (2011), ): Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains J.W. Barrett & E. Süli (M3AS, 22 (2012), 1 84): Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type bead-spring chains J.W. Barrett & E. Süli (J. Differential Equations, 253 (2012), ): Existence of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers with variable density and viscosity

26 13 / 31 J.W. Barrett & E. Süli (M3AS, 21 (2011), ): Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains J.W. Barrett & E. Süli (M3AS, 22 (2012), 1 84): Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type bead-spring chains J.W. Barrett & E. Süli (J. Differential Equations, 253 (2012), ): Existence of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers with variable density and viscosity M. Buĺıček, J. Málek & E. Süli (Communications in PDE, 38 (2013), ): Existence of global weak solutions to implicitly constituted kinetic models of incompressible homogeneous flows of dilute polymers DOI: /

27 14 / 31 Assumptions on the data For the Maxwellian M we assume that M C 0 (D) C 0,1 loc (D), and M > 0 on D. (4) For the initial velocity u 0 we assume that For ψ 0 := ψ 0 /M we assume, with O := Ω D, that u 0 L 2 0,div(Ω). (5) ψ 0 0 a.e. in O, ψ 0 log ψ 0 L 1 M(O), (6) and that the initial marginal probability density function M(q) ψ 0 (,q)dq L (Ω). (7) D

28 Theorem For d {2,3} let D i R d, i = 1,...,K, be bounded open balls centred at the origin in R d, let Ω R d be a bounded open Lipschitz domain and suppose f L r (0,T ;W 1,r 0,div (Ω)), r (1, ). Assume that A, given by G, is a maximal monotone r-graph satisfying (A1) (A4), the Maxwellian M : D R satisfies (4), and (u 0, ψ 0 ) satisfy (5) (7). Then, there exist (u,s v,s e, ψ) such that u L (0,T ;L 2 0,div(Ω) d ) L r (0,T ;W 1,r 0 (Ω)d ) W 1,r (0,T ;W 1,r 0,div (Ω)), S v L r (0,T ;L r (Ω) d d ), S e L 2 (0,T ;L 2 (Ω) d d ), ψ L (Q;LM(D)) 1 L 2 (0,T ;W 1,1 M (O)), ψ 0 a.e. in O (0,T ), M ψ W 1,1 (0,T ;W 1,1 (O)), ψ log ψ L (0,T ;LM(O)), 1 where r := min { r,2, ( d ) r } and r := r r / 31

29 16 / 31 Theorem (Continued...) Moreover, (1) is satisfied in the following sense: T 0 where T u t,w dt + T = 0 0 ( (u u, w) + (Sv, w) ) dt ( (Se, w) + f,w ) dt for all w L (0,T ;W 1, 0,div (Ω)), (S v (x,t),d(u(x,t))) A for a.e. (x,t) Q, and S e is given by the Kramers expression S e (x,t) = k B T K i=1 D M qi ψ(x,q,t) q i dq for a.e. (x,t) Q.

30 17 / 31 Theorem (Continued...) In addition, the Fokker Planck eqn (3) is satisfied in the following sense: T [ (M ψ)t,ϕ (M u ψ, ϕ) O (M ψ( u)q, q ϕ) O ] dt 0 T [ ] + (M ψ, ϕ)o + (M A q ψ, q ϕ) O dt = 0 0 for all ϕ L (0,T ;W 1, (O)), and the initial data are attained strongly in L 2 (Ω) d LM 1 (O), i.e., lim u(,t) u 0 ( ) ψ(,t) ψ 0 ( ) L 1 t 0 M (O) = 0. +

31 18 / 31 Theorem (Continued...) Further, for t (0,T ) the following energy inequality holds in a weak sense: ( ) ( ) d k dt M ψ log ψ dxdq O u (S v,d(u)) + 4k M ψ, ψ O ) + 4k (M A q ψ, q ψ f,u, with k := k B T. O

32 19 / 31 Proof STEP 1. Truncate ψ in the Kramers expression and in the drag term in the FP equation by replacing ψ with T l ( ψ), preserving the energy inequality.

33 19 / 31 Proof STEP 1. Truncate ψ in the Kramers expression and in the drag term in the FP equation by replacing ψ with T l ( ψ), preserving the energy inequality. STEP 2. We form a Galerkin approximation of the velocity and the probability density function, resulting in a system of ODEs in t.

34 19 / 31 Proof STEP 1. Truncate ψ in the Kramers expression and in the drag term in the FP equation by replacing ψ with T l ( ψ), preserving the energy inequality. STEP 2. We form a Galerkin approximation of the velocity and the probability density function, resulting in a system of ODEs in t. STEP 3. The sequence of Galerkin approximations satisfies an energy inequality, uniformly in the number of Galerkin basis functions and the truncation parameter l.

35 19 / 31 Proof STEP 1. Truncate ψ in the Kramers expression and in the drag term in the FP equation by replacing ψ with T l ( ψ), preserving the energy inequality. STEP 2. We form a Galerkin approximation of the velocity and the probability density function, resulting in a system of ODEs in t. STEP 3. The sequence of Galerkin approximations satisfies an energy inequality, uniformly in the number of Galerkin basis functions and the truncation parameter l. STEP 4. We extract weakly (and weak*) convergent subsequences, and pass to the limits in the Galerkin approximations.

36 19 / 31 Proof STEP 1. Truncate ψ in the Kramers expression and in the drag term in the FP equation by replacing ψ with T l ( ψ), preserving the energy inequality. STEP 2. We form a Galerkin approximation of the velocity and the probability density function, resulting in a system of ODEs in t. STEP 3. The sequence of Galerkin approximations satisfies an energy inequality, uniformly in the number of Galerkin basis functions and the truncation parameter l. STEP 4. We extract weakly (and weak*) convergent subsequences, and pass to the limits in the Galerkin approximations. STEP 5. We require strongly convergent sequences for passage to limit in l in the various nonlinear terms. This is the most difficult step to realize.

37 weak convergence strong convergence 20 / 31

38 20 / 31 weak convergence strong convergence ( ) ( ) d k dt M ψl log ψ l dxdq O ul (S l v,d(u l )) + 4k M ψ l, ψ l O ) + 4k (M A q ψ l, q ψ l f,u l, with k := k B T. O

39 20 / 31 weak convergence strong convergence ( ) ( ) d k dt M ψl log ψ l dxdq O ul (S l v,d(u l )) + 4k M ψ l, ψ l O ) + 4k (M A q ψ l, q ψ l f,u l, with k := k B T. O Velocity: strong convergence immediate by Aubin Lions Simon compactness theorem.

40 20 / 31 weak convergence strong convergence ( ) ( ) d k dt M ψl log ψ l dxdq O ul (S l v,d(u l )) + 4k M ψ l, ψ l O ) + 4k (M A q ψ l, q ψ l f,u l, with k := k B T. O Velocity: strong convergence immediate by Aubin Lions Simon compactness theorem. Probability density function: (much more difficult)

41 20 / 31 weak convergence strong convergence ( ) ( ) d k dt M ψl log ψ l dxdq O ul (S l v,d(u l )) + 4k M ψ l, ψ l O ) + 4k (M A q ψ l, q ψ l f,u l, with k := k B T. O Velocity: strong convergence immediate by Aubin Lions Simon compactness theorem. Probability density function: (much more difficult) Vitali s convergence theorem (a.e. convergence + L 1 equi-integrability);

42 20 / 31 weak convergence strong convergence ( ) ( ) d k dt M ψl log ψ l dxdq O ul (S l v,d(u l )) + 4k M ψ l, ψ l O ) + 4k (M A q ψ l, q ψ l f,u l, with k := k B T. O Velocity: strong convergence immediate by Aubin Lions Simon compactness theorem. Probability density function: (much more difficult) Vitali s convergence theorem (a.e. convergence + L 1 equi-integrability); Weak lower semicontinuity of convex functions (Feireisl & Novotný);

43 20 / 31 weak convergence strong convergence ( ) ( ) d k dt M ψl log ψ l dxdq O ul (S l v,d(u l )) + 4k M ψ l, ψ l O ) + 4k (M A q ψ l, q ψ l f,u l, with k := k B T. O Velocity: strong convergence immediate by Aubin Lions Simon compactness theorem. Probability density function: (much more difficult) Vitali s convergence theorem (a.e. convergence + L 1 equi-integrability); Weak lower semicontinuity of convex functions (Feireisl & Novotný); Murat Tartar Div Curl lemma;

44 20 / 31 weak convergence strong convergence ( ) ( ) d k dt M ψl log ψ l dxdq O ul (S l v,d(u l )) + 4k M ψ l, ψ l O ) + 4k (M A q ψ l, q ψ l f,u l, with k := k B T. O Velocity: strong convergence immediate by Aubin Lions Simon compactness theorem. Probability density function: (much more difficult) Vitali s convergence theorem (a.e. convergence + L 1 equi-integrability); Weak lower semicontinuity of convex functions (Feireisl & Novotný); Murat Tartar Div Curl lemma; Uniform interior estimates on Ω D (0,T ), obtained by function space interpolation from the energy inequality.

45 STEP 6. Identification of S e : the sequence of truncated Kramers expressions S l e converges to S e strongly in L q (0,T ;L q (Ω) d d ), q [1,2). 21 / 31

46 21 / 31 STEP 6. Identification of S e : the sequence of truncated Kramers expressions S l e converges to S e strongly in L q (0,T ;L q (Ω) d d ), q [1,2). STEP 7. The initial data are attained strongly in L 2 (Ω) d LM 1 (O), i.e., lim t 0 + u(,t) u 0 ( ) ψ(,t) ψ 0 ( ) L 1 M (O) = 0.

47 21 / 31 STEP 6. Identification of S e : the sequence of truncated Kramers expressions S l e converges to S e strongly in L q (0,T ;L q (Ω) d d ), q [1,2). STEP 7. The initial data are attained strongly in L 2 (Ω) d LM 1 (O), i.e., lim t 0 + u(,t) u 0 ( ) ψ(,t) ψ 0 ( ) L 1 M (O) = 0. STEP 8. Identification of S v : by a parabolic Acerbi Fusco type Lipschitz-truncation of Diening, Ružička & Wolf (2010) and STEP 6: lim l Q (S l v S (D(u))) D(u l u) α dxdt = 0 α (0,1). S is a measurable selection such that for any D we have (S (D),D) A.

48 22 / 31 Thus, for a subsequence, (S l v S (D(u))) D(u l u) 0 almost everywhere in Q. Moreover, using the energy inequality, we see that (S l v S (D(u))) D(u l u) dxdt C. Q We apply Chacon s Biting Lemma to find a nondecreasing countable sequence of measurable sets Q 1 Q k Q k+1 Q such that lim Q \ Q k 0 k and such that for any k there is a subsequence such that (S l v S (D(u))) D(u l u) converges weakly in L 1 (Q k ).

49 23 / 31 By Vitali s theorem we then deduce that (S l v S (D(u))) D(u l u) 0 strongly in L 1 (Q k ). The weak convergence of (S l v) to S v and (D(u l )) to D(u) implies that lim l (Sl v,d(u l )) Qk = (S v,d(u)) Qk. The assumption that A is a maximal monotone r-graph then implies that (S v,d(u)) A a.e. in Q k, k = 1,2,... Finally, by a diagonal procedure and lim k Q \ Q k 0 we deduce that (S v,d(u)) A a.e. in Q = Ω (0,T ).

50 24 / 31 Open problems The extension of these results to implicitly constituted kinetic models with variable density, density-dependent viscosity and drag is open.

51 24 / 31 Open problems The extension of these results to implicitly constituted kinetic models with variable density, density-dependent viscosity and drag is open. Special case: For Navier Stokes Fokker Planck systems with variable density and density-dependent dynamic viscosity and drag the existence of global weak solutions was shown in Barrett & Süli (Journal of Differential Equations, 2012).

52 24 / 31 Open problems The extension of these results to implicitly constituted kinetic models with variable density, density-dependent viscosity and drag is open. Special case: For Navier Stokes Fokker Planck systems with variable density and density-dependent dynamic viscosity and drag the existence of global weak solutions was shown in Barrett & Süli (Journal of Differential Equations, 2012). The numerical analysis of implicitly constituted kinetic models of polymers is open. Special cases: Barrett & Süli (M2AN, 2012) Diening, Kreuzer & Süli (SIAM J. Numer. Anal., 2013) Kreuzer & Süli (In preparation, 2014).

53 25 / 31 2D/4D: Flow around a cylinder Standard benchmark problem: flow around a cylinder Assume Stokes flow, parabolic inflow BCs on u x, no-slip on stationary walls and cylinder Steady state solution (computed on 8 processors): u x : u y : p :

54 26 / 31 2D/4D: Flow around a cylinder: extra stress tensor (S e ) 11 : (S e ) 12 : (S e ) 22 :

55 27 / 31 2D/4D: Flow around a cylinder: probability density fn. (a) (b) Figure : Configuration space cross-sections of ψ for x in (r, θ)-coordinates: (a) wake of cylinder, and (b) between cylinder and wall.

56 2D/4D: Flow around a cylinder: probability density fn. 28 / 31

57 29 / 31 3D/6D: Flow past a ball in a channel Pressure-drop-driven flow past a ball in hexahedral channel. P 2 /P 1 mixed FEM for (Navier )Stokes equation on a mesh with 3045 tetrahedral elements and Gaussian quadrature points. Fokker Planck equation solved using heterogenous ADI method in 6D domain Ω D D solves per time step in q = (q 1,q 2,q 3 ) D and D solves per time-step in x = (x,y,z) Ω. Computed using 120 processors; 45s/time step; 10 time steps; t = 0.05; λ = 0.5.

58 30 / 31 3D/6D: Flow past a ball in a channel: elastic part of the Cauchy stress (S e ) 11 (S e ) 12 (S e ) 13 (S e ) 22 (S e ) 23 (S e ) 33

59 HAPPY BIRTHDAY, EITAN! 31 / 31