Coagulation-Fragmentation Models

Transcription

1 KFU p. 1/45 Coagulation-Fragmentation Models Klemens Fellner Institute for Mathematics and Scientific Computing University of Graz

2 KFU p. 2/45 Outline of the course Overview 1. Motivation and modelling 2. The Smoluchowski equation : a model for coagulation 3. Coagulation-Fragmentation Models The Becker-Döring model Saturation phenomena and large-time asymptotics 4. Inhomogeneous coagulation-fragmentation models with diffusion The existence theory of Laurençot and Mischler A duality argument an entropy method A fast reaction limit

3 KFU p. 3/45 Modelling Introduction The Formation and the Break-up of Clusters/Polymers in Physics aerosols, rainsdrops, smoke, sprays Chemistry monomers/polymers Astronomy formation of galaxies Biology hematology, animal grouping

4 KFU p. 4/45 Background Introduction The Formation and the Break-up of Clusters/Polymers assume particles fully described by mass/size y Y. full/realistic models can quickly get very difficult

5 KFU p. 5/45 Background Introduction Levels of description: Microscopic description N 1 particles, stochastic events Mesoscopic description density f(t, y), mean-field equation Macroscopic description physical observations Linking limits micro meso Convergence of stochastic processes (Marcus-Lushnikov process), mean-field limits meso macro Fast-reaction-limits

6 KFU p. 6/45 The Smoluchowski equation The Smoluchowski coagulation equation [1916/17,1928] mesoscopic density of clusters/polymers f(t,y), y Y : t f(t,y) = Q coag (f,f)(y) = Q 1 (f,f) Q 2 (f,f) Q 1 (f,f): gain of particles of size y {y } + {y y } a(y,y y ) {y}, y < y Consider only binary interaction!

7 KFU p. 6/45 The Smoluchowski equation The Smoluchowski coagulation equation [1916/17,1928] mesoscopic density of clusters/polymers f(t,y), y Y : t f(t,y) = Q coag (f,f)(y) = Q 1 (f,f) Q 2 (f,f) Q 1 (f,f): gain of particles of size y {y } + {y y } a(y,y y ) {y}, y < y Q 2 (f,f): loss of particles of size y {y} + {y } a(y,y ) {y + y }, y Y

8 KFU p. 6/45 The Smoluchowski equation The Smoluchowski coagulation equation [1916/17,1928] mesoscopic density of clusters/polymers f(t,y), y Y : t f(t,y) = Q coag (f,f)(y) = Q 1 (f,f) Q 2 (f,f) Q 1 (f,f): gain of particles of continuous size y [, ) Q 1 (f,f) = 1 2 y a(y,y y )f(y y )f(y )dy Q 2 (f,f): loss of particles of size y {y} + {y } a(y,y ) {y + y }

9 KFU p. 6/45 The Smoluchowski equation The Smoluchowski coagulation equation [1916/17,1928] mesoscopic density of clusters/polymers f(t,y), y Y : t f(t,y) = Q coag (f,f)(y) = Q 1 (f,f) Q 2 (f,f) Q 1 (f,f): gain of particles of continuous size y [, ) Q 1 (f,f) = 1 2 y a(y,y y )f(y y )f(y )dy Q 2 (f,f): loss of particles of continuous size y [, ) f(y) a(y,y )f(y )dy

10 KFU p. 7/45 The Smoluchowski equation The continuous Smoluchowski coagulation equation Evolution (t ) of mesoscopic density f(t,y) of clusters/polymers of size y [, ): t f(t,y) = Q coag (f,f)(y) = Q 1 (f,f)(y) Q 2 (f,f)(y) Q coag (f,f)(y) = 1 2 y f(y) a(y,y y )f(y y )f(y )dy a(y,y )f(y )dy coagulation coefficient/kernel/rate a(y,y )

11 KFU p. 8/45 The Smoluchowski equation Examples of coagulation coefficients a(y,y ) = a(y,y) Colloidal particles: e.g. Smoluchowski α = γ = 1/3, β = 1 a(y,y ) = (y α + (y ) α ) β (y γ + (y ) γ ), α,β,γ, αβ 1 Ballistic kernel: a(y,y ) = (y α + (y ) α ) β y γ (y ) γ, α,β,γ, αβ + γ 1 Other kernels: a(y,y ) = y α (y ) β + (y ) α (y) β, α,β [, 1] e.g. Golovin kernel (α,β) = (, 1) (cloud droplets) e.g. Stockmeyer kernel α = β = 1 (branched-chain polymers)

12 KFU p. 9/45 The Smoluchowski equation The continuous Smoluchowski coagulation equation Weak formulation: Multiplication with a testfunction ϕ(y) and integration yields Q coag ϕ(y)dy = 1 2 y a(y,y y )f(y y )f(y )ϕ(y)dy dy a(y,y )f(y)f(y )ϕ(y)dy dy

13 KFU p. 9/45 The Smoluchowski equation The continuous Smoluchowski coagulation equation Weak formulation: Multiplication with a testfunction ϕ(y) and integration yields (formally) with Fubini for Q 1 (y)ϕ(y)dy Q coag ϕ(y)dy = y a(y,y y )f(y y )f(y )ϕ(y)dy dy a(y,y )f(y)f(y )ϕ(y)dy dy a(y,y)f(y )f(y)ϕ(y )dydy

14 KFU p. 9/45 The Smoluchowski equation The continuous Smoluchowski coagulation equation Weak formulation: Multiplication with a testfunction ϕ(y) and integration yields with y y = y and a(y,y ) = a(y,y) Q coag ϕ(y)dy = a(y,y )f(y )f(y )ϕ(y + y )dy dy a(y,y )f(y)f(y )ϕ(y)dy dy a(y,y )f(y)f(y )ϕ(y )dy dy

15 KFU p. 9/45 The Smoluchowski equation The continuous Smoluchowski coagulation equation Weak formulation: Multiplication with a testfunction ϕ(y) and integration yields (formally) with Fubini for Q 1 (y)ϕ(y)dy to Q coag ϕ(y)dy = 1 2 a(y,y )f(y)f(y )(ϕ ϕ ϕ )dydy Notation: ϕ = ϕ(y), ϕ = ϕ(y ), ϕ = ϕ(y + y )

16 KFU p. 1/45 The Smoluchowski equation Formal properties of the cont. Smoluchowski equation Weak formulation: ϕ(y), ϕ = ϕ(y ), ϕ = ϕ(y + y ) Q coag ϕ(y)dy = 1 2 a(y,y )f(y)f(y )(ϕ ϕ ϕ )dydy Moments: ϕ(y) = y k From sign(ϕ ϕ ϕ ) = sign(k 1) follows t ϕ(y)f(t,y)dy = ց k < 1 constant k = 1 ր k > 1

17 KFU p. 1/45 The Smoluchowski equation Formal properties of the cont. Smoluchowski equation Weak formulation: ϕ(y), ϕ = ϕ(y ), ϕ = ϕ(y + y ) Q coag ϕ(y)dy = 1 2 a(y,y )f(y)f(y )(ϕ ϕ ϕ )dydy Decreasing number of particles: ϕ = 1 stoichiometric coefficients 1 2,1 Formal conservation of mass: ϕ(y) = y d dt y f(t,y)dy = y Q coag dy =

18 KFU p. 1/45 The Smoluchowski equation Formal properties of the cont. Smoluchowski equation Weak formulation: ϕ(y), ϕ = ϕ(y ), ϕ = ϕ(y + y ) Q coag ϕ(y)dy = 1 2 a(y,y )f(y)f(y )(ϕ ϕ ϕ )dydy Rigorous?? Mass is non-increasing! test-function ϕ(y) = min{y,r} with ϕ ϕ ϕ for R > Then, t min{y,r}f(t,y)dy R Fatou y f(t,y)dy is non-increasing!

19 KFU p. 11/45 The Smoluchowski equation The kernel a(y,y ) = y y Coagulation kernel: a(y,y ) = y y Testing with ϕ(y) = 1 and for T >, d dt f(t,y)ϕ(y)dy = 1 2 yy f(y)f(y )(ϕ ϕ ϕ )dy dy

20 KFU p. 11/45 The Smoluchowski equation The kernel a(y,y ) = y y Coagulation kernel: a(y,y ) = y y Testing with ϕ(y) = 1 and for T >, and introducing the moments: M (t) = f(t,y)dy, M 1 (t) = y f(t,y)dy yields d dt M (t) = 1 2 yy f(y)f(y )dy dy = 1 2 M2 1(t)

21 KFU p. 11/45 The Smoluchowski equation The kernel a(y,y ) = y y Coagulation kernel: a(y,y ) = y y Testing with ϕ(y) = 1 and for T >, Moments: M (t) = f(t,y)dy, M 1 (t) = y f(t,y)dy. M (T) T M implies M 1 L 2 ((, ))! M 2 1 dt = M () Gelation: M 1 (t) < M 1 () for some finite t Formation of clusters of infinite size, Phase separation

22 KFU p. 12/45 The Smoluchowski equation The kernel a(y,y ) = y y Gelation time: T g := inf{t : M 1 (t) < M 1 ()} Second moment M 2 = y 2 f(t,y)dy. Then, d dt M 2(t) = M 2 2(t) blows-up at time T 2 = 1 1 M 2 (recall M () 2 (t) = 1 Is this the gelation time T g = T 2? M 2 () t) Yes, for a(y,y ) = y y. In general open problem!

23 KFU p. 13/45 The Smoluchowski equation The kernel a(y,y ) = y y Formal Laplace-type-transform: E(t,p) = e py yf(t,y)dy, p E(t,p) = e py y 2 f(t,y)dy Note that E(t, ) = M 1 (t) and p E(t, ) = M 2 (t). Thus, t E(t,p) + (E(p) E()) p E(t,p) = Note also p E(t,p) p E(t, ) = M 2 (t) and the above Burgers equation develops a first shock at p = and gelation occurs at T g = T 2 : E(t, ) = M 1 (t) < M 1 (), t > T g = T 2

24 KFU p. 14/45 Coagulation-Fragmentation Models Coagulation-fragmentation models with diffusion Evolution of a polymer/cluster density f(t,x,y) t f d(y) x f = Q coag (f,f) + Q frag (f) Time t, space x Ω normalised with Ω = 1 y Y cluster size, size-dependent diffusion coefficients d(y) Homogeneous Neumann B.C.: Non-negative initial data: x f(t,x,y) ν(x) = on Ω f (x,y) on Ω

25 KFU p. 15/45 Discrete coagulation-fragmentation models Discrete in size models discrete size y = i N = Y, f(t,y) = c i (t), c = (c i ) d t c i (t) = Q coag (c,c) + Q frag (c) Binary coagulation: = Q 1 (c,c) Q 2 (c,c) + Q 3 (c) Q 4 (c) Q 1 (c,c): gain of particles of size i {i j} + {j} a i j,j {i}, j < i Q 2 (c,c): loss of particles of size i {i} + {j} a i,j {i + j}, j 1

26 KFU p. 15/45 Discrete coagulation-fragmentation models Discrete in size models discrete size y = i N = Y, f(t,y) = c i (t), c = (c i ) Fragmentation: d t c i (t) = Q coag (c,c) + Q frag (c) Q 3 (c): gain of particles of size i = Q 1 (c,c) Q 2 (c,c) + Q 3 (c) Q 4 (c) {i + j} B i+jβ i+j,i {i} + {j}, j > 1 Q 4 (c): loss of particles of size i {i} B i all pairs {i j} + {j} with j < i

27 KFU p. 16/45 Discrete models Introduction Discrete in size models y = i N f(y) = c i, d(y) = d i t c i d i x c i = Q i,coag (c,c) + Q i,frag (c), i N, i 1 Q i,coag = 1 2 a i j,j c i j c j a i,j c i c j, j=1 j=1 Q i,frag = 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), (mass conservation) i = i 1 i,j, j=1 (i N,i 2).

28 KFU p. 17/45 Discrete coagulation-fragmentation models Weak formulation, conservation of mass Test-sequence ϕ i, ϕ i Q i,coal = 1 2 i=1 ϕ i Q i,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 Ω i=1 Ω i=1 ic i(x)dx = ρ L 1

29 KFU p. 18/45 The Becker-Döring model Interactions between monomers and polymers only discrete size i N, c i (t), c = (c i ) d t c i (t) = Q coag (c,c) + Q frag (c) = Q 1 (c,c) Q 2 (c,c) + Q 3 (c) Q 4 (c) Binary coagulation between monomers and polymers Q 1 (c,c): gain of particles of size i {i 1} + {1} a i 1 {i}, 1 < i Q 2 (c,c): loss of particles of size i {i} + {1} a i {i + 1}, 1 i

30 KFU p. 18/45 The Becker-Döring model Interactions between monomers and polymers only discrete size i N, c i (t), c = (c i ) d t c i (t) = Q coag (c,c) + Q frag (c) = Q 1 (c,c) Q 2 (c,c) + Q 3 (c) Q 4 (c) Fragmentation of monomers from polymers Q 3 (c): gain of particles of size i {i + 1} b i+1 {i} + {1}, 1 i Q 4 (c): loss of particles of size i {i} b i {i 1} + {1}, 1 < i

31 KFU p. 19/45 The Becker-Döring model Interaction between monomers and polymers In the Becker-Döring model all coagulation and fragmentation events involve monomers/clusters-of-size-one. System of a monomer-equation and polymer-equations: d t c 1 = W 1 (c) i=1 W i(c), d t c i = W i 1 (c) W i (c), i 2 where W i (c) = a i c 1 c i b i+1 c i+1 a little knows about C-F models except with detailed balance a a 1 = a 1,2 /2, b 2 = b 1,1 /2, and a i = a i,1, b i + 1 = b i,1, i 2

32 KFU p. 2/45 Coagulation-Fragmentation models Detailed balance condition : continuous and discrete non-negative equilibrium E(y) L 1 1(Y ) := L 1 (Y, (1 + y)dy): a(y,y )E(y)E(y ) = b(y,y )E(y + y ), (y,y ) Y Y This equation is also satisfied by all E z (y) = E(y)z y, y Y, for z but E z not necessarily in L 1 1(Y ). Thus, z s := sup{z : E z L 1 1(Y )} [1, ] ρ s := M 1 (E zs (y)) [, ]. ρ s is called the saturation mass

33 KFU p. 21/45 Coagulation-Fragmentation models Entropy and detailed balance Entropy functional: H(f E) = Y f ( ln( f E ) 1) dy H-Theorem f = f(y ), f = f(y + y ) d dt H(f E) = 1 2 D(f), D(f) = (aff bf )(ln(aff ) ln(bf ))dydy Y Y Dissipation D(f) = vanishes only for equilibria, conjecture f(t,y) t z : M 1 (E z )) = M 1 (f ) M 1 (f ) z s E z (y), z s M 1 (f ) > z s Becker-Döring and generalisations, strong fragmentation, Aizenman-Bak

34 KFU p. 22/45 The Becker-Döring model Large time saturation phenomenon in Becker-Döring assume initial mass ρ = M 1 (c ) larger than ρ s = M 1 (E zs ) < Expect c i (t) E i zs i as t. The remaining mass ρ ρ s should go to larger and larger clusters as t. O. Penrose The Becker-Döring equations at large times and their connection with the LSW theory of coarsening J. Stat. Phys. 87 (1997) pp B. Niethammer On the evolution of large clusters in the Becker-Döring model J. Nonlinear Science 13 (23) pp

35 KFU p. 23/45 Existence of C-F with diffusion (Global) weak solutions of continuous C-F with diffusion Let T (, ], initial data f in L 1 (Ω R + ; (1 + y)dxdy) A weak solution of C-F on [,T) is a non-negative function f C((,T);L 1 (Ω R + )) L (,T;L 1 (Ω R + ;ydydx)) with f() = f in, f L 1 ((,T) (1/R,R);W 1,1 (Ω)), R R + and Q 1,2,3,4 (f) L 1 ((,T) Ω (,R)), which satisfies Ω ( ) t ψ(t)f(t) ψ()f in dydx + +d(y) f ψ) dydxds = 1 2 t Ω Ω ( f t ψ Q(f)ψ dydxds t (,T) and compactly supported ψ C 1 ([,T] Ω R + ).

36 KFU p. 24/45 Existence of C-F with diffusion A priori estimate and weak compactness Weak compactness lemma: Let T R + and (f n ) a sequence such that for all n 1 ( ( ) ) sup f n (t) 1 + y + ln fn (t) dydx C T t [,T] Ω Y E(y) t (a n f n f n b n f n)(ln(a n f n f n) ln(b n f n))dydy dxds < C T Ω Y Y Then, (f n ) weakly compact in L 1 ((,T) Ω R + ) and (Q i (f n )) weakly compact in L 1 ((,T) Ω (,R)) for i 1, 2, 3, 4 and R R +. Moreover, exists a weakly compact subset K w L 1 (Ω R + ) such that (f n (t)) K w for all t [,T] and n 1.

37 KFU p. 24/45 Existence of C-F with diffusion A priori estimate and weak compactness Dunford-Pettis theorem: A sequence (f n ) is contained in a weakly compact subset K w L 1 (Ω R + ) if (f n ) bounded in L 1 (Ω R + ) and for all ε >, there exists a measurable S Ω R + with S < such that sup n 1 (Ω R + )\S f n ε

38 KFU p. 25/45 Discrete C-F with diffusion 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 (Ω)), i N, T > sup t Ω a i,j c i c j L 1 ([,T] Ω), j=1 [ ] ic i (t,x) dx i=1 Ω [ i=1 ] ic i(x) dx, Ph. Laurençot, S. Mischler Global existence for the discrete diffusive coagulation-fragmentation equation in L 1 Rev. Mat. Iberoamericana 18 (22) pp

39 Discrete C-F with diffusion Lemma: L 2 estimates via duality Assume coefficients like above and ρ = i=1 ic i L 2 (Ω) Then, for all T > ρ L 2 (Ω T ) ( 1 + sup ) i{d i } T ρ L inf i {d i } 2 (Ω), or for degenerate diffusion T Ω [ i=1 ][ id i c i (t,x) i=1 ] ic i (t,x) 4T sup i N {d i } ρ L 2 (Ω). J.A. Cañizo, L. Desvillettes, K. F. Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion, Ann. Inst. H. Poincaré (C) Anal. Non Linéaire, 27 no.2 (21) pp KFU p. 26/45

40 KFU p. 27/45 Discrete coagulation-fragmentation models Proof of duality bounds Denoting M(t,x) = 1 ρ i=1 id i c i, then M sup i N {d i } and t ρ x (M ρ) =. Multiplying with the solution w of the dual problem: ( t w + M x w) = Φ M, x w n(x) Ω =, w(t, ) = for any smooth testfunction Φ := Φ(t,x) leads to T Ω Φ(t,x) M(t,x)ρ(t,x)dxdt = Ω w(,x)ρ(,x)dx.

41 KFU p. 27/45 Discrete coagulation-fragmentation models Proof of duality bounds (Formally) multiplying ( t w + M x w) = Φ M by x w yields Similar, Hence, w(,x) 2 T T Ω ( T Ω M ( x w) 2 dxdt t w 2 M dxdt 4 T T Ω Ω Φ 2 dxdt. Φ 2 dxdt. M t w M dt) 2 4T M L (Ω) T Ω Φ 2 dxdt.

42 KFU p. 27/45 Discrete coagulation-fragmentation models Proof of duality bounds Recalling above T Ω Φ M ρdxdt ρ(, ) L 2 (Ω) w(, ) L 2 (Ω) 2 T M L (Ω) Φ L 2 ([,T] Ω) ρ(, ) L 2 (Ω). for all nonnegative smooth functions Φ, we obtain by duality that T Ω M ρ L 2 (Ω) 2 [ i=1 ][ d i ic i (t,x) T M L (Ω) ρ(, ) L 2 (Ω), i=1 ] ic i (t,x) 4T sup i N {d i } ρ 2 L 2 (Ω).

43 KFU p. 28/45 Discrete C-F with diffusion Absence of gelation Theorem: 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 [,T] for all T >, i.e. for a test-sequence {ψ i } i 1 with lim i ψ i, we have: iψ i c i C(T) Ω i=1 As a consequence, the mass is conserved ρ(t,x)dx = ρ (x)dx for all t. Ω Ω

44 Continuous C-F with diffusion Inhomogeneous C-F with normalised rates Continuous in size density f(t,x,y), t, x Ω, y [, ) coagulation-fragmentation coefficients a(y,y ) = b(y) = 1 t f d(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 d(y) d (δ), y [δ,δ 1 ], < d 1 + y d(y), y [, ) M. Aizenman, T. Bak Convergence to equilibrium in a system of reacting polymers Comm. Math. Phys. 65 (1979) pp KFU p. 29/45

45 KFU p. 3/45 Inhomogeneous C-F with normalised rates Macroscopic densities amount of monomers or mass density N, number density M N = y f(y )dy, M = f(y )dy conservation of the total mass ( ) t N x d(y )y f(y )dy = ( ) t M x d(y )f(y )dy = N M 2 Large time behaviour?

46 KFU p. 31/45 Large Time Behaviour Entropy Method E entropy functional, nonincreasing D entropy dissipation d dt (E E ) = D provided conservation laws: D = steady state D Φ(E E ), Φ() =, Φ convergence in entropy, exponential if Φ () > convergence in L 1 : Cziszár-Kullback-Pinsker type inequalities

47 KFU p. 32/45 Advantages Entropy Method based on functional inequalities "robust" avoids linearisation "global" results explicit constants (non)linear diffusion: [O], [CJMTU], [AMTU],... inhomogeneous kinetic equations: [DV] reaction-diffusion systems: [DF] (no Bakry-Emery for systems)

48 KFU p. 33/45 Inhomogeneous C-F with normalised rates Entropy (free energy functional) Entropy functional H(f)(t,x) = (f ln f f) dy, Entropy dissipation: f = f(y), f = f(y ), f = f(y + y ) d H(f)dx = D H (f) dt Ω D H (f) = d(y) xf 2 dy dx Ω f ( ) f + (f ff ) ln dy dy dx ff Ω

49 KFU p. 34/45 Inhomogeneous C-F with normalised rates 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 Lower bound of entropy dissipation D H (f) d(y) xf 2 f Ω dy dx + M H(f f N,N ) + 2(M N) 2 Relative entropy H(f f N,N ) = H(f) H(f N,N )

50 KFU p. 35/45 Inhomogeneous C-F with normalised rates Local and global equilibria Intermediate equilibria with the moments N and M = N f N,N = e 1 N y Global equilibrium f = e y N constant in x satisfying M 2 = N preserves the initial mass N = N(x)dx

51 KFU p. 36/45 Inhomogeneous C-F with normalised rates Existence results Global existence of weak solutions satisfying the entropy dissipation inequality Ω H(f(t))dx + t D H (f(s))ds Diffusivity d(y) L ([1/R,R]) for all R > Equilibria attract all global weak solutions (no rate, De la Salle principle) Ω H(f )dx Ph. Laurençot, S. Mischler, The continuous coagulation-fragmentation equation with diffusion Arch. Rat. Mech. Anal. 162 (22), pp

52 Inhomogeneous C-F with normalised rates Exponential convergence to equilibrium Nonnegative initial data (1 + y + ln f )f L 1 ((, 1) (, )) with positive initial mass 1 N (x)dx = N > on Ω = (, 1). At most linearly degenerating diffusion coefficients. 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, L. Desvillettes, K. F. Large time asymptotics for a Continuous Coagulation-Fragmentation Model with Degenerate Size-dependent Diffusion, SIAM J. Math. Anal. 41 no.6 (29) pp KFU p. 37/45

53 KFU p. 38/45 inhomogeneous CF with constant kernel 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 Ω Ω 2p. Then, for all A 1 and p > 1 C D 1 (f) H(f f )dx C M 2p A M L x Ω A, 2p+1 with a constant C = C(M,N,d,P(Ω)) depending only on M, N, d, and the Poincaré constant P(Ω).

54 KFU p. 39/45 Inhomogeneous Aizenman-Bak Fast-reaction limit t f ε d(y) x f ε = 1 ε (Q coag(f ε,f ε ) + Q frag (f ε )) formal limit: f ε e y N (t,x) satisfying the nonlinear, nondegenerate diffusion equation t N (t,x) x n(n (t,x)) = where with n(n) := d(y)y e y N dy < inf {d(y)}n n(n) sup {d(y)}n. [, ) [, )

55 KFU p. 4/45 Inhomogeneous Aizenman-Bak Fast-reaction limit Theorems: convergence without rate using compactness assuming lower bound: convergence with rate in ε a a J.A. Carrillo, L. Desvillettes, K. F., Rigorous Derivation of a Nonlinear Diffusion Equation as Fast-Reaction Limit of a continuous Coagulation Fragmentation Model with Diffusion, Comm. Partial Differential Equations 34 no.1-12 (29) pp J.A. Carrillo, L. Desvillettes, K. F., Fast-Reaction Limit for the Inhomogeneous Aizenman-Bak Model. Kinetic and Related Models 1 no. 1 (28) pp

56 KFU p. 41/45 Coagulation-Fragmentation Models THANK YOU VERY MUCH!

57 KFU p. 42/45 References : Motivation and Modelling The beginning M. Smoluchowski Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen Physik. Zeitschr. 17 (1916) pp M. Smoluchowski Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen Zeitschrift f. physik. Chemie 92 (1917) pp H. Müller Zur allgemeinen Theorie der raschen Koagulation Kolloidchemische Beihefte 27 (1928) pp Ph. Laurençot, S. Mischler On coalescence equations and related models In Modeling and computational methods for kinetics equations, Model. Simul. Sci. Eng. Technol. Birkhäuser Boston, Boston, MA (24) pp

58 KFU p. 43/45 References : Motivation and Modelling Kernels R.L. Drake A general mathematical survay of the coagulation equation International Reviews in Aerosol Physics and Chemistry, Oxford (1972) pp A.M. Golovin The solution of the coagulation equation for cloud droplets in a rising air current Izv. Geophys. Ser. 5 (1963) pp W.H. Stockmayer Theory of molecular size distribution and gel formation in branched-chain polymers J. Chem. Phys. 11 (1943) pp F. Leyvraz, H.R. Tschudi Singularities in the kinetics of coagulation processes J. Phys. A 34 (1981) pp

59 KFU p. 44/45 References : Motivation and Modelling Absence of gelation and gelation time W.H. White A global existence theorem for Smoluchowski s coagulation equations Proc. Amer. Math. Soc. 8 (198) pp J.M. Ball, J. Carr The discrete coagulation-fragmentation equations : basic properties, uniqueness, and density conservation J. Stat. Phys. 61 (199) pp M.H. Ernst, R.M. Ziff, E.M. Hendricks Coagulation processes with a phase transition J. Colloid Interface Sci. 97 (1984) pp J.R. Norris Cluster coagulation Comm. Math. Phys. 29 (2)

60 KFU p. 45/45 References : Motivation and Modelling Profiles and Scalings M. Escobedo, S. Mischler, B. Perthame Gelation in coagulation and fragmentation models Comm. Math. Phys. 231 (22) pp P.G.J. van Dongen, M.H. Ernst Scaling solutions of Smoluchowski s coagulation equation J. Stat. Phys. 5 (1988) pp