Coagulation-Fragmentation Models with Diffusion

Transcription

1 Coagulation-Fragmentation Models with Diffusion Klemens Fellner DAMTP, CMS, University of Cambridge Faculty of Mathematics, University of Vienna Jose A. Cañizo, Jose A. Carrillo, Laurent Desvillettes UVIC p.1/34

2 Introduction Coagulation-fragmentation models with diffusion evolution of a polymer/cluster density f(t,x,y) t f a(y) x f = Q coag (f,f) + Q frag (f) time t, space x normalised with = 1 homogeneous Neumann x f(t,x,y) ν(x) = on non-negative initial data f (x,y) size-dependet diffusion coefficients a(y) UVIC p.2/34

3 Introduction Continuous and discreet models discrete in size models y = i N, f(y) = c i Q coag = 1 2 Q frag = i 1 j=1 a i j,j c i j c j a i,j c i c j, j=1 B i+j β i+j,i c i+j B i c i. j=1 coagulation-fragmentation coefficients a i,j = a j,i, β i,j, (i,j N), B 1 =, B i, (i N), i = i 1 i,j, j=1 (i N,i 2). UVIC p.3/34

4 Introduction Continuous and discreet models continuous in size model y [, ) Q coag (f,f) = Q frag (f) = 2 y y f(y y )f(y )dy 2f(y) f(y )dy y f(y) f(y )dy assume constant coagulation-fragmentation coefficients [Aizenman, Bak] 79 UVIC p.3/34

5 Overview Introduction Questions: existence of solutions (weak, strong) conservation of mass or gelation degenerate diffusion coefficients large-time behaviour fast-reaction limit Main tools: duality arguments entropy methods (detailed balance) UVIC p.4/34

6 Discrete coagulation-fragmentation models Weak formulation, conservation of mass test-sequence ϕ i, ϕ i Q coal = 1 2 i=1 ϕ i Q frag = i=1 i=1 i=2 a i,j c i c j (ϕ i+j ϕ i ϕ j ), j=1 B i c i (ϕ i i 1 j=1 β i,j ϕ j ). conservation of total mass or gelation ρ(t, ) L 1 = ic i (t,x)dx i=1 i=1 ic i(x)dx = ρ L 1 UVIC p.5/34

7 Discrete coagulation-fragmentation models Existence of global weak solutions in L 1 assumptions on coefficients lim j + a i,j j = lim j + B i+j β i+j,i i + j =, (for fixed i 1), Then, global weak solutions c i C([,T];L 1 ()), i N sup t a i,j c i c j L 1 ([,T] ), j=1 [ i=1 ] ic i (t,x) dx [ i=1 ] ic i(x) dx, [Laurençot, Mischler] 2 UVIC p.6/34

8 Discrete coagulation-fragmentation models L 2 estimates via duality Assume coefficients like above and ρ = i=1 ic i L 2 () Then, for all T > ρ L 2 ( T ) ( 1 + sup ) i{d i } T ρ L inf i {d i } 2 (), or for degenerate diffusion T [ i=1 ][ id i c i (t,x) i=1 ] ic i (t,x) 4T sup i N {d i } ρ L 2 (). [Cañizo, Desvillettes, F.] 9 preprints UVIC p.7/34

9 Discrete coagulation-fragmentation models Proof of duality bounds Denoting A(t,x) = 1 ρ i=1 id i c i, then A sup i N {d i } and t ρ x (Aρ) =. Multiplying with the solution w of the dual problem: ( t w + A x w) = H A, x w n(x) =, w(t, ) = for any smooth function H := H(t,x) leads to T H(t,x) A(t,x) ρ(t,x)dxdt = w(,x)ρ(,x)dx. UVIC p.8/34

10 Discrete coagulation-fragmentation models Proof of duality bounds Multiplying the dual problem by x w yields T A ( x w) 2 dxdt T H 2 dxdt. and Hence, w(,x) 2 T ( T t w 2 A dxdt 4 T H 2 dxdt. A t w A dt) 2 4T A L () T H 2 dxdt. UVIC p.8/34

11 Discrete coagulation-fragmentation models Proof of duality bounds Recalling above T H Aρdxdt ρ(, ) L 2 () w(, ) L 2 () 2 T A L () H L 2 ([,T] ) ρ(, ) L 2 (). for all (nonnegative smooth) functions H, we obtain by duality that Aρ L 2 () 2 T A L () ρ(, ) L 2 (). UVIC p.8/34

12 Discrete coagulation-fragmentation models Absence of gelation Assume a i,j (i + j)θ(j/i) for all j i N for a bounded function θ(x) as x +. Then, a superlinear moment is bounded on bounded time intervals, i.e. for a test-sequence {ψ i } i 1 with lim i ψ i : iψ i c i C(T) i=1 As a consequence, the mass is conserved ρ(t,x)dx = ρ (x)dx for all t. [Cañizo, Desvillettes, F.] 9 preprints UVIC p.9/34

13 Discrete coagulation-fragmentation models Specialised proof for absence of gelation In particular a i,j = ij and B i =, i= i log ic i(,x)dx. < Then, (using log(1 + x) C x) d dt = i log ic i dx i=1 i=1 j=1 ij ci c j ( i log(1 + j i ) + j log(1 + i j ) )dx 2 i=1 ij c i c j dx 2 j=1 ρ(t,x) 2 dx. UVIC p.1/34

14 Discrete coagulation-fragmentation models Specialised proof for absence of gelation In particular a i,j = ij and B i =, i= i log ic i(,x)dx. < As a consequence, we have for all T > i log ic i (T,x)dx i= + 2 i log ic i (,x)dx i= T ρ(t,x) 2 dxdt, and the propagation of the moment i= i log ic i(,x)dx ensures the conservation of the mass. UVIC p.1/34

15 Discrete coagulation-fragmentation models Degenerate Diffusion: Absence of gelation Assume d i C i γ, a i,j C (i α j β + i β j α ), with α + β + γ 1, α,β [, 1), γ [, 1], Then, the mass is conserved ρ(t,x)dx = ρ (x)dx for all t. UVIC p.11/34

16 Discrete coagulation-fragmentation models Existence theory via duality we have uniform L 2 -bound of approximating systems independent of a i,j the assumption lim j + a i,j j = is needed for the limit of Q,M coag = cm i j=1 a i,j c M j. as c M i converges to c i weak- in L ( T ), we need j=1 a i,j c M j j=1 a i,j c j strongly in L 1 ( T ) T a i,j (c M j c j ) dxdt 2 sup a i,j j ρ+sup c M j c j L 1 ( T ) j J j j J UVIC p.12/34

17 Discrete coagulation-fragmentation models Existence theory existence of generalised quadratic models t c i d i x c i = a k,l c k c l k+l=i k,l=1 i<max{k,l} a i,k c i c k k=1 b k,l c k c l β i,k,l k=1 b i,k c i c k global L 1 -existence in 1D provided lim l a k,l l b k,l =, lim l l =, lim l sup k { } bk,l kl β i,k,l = k,i N [Cañizo, Desvillettes, F.] 9 preprints UVIC p.12/34

18 Continuous coagulation-fragmentation models [Aizenman, Bak] 79 inhomogeneous Continuous in size density f(t,x,y) with y [, ) t f a(y) x f = y + 2 f(y y )f(y )dy 2f(y) y f(y )dy y f(y) f(y )dy homogeneous Neumann, non-negative initial density f (x,y) diffusion may degenerate at most linearly for large sizes a(y) a (δ), y [δ,δ 1 ], < a 1 + y a(y), y [, ). UVIC p.13/34

19 [Aizenman, Bak] 79 inhomogeneous Macroscopic densities amount of monomers N, number density M N = y f(y )dy, M = f(y )dy conservation of the total mass ( ) t N x y a(y )f(y )dy = ( ) t M x a(y )f(y )dy = N M 2 UVIC p.14/34

20 [Aizenman, Bak] 79 inhomogeneous Entropy (free energy functional) Entropy H(f)(t,x) = (f ln f f) dy, Entropy dissipation d H(f)dx = D H (f) dt D H (f) = a(y) xf 2 dy dx f ( ) f + (f ff ) ln ff dy dy dx UVIC p.15/34

21 [Aizenman, Bak] 79 inhomogeneous Inequality by [Aizenman, Bak] 79 (f(y + y ) f(y)f(y )) ln ( ) f(y + y ) dy dy f(y)f(y ) M H(f f N,N ) + 2(M N) 2 Entropy dissipation D H (f) a(y) xf 2 f dy dx +M H(f f N,N ) + 2(M N) 2 UVIC p.16/34

22 [Aizenman, Bak] 79 inhomogeneous Local and global equilibria Intermediate equilibria with the very moments N and M 2 = N f N,N = e 1 N y the global equilibrium f = e y N is constant in x satisfying M 2 = N preserves the initial mass N = N(x)dx UVIC p.17/34

23 [Aizenman, Bak] 79 inhomogeneous relative entropy, additivity relative entropy H(f g) = H(f) H(g) additivity H(f f ) = H(f f N,N ) + H(f N,N f ) f N,N and f do not need to have the same L 1 y norm, but nevertheless ( H(f N,N f )dx = 2 N dx ) N dx UVIC p.18/34

24 [Aizenman, Bak] 79 inhomogeneous Existence results [Am, AW] global existence and uniqueness of classical solutions (1D, not [Aizenman, Bak]) [LM] global existence of weak solutions satisfying the entropy dissipation inequality H(f(t))dx + t D H (f(s))ds H(f )dx Diffusivity a(y) L ([1/R,R]) for all R > Without rate: Equilibrium states attract all global weak solutions UVIC p.19/34

25 [Aizenman, Bak] 79 inhomogeneous Theorems Nonnegative initial data (1 + y + ln f )f L 1 ((, 1) (, )) with positive initial mass 1 N (x)dx = N > on = (, 1). As above at most linearily degenerate diffusion. Then, for β < 2 and t > and for all t t >. f(t,, ) f L 1 x,y C β e (ln t)β, (1 + y) q f(t,,y) f (y) L x dy C β,q e (ln t)β, UVIC p.2/34

26 Entropy Entropy Dissipation Estimate needs M L x, M L t (L 1 x), M > Step 1) Additivity 1 H(f f )dx = 1 H(f N,N f )dx + 2 ( N ) N UVIC p.21/34

27 Entropy Entropy Dissipation Estimate needs M L x, M L t (L 1 x), M > Step 2) "Reacting" Moments N and M 1 H(f f )dx = 1 H(f N,N f )dx + 2 ( N ) N N N 2 [ M ] N 2 L + M 2 M 2 N x L. 2 x UVIC p.21/34

28 Entropy Entropy Dissipation Estimate needs M L x, M L t (L 1 x), M > Step 2) "Reacting" Moments N and M > M > 1 H(f f )dx C C [ 1 MH(f f N,N )dx + 2 M ] N 2 L 2 x + 4 N M M 2 L 2 x N M M 2 L 2 x, ( ) f (f ff ) ln dy dy dx ff UVIC p.21/34

29 Entropy Entropy Dissipation Estimate needs M L x, M L t (L 1 x), M > Step 3) Diffusion For a cut-off size A >, denote M A (t,x) := A f(t,x,y)dy and Mc A (t,x) := f(t,x,y)dy A M M 2 L = ( ) 2 MA M x A + MA c MA c 2 dx 2 M A M A 2 L + 4 ( 2 y f(y)dy) p dx 2 x C(P,a )A M L x + 4 A 2p M L x M 2p A 2p a(y) xf 2 f dydx for any p > 1. UVIC p.21/34

30 Entropy Entropy Dissipation Estimate needs M L x, M L t (L 1 x), M > Entropy Entropy Dissipation Estimate Let f := f(x,y) be measurable with moments < M M(x) = f(x,y)dy M L x, < N = y f(x,y)dydx, y 2p f(x,y)dxdy M 2p. Then, for all A 1 and p > 1 C D 1 (f) H(f f )dx C M 2p, (-1) A M L x A2p+1 with a constant C = C(M,N,a,P()) depending only on M, N, a, and the Poincaré constant P(). UVIC p.21/34

31 A-priori Estimates (L 1 L 2 ) + L bounds in 1D Lemma M(t, ) L x m + m 2 (t). Proof: f(t,x,y) f(t, x,y) = 2 x x f(t,ξ,y) x f(t,ξ,y)dξ [ 2 1 f(t,x,y) f(t, x,y)d x dy 1 f(t,x,y) a(y) dxdy ]1 2 [ 1 a(y) x f(t,x,y) 2 dxdy ]1 2 M(t,x) 1 M(t, x)d x + a 1/2 (M + N ) 1/2 D(f(t)) 1/2. UVIC p.22/34

32 Lower A-priori Estimates M(t,x)dx bound Lemma: M(t,x)dx M d dt 1 M(t,x)dx 1 N (x)dx (m + m 1 (t)) 1 M(t,x)dx Idea: 1 M(t,x)dx e µ 1 M (x)dxe R t (m +m 1 (σ)) dσ t N (x)dx e R t s (m +m 1 (σ)) dσ ds [e m t M L1x + 1 ] e m t N L 1 x m UVIC p.23/34

33 Moments M p (f)(t) := 1 A-priori Estimates y p f dy dx Lemma: For p > 1 and for a.a. t t > M p (f)(t) (2 2p C) p =: M p for C = C(t,f ) depending only on the initial datum and t. Idea: fragmentation produces moments d dt M p(f)(t) (2 p 2)M p (f)(t) [m + m 2 (t)] p 1 p + 1 M p+1(f)(t). interpolation p 1 p+1 M p+1(f) ǫ p p+1 N p p+1 ǫ 1 M p (f) use Duhamel s formula, t t m 2 ds µ 2 t t, Vallée-Poussin UVIC p.24/34

34 A-priori Estimates Lower bound M(t,x) M > Lemma: Let t > be given. Then, there is a strictly positive constant M (depending on t,a and a (δ)) M(t,x) M. Idea: linear lower bound for lost terms t f a(y) xx f = g 1 y f M(t, ) L x f where g 1 is nonnegative, then ( ) ( t + a(y) xx ) f e ty+r t M(s, ) L x ds = g 2 where g 2 is nonnegative. UVIC p.25/34

35 A-priori Estimates Lower bounds M(t,x) M and N(t,x) N Fourier series and Poisson s formula for t h a xx h = G L 1 with homogeneous Neumann boundaries on (, 1) h(t,x) = 1 2 π π h(, z) 1 t 1 1 k= G(s, z) 1 e (2k+x z)2 4a t at k= dz 1 e (2k+x z) 4a (t s) a (t s) 2 dzds h and G "mirrored" around UVIC p.26/34

36 A-priori Estimates Lower bounds M(t,x) M and N(t,x) N f(t 1 + t,x,y) C 1 f(t 1,z,y)e (2t + 1 a t ) y dz, Moment bound 1 y 2 f(t,x,y)dy dx M 2 M(t 1 + t,x) C e (2t + 1 a t )1 δ C e (2t + 1 a t )1 δ 1 1/δ ( M δ N K δ δ f(t 1,z,y)dydz H(f)dx/ ln K ). Choosing δ and K, we get that M(t 1 + t,x) M. UVIC p.27/34

37 Proof of Theorem Algebraic rate for all p > 1 Algebraic rate for all p > 1 We have for any A > 1 d dt 1 H(f f )dx C M L x 1 A 1 H(f f )dx + C p 2 8p2 A 2p+1, where M L x (t) m + m 2 (t). balance r.h.s. by choosing A = A(t) > 2 as 1 A C 1/2 3 ( 1 C4 H(f f )dx M L x 2 8p2 ) 1 2p, Thus, Gronwall yields algebraic rate for all p > 1. UVIC p.28/34

38 Proof of Theorem Algebraic rate for all p > 1 Faster than polyniminal rate Then, by summing w.r.t. p N 1 H(f(t) f )dx L(t C 7 ), where (for all 1 < α < 2) L 1 (t) = q 1, even t q (C 1 q) q 2 2q2 = q 1, even t q e 2q2 ln 2 q ln(q C 1 ) C(α)e [ln2(α 1) (t)] for all t large enough and 1 < α < 2. UVIC p.28/34

39 formal Fast-Reaction limit t f ε a(y) x f ε = 1 ε (Q coag(f ε,f ε ) + Q frag (f ε )) formal limit: f ε e where y N t N x n(n ) = n(n) := a(y)ye y N dy < a N n(n) a N, < a n (N) a UVIC p.29/34

40 Fast-Reaction limit Entropy, entropy dissipation ε d dt H(f ε )dx M ε H(f ε f M ε,n ε )dx + 2 ((M ε ) N ε ) 2 dx Thus M ε H(f ε f N ε,n ε )dxdt εc Assumption M ε M f ε e y Nε 2 L 2 t (L1 x,y) εc(m ) UVIC p.3/34

41 Expansion Fast-Reaction limit Looking for f ε 1 L 2 t,x(l 1 y((1 + y)dy)) x f1 ε ν(x) = and θ > Then f ε = e Nε + ε θ f1 ε y t N ε x n(n ε ) = ε θ x a(y)yf ε 1 dy := ε θ x g ε where g ε L 2 t,x with xg ε ν(x) = UVIC p.31/34

42 Compactness Fast-Reaction limit Let g ε L 2 t,x with x g ε ν(x) =. Take initial data N in L 2 x. Then, the solutions of the nonlinear diffusion equation t N ε x n(n ε ) = ε θ x g ε x N ε ν(x) = converge in L 2 t,x to the solution N of t N x n(n) = x N ν(x) = UVIC p.32/34

43 Fast-Reaction limit Compactness : proof Duality argument : w, w(t) =, x w ν(x) = t w n(nε ) n(n) N ε N xw = H Holds that x w L 2 ([,T] ) C H L 2 ([,T] ) Then T (Nε N)Hdxdt ε θ g ε L 2 t,x x w L 2 ([,T] ) Since H C ([,T] ) is arbitrary N ε N L 2 t,x C ε θ g ε L 2 t,x C ε θ UVIC p.33/34

44 Non-local evolution equations THANK YOU! UVIC p.34/34

45 References [AB] M. Aizenman, T. Bak, Convergence to Equilibrium in a System of Reacting Polymers, Commun. math. Phys. 65 (1979), [Am] H. Amann, Coagulation-fragmentation processes, Arch. Rational Mech. Anal. 151 (2), [AW] Amann, H., Walker, C., Local and global strong solutions to continuous coagulation-fragmentation equations with diffusion, J. Differential Equations 218 (25), [LM] Ph. Laurençot, S. Mischler, The continuous coagulationfragmentation equation with diffusion, Arch. Rational Mech. Anal. 162 (22),