Coagulation-Fragmentation Models

Transcription

1 Porto Ercole June 212 p. 1/77 Coagulation-Fragmentation Models Klemens Fellner Institute for Mathematics and Scientific Computing University of Graz

2 Porto Ercole June 212 p. 2/77 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 Porto Ercole June 212 p. 3/77 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 Porto Ercole June 212 p. 4/77 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 Porto Ercole June 212 p. 5/77 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 Porto Ercole June 212 p. 6/77 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 Porto Ercole June 212 p. 6/77 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 Porto Ercole June 212 p. 6/77 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 Porto Ercole June 212 p. 6/77 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 Porto Ercole June 212 p. 7/77 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 ) = a(y,y)

11 Porto Ercole June 212 p. 8/77 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 Porto Ercole June 212 p. 9/77 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 Porto Ercole June 212 p. 9/77 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 Porto Ercole June 212 p. 9/77 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 Porto Ercole June 212 p. 9/77 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 Porto Ercole June 212 p. 1/77 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 Porto Ercole June 212 p. 1/77 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 Porto Ercole June 212 p. 1/77 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 Porto Ercole June 212 p. 1/77 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 All the above kernels a(y,y ) satisfy a(y,y ) a(y,y + y ) + a(y,y + y ), y,y Y from which L p -Norm: ϕ(y) = pf(t,y) p 1 t f(t, ) L p non-increasing for p 1

20 Porto Ercole June 212 p. 11/77 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

21 Porto Ercole June 212 p. 11/77 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)

22 Porto Ercole June 212 p. 11/77 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

23 Porto Ercole June 212 p. 12/77 The Smoluchowski equation More gelation for other examples of kernels Kernel a(y,y ) = y α (y ) β + (y ) α (y) β, α,β [, 1] For λ = α + β (1, 2]: M k = y k f(t,y)dy L 2 ((, )) for k ( λ 2, 1 + λ ) 2 Gelation: have M 1 L 2 ((, ) for λ > 1

24 Porto Ercole June 212 p. 13/77 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 f(t,y)y 2 dy = 1 2 yy f(y)f(y ) ( (y + y ) 2 y 2 (y ) 2) dy dy

25 Porto Ercole June 212 p. 13/77 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) = y 2 (y ) 2 f(y)f(y )dy dy

26 Porto Ercole June 212 p. 13/77 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!

27 Porto Ercole June 212 p. 14/77 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). Testing with ϕ(y) = e py y for p [, ) d dt f(t,y)ϕ(y)dy = 1 2 yy f(y)f(y ) (ϕ ϕ ϕ ) dy dy

28 Porto Ercole June 212 p. 14/77 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). d dt E(t,p) = 1 2 yy f(y)f(y ) ) (e py e py (y + y ) e py y e py y dy dy

29 Porto Ercole June 212 p. 14/77 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). d dt E(t,p) = e py y 2 f(y)dy e py y f(y )dy e py y 2 f(y)dy y f(y )dy

30 Porto Ercole June 212 p. 14/77 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

31 Porto Ercole June 212 p. 15/77 The Smoluchowski equation Subcritical kernels Absence of gelation, Conservation of mass a(y,y ) C(1 + y + y ) a(y,y ) = y α (y ) β + (y ) α (y) β and λ = α + β 1 See below

32 Porto Ercole June 212 p. 16/77 The Smoluchowski equation Loss of mass in infinite time Number of particles is reduced by coagulation. Rigorous: Assume a(y,y ) > for (y,y ) Y Y. Then, M k (t) = y k f(t,y)dy as t, k [, 1) Loss of mass in infinite time (even if M 1 is conserved) Speed: Assume λ [, 1) such that a(y,y ) (yy ) λ and f in a.e. on (,δ > ). Then, M k (t) C k t k for all k (, 1). J. Carr, F.P. da Costa Asymptotic behaviour of solutions to the coagulationfragmentation equations. II. Weak fragmentation J. Stat. Phys. 77 (1994) pp

33 Porto Ercole June 212 p. 17/77 The Smoluchowski equation Gelation profiles, many open problems Conjecture: For the kernel a(y,y ) = y α (y ) β + (y ) α (y) β, α,β [, 1], if λ = α + β (1, 2]: f(t g,y) y 3/2 λ/2 as y Dynamical scaling hypothesis: As mean particle size s(t) as t T f(t,y) 1 ( ) y s(t) ϕ as τ s(t) t T If a(ξy,ξy ) = ξ λ a(y,y ) conjecture that 1 s(t) τ ϕ ( y s(t)) self-similar solution (τ = 2 for T = and τ = (λ + 3)/2 for T = T g ).

34 Porto Ercole June 212 p. 18/77 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

35 Porto Ercole June 212 p. 19/77 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

36 Porto Ercole June 212 p. 2/77 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)

37 Porto Ercole June 212 p. 21/77 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

38 Porto Ercole June 212 p. 22/77 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

39 Porto Ercole June 212 p. 22/77 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

40 Porto Ercole June 212 p. 23/77 Discrete coagulation-fragmentation models Discrete in size models discrete size y = i N = Y, c i (), c = (c i ) d t c i = Q coag (c,c) + Q frag (c) = 1 2 i 1 a i j,j c i j c j a i,j c i c j + B i+j β i+j,i c i+j B i c i j=1 j=1 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 j=1 j β i,j, (i N,i 2).

41 Porto Ercole June 212 p. 24/77 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

42 Porto Ercole June 212 p. 24/77 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

43 Porto Ercole June 212 p. 25/77 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 little knows about C-F models except with detailed balance 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

44 Porto Ercole June 212 p. 26/77 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

45 Porto Ercole June 212 p. 27/77 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

46 Porto Ercole June 212 p. 28/77 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. How does this work? 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

47 Porto Ercole June 212 p. 28/77 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 ) < [Niethammer] extended [Penrose] by considering the coefficients a i = a 1 i α, b i = a i (z s + qi γ ), i 2 with α (, 1], γ [, 1), a 1 >, z s >, q > rescale time τ = ε 1 α+γ t cut-off i ε where i ε and εi ε as ε Goal is to capture saturation mass in i ε i=1 ic i(τ) ρ s

48 Porto Ercole June 212 p. 28/77 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 ) < alternative Becker-Döring i=1 ic i(τ) = ρ d τ c i = 1 (W ε 1 α+γ i 1 (c) W i (c)), i 2 where ( W i (c) = a i c 1 b ) i c i (b i+1 c i+1 b i c i ) a i = a 1 i α ( c 1 z s qi γ) (b i+1 c i+1 b i c i )

49 Porto Ercole June 212 p. 28/77 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 ) < Continuum approximation Rewrite for (τ,x) (, ) ((i 1/2)ε, (i + 1/2)ε) f(τ,x) = 1 ε 2c i(τ), W(τ,x) = 1 ε 2W i(f(τ)) Then, τ f = x W(f), W(f)(τ,x) a 1 ( x α u(τ) qx α γ) where u(τ) = ε γ (c 1 (τ) z s ) with c 1 (τ) E 1 z s and E i E i+1 = z s E 1 for large i

50 Porto Ercole June 212 p. 28/77 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 ) < for i i ε ln(ε): continuum approximation x iε One can show that for ε, i ε, x i ε ε i ε i=1 ic i (τ) ρ s, xf(τ,x)dx = ρ ρ s. For general models saturation is an open problem.

51 Porto Ercole June 212 p. 29/77 Discrete C-F with diffusion 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) = Q 1 (f,f) Q 2 (f,f) + Q 3 (f) Q 4 (f) time t, size y Y position x Ω, normalised with Ω = 1 homogeneous Neumann x f(t,x,y) ν(x) = on Ω non-negative initial data f (x,y) size-dependent diffusion coefficients d(y)

52 Porto Ercole June 212 p. 3/77 Discrete C-F with diffusion Discrete C-F models discrete in size models y = i N, f(y) = c i, c = (c i ) Q coag (c,c) = 1 2 Q frag (c) = 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 (mass conservation) a i,j = a j,i, β i,j, (i,j N) B 1 =, B i, (i N) i = i 1 j=1 j β i,j, (i N,i 2)

53 Porto Ercole June 212 p. 31/77 Discrete C-F with diffusion 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 (ϕ i = i) ρ(t, ) L 1 = ic i (t,x)dx Ω i=1 ic i(x)dx = ρ L 1 Ω i=1

54 Porto Ercole June 212 p. 32/77 Discrete C-F with diffusion Existence theory natural initial data: f in (y) L 1 1(Y ) = L 1 (Y, (1 + y)dy) two basic approaches: fixed-point and compactness methods in spaces of continuous functions J.B. McLeod On the scalar transport equation Proc. London Math. Soc. (3) 14 (1964) pp Z.A. Melzak A scalar transport equation Trans. Amer. Math. Soc. 85 (1957) pp P.B. Dubovskii, I. Stewart Existence, uniqueness and mass conservation for the coagulation-fragmentation equation Math. Meth. Appl. Sci. 19 (1996) V.A. Galkin, P.B. Dubovskii Solution of the coagulation equation with unbounded kernels Differential Equations 22 (1986) pp

55 Porto Ercole June 212 p. 32/77 Discrete C-F with diffusion Existence theory natural initial data: f in (y) L 1 1(Y ) = L 1 (Y, (1 + y)dy) two basic approaches: weak and strong compactness methods in L 1 (Y ) J.M. Ball, J. Carr, O. Penrose The Becker-Döring cluster equation : basic properties, uniqueness and asymptotic behaviour of solutions Comm. Math. Phys. 14 (1986) pp I.W. Stewart A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels Math. Methods Appl. Sci 11 (1989) pp M. Escobedo, Ph. Laurençot, S. Mischler, P. Berthame Gelation and mass conservation in coagulation-fragmentation models J. Differential Equations Ph. Laurençot, S. Mischler The continuous coagulation-fragmentation equation with diffusion Arch. Rat. Mech. Anal. 162 (22) pp

56 Porto Ercole June 212 p. 33/77 Discrete C-F with diffusion (Global) weak solutions of discrete C-F with diffusion Let T (, ], initial data c in i L 1 (Ω), i=1 i c i 1 < A weak solution of C-F on [,T) is a non-negative function c i C([,T);L 1 (Ω)), sup t [,T) i=1 i c i 1 < C(c in i ) with Q 2 (c) = j=1 a i,jc i c j L 1 ((,T) Ω),... Moreover, c i are mild solutions of c i (t) = e d i x t c in i + t e d i x (t s) (Q coag + Q frag )(c(s))ds and e d i x t is the C -semigroup of d i x in L 1 (Ω) with homogeneous Neumann conditions.

57 Porto Ercole June 212 p. 34/77 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 + ).

58 Porto Ercole June 212 p. 35/77 Existence of C-F with diffusion Stability principle for (global) weak solutions in L 1 For T (, ) let (f n ) be weak solutions of C-F with coefficients a n a, b n b and d n d and initial datum f in. For all n N let K w L 1 (Ω R + ) be a weakly compact with f n (t) K w, for each t [,T) and suppose moreover for all R > and i {1, 2, 3, 4} that f n (t)(1 + y)dydx C T, sup t [,T] Ω Q i,n (f n ) weakly compact in L 1 ((,T) Ω (,R)) Ph. Laurençot, S. Mischler The continuous coagulation-fragmentation equation with diffusion Arch. Rat. Mech. Anal. 162 (22) pp

59 Porto Ercole June 212 p. 36/77 Existence of C-F with diffusion Stability principle for (global) weak solutions in L 1 Then, there exists a subsequence (f nk ) and f such that f nk f in C([,T);w L 1 (Ω R + )) Q i,nk (f nk ) Q i (f) weakly in L 1 ((,t) Ω (,R)) for R R +, i {1, 2, 3, 4}. Thus, f is a weak solution of C-F on [,T). Moreover, ψ(y)f nk dy ψ(y)f dy in L 1 ((,T) Ω) for ψ D(R + ). Thus, M 1 (t) M 1 ().

60 Porto Ercole June 212 p. 37/77 Existence of C-F with diffusion A priori estimate with detailed balance Entropy functional: H(f E(y)) = Ω Y f ( ln( f E ) 1) dy H-Theorem f = f(y), f = f(y ), f = f(y + y ) d dt H(f E) Ω Total mass Y Y Ω Y d(y) f 2 f dydx (aff bf )(ln(aff ) ln(bf ))dydy dx = C := sup Ω t [, ) Y yf(t,x,y)dydx <

61 Porto Ercole June 212 p. 37/77 Existence of C-F with diffusion A priori estimate with detailed balance Initial data d dt Ω Y H(f in E) < C < ( ( f ) f ln 1 dy + d(y) E) f 2 dydx Ω Y f (aff bf )(ln(aff ) ln(bf ))dydy dx = Ω Y Y A priori estimates H(f(t) E) < C t t Ω Ω Y Y f 2 d(y) f dydxds < C Y (aff bf )(ln(aff ) ln(bf ))dydy dxds < C for C = C(Ω,E(y),H(f in E),C ).

62 Porto Ercole June 212 p. 38/77 Existence of C-F with diffusion A priori estimate and weak compactness Lemma: Let ξ {, 1} be measurable on R + Ω R + and α e 2. Then, Ω Y ξ(t)f(t)dydx 2(α + e 1 ) Lemma: For t R + holds with f(t) ln f(t) Ω Y f(t) ( 1 + Ω Y E(y) ξ(t)e dydx ( ) ) f(t) ln dydx C E(y) + 2 ln(α) H(f(t) E) f(t) ln f(t) + 2E E(y) e

63 Porto Ercole June 212 p. 38/77 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.

64 Porto Ercole June 212 p. 38/77 Existence of C-F with diffusion A priori estimate and weak compactness Proof of weak compactness lemma: For S Ω R + measurable, S <, α e 2 follows from above S Ω f n (t)dydx 4α α S f n (t)dydx C T α E(y)dydx + 2C T ln(α) C T(E, Ω,α) Then, f n (t) K w L 1 (Ω R + ) for all n 1 with K w defined that g K w satisfying the above equations for all measurable S Ω R + with S < and α e 2. By the Dunford-Pettis theorem is K w weakly compact.

65 Porto Ercole June 212 p. 38/77 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

66 Porto Ercole June 212 p. 38/77 Existence of C-F with diffusion A priori estimate and weak compactness Proof of weak compactness lemma: For all R R + Q 4,n (f n ) = f n(t,x,y) 2 y b n (y,y y )dy R b n f n Therefore, the sequence (Q 4,n (f n )) (where Q 4,n may be an approximation of Q 4 with coefficients a n a, b n b and d n d) is weakly compact in L 1 ((,T) Ω (,R)) since (f n ) is weakly compact and b n is assumed bounded.

67 Porto Ercole June 212 p. 38/77 Existence of C-F with diffusion A priori estimate and weak compactness Proof of weak compactness lemma: For all α e 2, the elementary inequality η αξ+ yields for measurable S (,T) Ω (,R) S (η ξ) ln(η/ξ) ln(α) a n (y,y y )f n (y )f n (y y )dydxdt α sup Q 4 (f n )dydxdt + C T n 1 ln(α) and letting α shows that the sequence (Q 1,n (f n )) is weakly compact in L 1 ((,T) Ω (,R)) for all R R +. S

68 Porto Ercole June 212 p. 38/77 Existence of C-F with diffusion A priori estimate and weak compactness Proof of weak compactness: For all α 2R R + α Q 3,n (f n )dydxdt b n (y,y y )f n (y )dy dydxdt S S + b n L (a R, ) f n (y )dy dydxdt S α α C f n (y )dy dxdt + b n L (a R, )C(T,R) S and the sequence (Q 3n (f n )) is weakly compact in L 1 ((,T) Ω (,R)) since (f n ) is weakly compact.

69 Porto Ercole June 212 p. 38/77 Existence of C-F with diffusion A priori estimate and weak compactness Proof of weak compactness: From t Ω Y Y ( ) (a n f n f n b nf n ) ln an f n f n dydy dxds < C b n f n T follows also that (Q 2,n (f n )) is weakly compact in L 1 ((,T) Ω (,R)) from the weak compactness of (Q 3,n (f n )) in a similar argument as above showing the weak compactness of (Q 1,n (f n )).

70 Porto Ercole June 212 p. 39/77 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

71 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 Porto Ercole June 212 p. 4/77

72 Porto Ercole June 212 p. 41/77 Discrete C-F with diffusion 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) Ω =, w(t, ) = for any smooth function H := H(t,x) leads to T Ω H(t,x) A(t,x)ρ(t,x)dxdt = Ω w(,x)ρ(,x)dx.

73 Porto Ercole June 212 p. 41/77 Discrete C-F with diffusion Proof of duality bounds Multiplying the dual problem by x w yields T Ω t ( w 2 /2)dxdt + T C ε Ω T T Ω A ( x w) 2 dxdt H A( x w)dxdt. Ω H 2 dxdt + ε T Ω A( x w) 2 dxdt and with w(t) = T Ω A ( x w) 2 dxdt T Ω H 2 dxdt.

74 Porto Ercole June 212 p. 41/77 Discrete C-F with diffusion Proof of duality bounds Multiplying the dual problem by x w yields and T T Ω Ω A ( x w) 2 dxdt t w 2 Hence, w(,x) 2 ( T Ω A dxdt 4 T T Ω A t w A dt ) 2 and w(,x) 2 dx 4T A L (Ω) Ω H 2 dxdt. H 2 dxdt. T Ω H 2 dxdt.

75 Porto Ercole June 212 p. 41/77 Discrete C-F with diffusion Proof of duality bounds Recalling above T Ω H Aρdxdt ρ(, ) L 2 (Ω) w(, ) L 2 (Ω) 2 T A L (Ω) H L 2 ([,T] Ω) ρ(, ) L 2 (Ω). for all (nonnegative smooth) functions H, we obtain by duality that Aρ L 2 (Ω) 2 T A L (Ω) ρ(, ) L 2 (Ω). L. Desvillettes, K. F., M. Pierre, J. Vovelle About Global Existence for Quadratic Systems of Reaction-Diffusion, J. Advanced Nonlinear Studies 7 no 3. (27) pp

76 Porto Ercole June 212 p. 42/77 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

77 Porto Ercole June 212 p. 43/77 Discrete C-F with diffusion Proof of absence of gelation in special case Consider a i,j = ij and B i =, Then, (using log(1 + x) C x) Ω i= i log ic i (,x)dx < d dt = Ω i log ic i dx i=1 Ω i=1 j=1 ij ci c j ( i log(1 + j i ) + j log(1 + i j ) )dx 2 Ω i=1 ij c i c j dx 2 j=1 Ω ρ(t,x) 2 dx.

78 Porto Ercole June 212 p. 43/77 Discrete C-F with diffusion Proof of absence of gelation in special case Consider a i,j = ij and B i =, Ω i= As a consequence, we have for all T > Ω i log ic i (T,x)dx i= Ω + 2 i log ic i (,x)dx < i log ic i (,x)dx i= T Ω ρ(t,x) 2 dxdt, and the propagation of the moment i= i log ic i(,x)dx ensures the conservation of the mass.

79 Porto Ercole June 212 p. 44/77 Discrete C-F with diffusion Degenerate Diffusion: Absence of gelation Assume decaying diffusion coefficients with γ [, 1] d i C i γ, and coagulation coefficients bounded like ) a i,j C (i α j β + i β j α, with α + β + γ 1, α,β [, 1). Then, the mass is conserved Ω ρ(t,x)dx = Ω ρ (x)dx, t. S. Simons, D.R. Simpson The effect of particle coagulation on the diffusive relaxation of a spatially inhomogeneous aerosol J. Phys. A 21 (1988) pp

80 Porto Ercole June 212 p. 45/77 Discrete C-F with diffusion Existence theory of discrete C-F with diffusion By duality we have uniform L 2 -bound of approximating systems independent of a i,j. The assumption lim j + a i,j j Q,M coag = cm i = is needed for the limit of j=1 a i,j c M j. as c M i converges to c i weak- in L (Ω T ), we need j=1 a i,j c M j j=1 a i,j c j strongly in L 1 (Ω T ) T a i,j (c M j c j ) dxdt 2sup a i,j j ρ 2+sup c M j c j L 1 j J Ω j j J

81 Porto Ercole June 212 p. 46/77 Discrete C-F with diffusion Existence theory generalised quadratic C-F models t c i d i x c i = a k,l c k c l a i,k c i c k k+l=i k=1 b k,l c k c l β i,k,l k,l=1 i<max{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 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

82 Porto Ercole June 212 p. 47/77 Discrete C-F with diffusion Strong solutions of C-F with diffusion? Reversible reaction-diffusion of 4 species A 1 + A 2 A 3 + A 4 concentrations a i, individual diffusivities d i > t a 1 d 1 x a 1 = a 1 a 2 + a 3 a 4 t a 2 d 2 x a 2 = a 1 a 2 + a 3 a 4 t a 3 d 3 x a 3 = +a 1 a 2 a 3 a 4 t a 4 d 4 x a 4 = +a 1 a 2 a 3 a 4 bounded, smooth domain Ω R N, Ω = 1 homogeneous Neumann boundary conditions mass action kinetics (quadratic, no invariant regions) integrable and bounded initial data a i, (x)

83 Porto Ercole June 212 p. 48/77 Discrete C-F with diffusion Strong solutions of C-F with diffusion? 1D: global classical solutions (Amann) exponential convergence to equilibrium in all Sobolev norms. 2D: global classical (De Giorgi s method) 2D: global classical (duality method) alld: global weak L 2 -solutions via duality explicit exponential decay (rates) in L p, 1 p < 2 T. Goudon, A. Vasseur Regularity analysis for systems of reaction-diffusion equations Annales de l École Normale Supérieure J.A. Cañizo, K.F., L. Desvillettes, F. Otto, in preparation L. Desvillettes, K. F., Revista Mat. Ibero. (28), Proc. Equadiff 27

84 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 Porto Ercole June 212 p. 49/77

85 Porto Ercole June 212 p. 5/77 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?

86 Porto Ercole June 212 p. 51/77 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

87 Porto Ercole June 212 p. 52/77 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)

88 Porto Ercole June 212 p. 53/77 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

89 Porto Ercole June 212 p. 54/77 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 )

90 Porto Ercole June 212 p. 55/77 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

91 Porto Ercole June 212 p. 56/77 Inhomogeneous C-F with normalised rates 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 Ω Ω N dx ) N dx Ω

92 Porto Ercole June 212 p. 57/77 Inhomogeneous C-F with normalised rates Existence results Global existence and uniqueness of classical solutions (in 1D with coefficients not quite [Aizenman, Bak]) H. Amann Coagulation-fragmentation processes, Arch. Rat. Mech. Anal. 151 (2), pp H. Amann, C. Walker Local and global strong solutions to continuous coagulationfragmentation equations with diffusion, J. Differential Equations 218 (25), pp

93 Porto Ercole June 212 p. 58/77 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

94 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 Porto Ercole June 212 p. 59/77

95 Porto Ercole June 212 p. 6/77 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 f N,N ( N )dx + 2 ) N

96 Porto Ercole June 212 p. 6/77 Entropy Entropy-Dissipation Estimate needs M L x, M L t (L 1 x), M > Step 2) "Reacting" moments M N 1 H(f f )dx = 1 H(f f N,N ( N )dx + 2 ) N N N 2 [ M ] N 2 L + M 2 M 2 N x L 2 x This inequality quantifies how a reversible reaction of two species M N passes the diffusion effects from M to N.

97 Porto Ercole June 212 p. 6/77 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

98 Porto Ercole June 212 p. 6/77 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 Ω 2 M A M A 2 L + 4 ( 2 y f(y)dy) p dx 2 x for any p > 1. C(P,d )A M L x + 4 A 2p M L x M 2p A 2p Ω Ω d(y) xf 2 f dydx

99 Porto Ercole June 212 p. 6/77 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 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).

100 Porto Ercole June 212 p. 61/77 Proof of Theorem 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). Balancing the r.h.s. (e.g. positive term = 1/2 negative term) by choosing A = A(t) > 2 yields 1 A C 1/2 ( 1 C H(f f )dx M L x 2 8p2 ) 1 2p, Thus, Gronwall yields algebraic rate 2p for all p > 1.

101 Porto Ercole June 212 p. 61/77 Proof of Theorem Algebraic rate for all p > 1 Faster than polynominal rate Then, by summing w.r.t. p N 1 H(f(t) f )dx L(t C), where (for all 1 < α < 2) L 1 (t) = q 1, even t q (C q) q 2 2q2 = q 1, even t q e 2q2 ln2 q ln(q C) C(α)e ln2(α 1) (t) for all t large enough and any 1 < α < 2.

102 Porto Ercole June 212 p. 62/77 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) d(y) dxdy ]1 2 [ 1 d(y) x f(t,x,y) 2 dxdy ]1 2 M(t,x) 1 M(t, x)d x + d 1/2 (M + N ) 1/2 D(f(t)) 1/2.

103 Porto Ercole June 212 p. 63/77 Lemma: Ω M(t,x)dx M A-priori Estimates d dt 1 M(t,x)dx = 1 1 (N M 2 )dx N in (x)dx (m + m 1 (t)) 1 M(t,x)dx 1 M(t,x)dx e µ 1 M in (x)dxe R t (m +m 1 (σ)) dσ t N in (x)dx e R t s (m +m 1 (σ)) dσ ds [e m t M in L1x + 1 ] e m t N in L 1 x m

104 Porto Ercole June 212 p. 64/77 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 in ) 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

105 Porto Ercole June 212 p. 65/77 Moments M p (f)(t) := 1 A-priori Estimates y p f dy dx How start with (1 + y)f in L 1?? Vallée-Poussin lemma: for any f L 1 exists ϕ ր such that ϕf L 1. Idea: F = f(t)dt and ϕ = F α for α < 1 x ϕf dy = F F α dy = 1 1 α F() α+1 Then, for a regularised version of ϕ(y), calculate "y ϕ(y)-moment" and y ϕ(y)q frag C 1 y 1+δ

106 Porto Ercole June 212 p. 66/77 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,d and d (δ)) M(t,x) M. Idea: linear lower bound for lost terms t f d(y) xx f = g 1 y f M(t, ) L x f where g 1 is nonnegative, then ( ) ( t + d(y) xx ) f e ty+r t M(s, ) L x ds = g 2 where g 2 is nonnegative.

107 Porto Ercole June 212 p. 67/77 A-priori Estimates Lower bounds M(t,x) M and N(t,x) N Fourier series and Poisson s formula for t h d 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 4d t dt k= dz 1 e (2k+x z) 4d (t s) d (t s) 2 dzds h and G "mirrored" around

108 Porto Ercole June 212 p. 68/77 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 d 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 d t )1 δ C e (2t + 1 d t )1 δ 1 1/δ δ f(t 1,z,y)dydz ( M δ N K δ Choosing δ and K, we get that M(t 1 + t,x) M. 1 H(f)dx ln K )

109 Porto Ercole June 212 p. 69/77 Regularity, Interpolation (1 + y)q f(t,,y) f (y) L x dy C e α t Moment control implies T 1 (1 + y)q Q + (f,f)dy dxdt C T Regularising effect of 1D heat equation: t f d(y) xx f = g for all q [1, 3): 1 r + 1 = 1 p + 1 q f L r ([,T] Ω) C T d(y) 1 q 2q fin L p x + C T d(y) 1 q 2q g L p t,x and while d(y) 1 q 2q (1 + y) 1/3 for y large. Thus, f(,,y) L 3 ε ([t,t] Ω) C T ( f(,,y) L 1 x + Q + (f,f)(,,y) L 1 ([,T] Ω) )

110 Porto Ercole June 212 p. 7/77 Regularity, Interpolation (1 + y)q f(t,,y) f (y) L x dy C e α t A bootstrap yields after three iteration steps (1 + y) q f(t,,y) H 1 x dy C T Interpolation (1 + y) q f(t,,y) f (y) L x dy ( (1+y) q f(t,,y) f (y) 3/4 H 1 x ) f(t,,y) f (y) 1/4 L 1 x dy C 3/4 T e α T

111 Porto Ercole June 212 p. 71/77 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. [, ) [, )

112 Porto Ercole June 212 p. 72/77 Inhomogeneous Aizenman-Bak Fast-reaction limit Theorems: convergence without rate using compactness assuming lower bound: convergence with rate in ε 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

113 Porto Ercole June 212 p. 73/77 Fast-Reaction limit Dissipation of entropy ε d dt Ω H(f ε )dx Ω M ε H(f ε f N ε,n )dx ε + 2 ((M ε ) N ε ) 2 dx Ω Thus Ω M ε H(f ε f N ε,n ε )dxdt εc Assuming M ε M it follows from Cziszár-Kullback-Pinsker f ε e y Nε 2 L 2 t (L1 x,y) εc(m )

114 Porto Ercole June 212 p. 74/77 Expansion Fast-Reaction limit However, want f1 ε L 2 t,x(l 1 y((1 + y)dy)) An interpolation shows that for a < θ < 1, there exists Then y f ε = e Nε + ε θ f1, ε with x f1 ε ν(x) = on Ω t N ε x n(n ε ) = ε θ x d(y)yf ε 1 dy := ε θ x g ε where g ε L 2 t,x with x g ε ν(x) = on Ω.

115 Porto Ercole June 212 p. 75/77 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) Ω = converge in L 2 t,x as ε to the solution N of t N x n(n) = x N ν(x) Ω =

116 Porto Ercole June 212 p. 76/77 Fast-Reaction limit Compactness : proof Duality argument: w, w(t) =, x w ν(x) Ω = t w n(nε ) n(n) N ε N xw = H Holds that x w L 2 ([,T] Ω) C H L 2 ([,T] Ω) Then T Ω (Nε N)Hdxdt ε θ g ε L 2 t,x x w L 2 ([,T] Ω) Since H C ([,T] Ω) is arbitrary N ε N L 2 t,x C ε θ g ε L 2 t,x C ε θ

117 Porto Ercole June 212 p. 77/77 Coagulation-Fragmentation Models THANK YOU! THANKS TO THE ORGANISERS! (now and in the future)