Analysis of Diffuse Interface Models

Transcription

1 Analysis of Diffuse Interface Models Helmut Abels Faculty of Mathematics University of Regensburg Juli 8, 2010 Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

2 Modeling (I) We consider two (macroscopically) immiscible incompressible, viscous fluids like oil and water. Classical Models: Interface is a two-dimensional surface. Surface tension is proportional to the mean curvature. But: Sharp interface is an idealization (van der Waals). Fluid mix in a thin interfacial region. Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

3 Overview 1 Phase Separation and Cahn-Hilliard Equation 2 Analysis of the Navier-Stokes Equations 3 Diffuse Interface Model for Matched Densities 4 Other Diffuse Interface Models Outlook Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

4 Overview 1 Phase Separation and Cahn-Hilliard Equation 2 Analysis of the Navier-Stokes Equations 3 Diffuse Interface Model for Matched Densities 4 Other Diffuse Interface Models Outlook Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

5 Free Energy of a Two-Component Mixture Ansatz: We assume the fluids to be (partly) miscible. Let c j : Ω R be the concentration of the component j = 1, 2, c = c 1 c 2, and let E mix (c) = ε c(x) 2 dx + ε 1 f (c(x)) dx 2 Ω Ω be the free energy of the mixture, where Ω R d, d = 1, 2, 3, ε > 0 and f : R [0, ) with f (c) = 0 c = ±1. Moreover, we assume 1 c(x) dx = c ( 1, 1) if Ω <. Ω Ω Example: f (c) = 1 8 (1 c2 ) 2 Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

6 Remarks A typical profile of a diffuse interface is c(x) = tanh x 2ε, x R, which minimizes E mix in the case Ω = R with constraint c(x) x ± ±1. Modica-Mortola 77, Modica 87 proved E mix E mix,ε ε 0 σp in the sense of Γ-convergence (w.r.t. L 1 ), where { H d 1 ( E) = area( E) if v = 2χ E 1 P(v) = + else. and σ = σ(f ). Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

7 Critical Points Lemma Let Ω R d be bounded and { } c V := u L 2 (Ω) : u L 2 1 (Ω), E mix (u) <, u dx = c Ω Ω be a local minimum (a critical point) of E mix : V [0, ). Then δe mix δc := ε c + ε 1 f (c) const. Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

8 Proof of Lemma 1 Let ϕ C 1 (Ω) such that ϕ Ω = 0 and Ω ϕ dx = 0. Then c + tϕ V for all t R, and F (t) := E mix (c + tϕ), t R has a local minimum at t = 0. F (0) = 0 where F (0) = d dt ε c + t ϕ 2 dx Ω 2 + d t=0 dt ε 1 f (c + tϕ) dx Ω = ε c ϕ dx + ε 1 f (c)ϕ dx Ω Ω = ( ε c + ε 1 f (c))ϕ dx Ω Since ϕ C 1 (Ω) with Ω ϕ dx = 0 and ϕ Ω = 0 was arbitrary, we conclude δe mix := ε c + ε 1 f (c) const. δc (Choose e.g. ϕ = ψ 1 Ω Ω ψ dx with ψ C 1 (Ω).) t=0 Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

9 Cahn-Hilliard Equation (I) Let J : Ω (0, ) R d be the mass flux, i.e. d c(x, t) dx = n J(x, t) dσ(x) = dt V for all V Ω, t 0. Then V V div J(x, t) dx t c(x, t) = div J(x, t) for (x, t) Ω (0, ). Assumption (Cahn-Hilliard 58): For some m > 0 we have J = m µ µ = δe mix δc Remark: µ = δe mix δc const. J 0 (generalized Fick s law) = ε c + ε 1 f (c) (chemical potential) Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

10 Cahn-Hilliard Equation (II) We consider t c = m µ in Ω (0, ), (1) µ = ε c + ε 1 f (c) in Ω (0, ) (2) with the initial and boundary conditions n c Ω = n µ Ω = 0 on Ω (0, ), (3) c t=0 = c 0 in Ω. (4) Remark: For every smooth solution we have: d dt E mix(c(t)) = m µ(x, t) 2 dx. Questions: Does a unique solution c(t, x) exist for all t > 0? Does c(t, x) converge as t to a critical point of E mix? Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33 Ω

11 Well-Posedness and Convergence If f (c) is smooth: Existence: Elliott & Zheng 86, Convergence: Hoffmann & Rybka 99 Problem: Does c(t, x) [ 1, 1] hold if c 0 (x) [ 1, 1]? One solution: Use a singular free energy density as e.g. f (c) = θ((1 c) log(1 c) + (1 + c) log(1 + c)) θ c c 2, c [ 1, 1], with 0 < θ < θ c, cf. Cahn & Hilliard 58. Existence: Elliott & Luckhaus 91, Debussche & Dettori 95, Kenmochi et al. 95 Convergence: A. & Wilke 07 Remark: For every solution c(t, x) ( 1, 1) a.e. Other results: Existence of weak solutions for degenerate mobility (Elliott & Garcke 96) and double obstacle potential (Blowey & Elliott 91) Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

12 Cahn-Hilliard Equation with Singular Free energies Use that f (c) = f 0 (c) θ c 2 c2, where f 0 is convex. Then (1)-(2) are equivalent to t c ( c + f 0(c)) = } {{ } θ c c } {{ } monotone operator Lipschitz perturbation Existence of solutions: General result on perturbations of (maximal) monotone operators. Fundamental idea for the regularity theory: Use that implies c + f 0(c) = µ + θ c c c 2 L 2 (Ω) + f Ω ( c) 0 (c) c 2 dx C µ + θ } {{ } c c 2 L 2 (Ω). 0 Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

13 Convergence to Stationary Solutions (A. & Wilke 07) Proof and use the Lojasiewicz-Simon inequality E mix (c) E mix (c ) 1 θ C DE mix (c) H 1 (0) for all c in a neighborhood of a critical point c and some θ (0, 1 2 ]. (NB: f : [ 1, 1] R is analytic in ( 1, 1).) Essential: Show that c(t, x) [ 1 + ε, 1 ε] for all t T 1 >> 1. Standard calculation t c L 1 (0, ; H 1 (0) ) convergence. Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

14 Coarse Graining/Ostwald Ripening Question: What is the asymptotic behavior of c(t) as t? Sternberg & Zumbrun 98: For every stable critical point of Ω the diffuse interface is connected. Question: Is there the effect of Ostwald ripening present in the coupled Navier-Stokes/Cahn-Hilliard system below? How strong is it in dependence of the parameters? Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

15 Helmut Abels (U Regensburg) Diffuse Interface Models Simulation by S. Bartels Juli 8, / 33 Coarse Graining/Ostwald Ripening Question: What is the asymptotic behavior of c(t) as t? Sternberg & Zumbrun 98: For every stable critical point of Ω the diffuse interface is connected. Question: Is there the effect of Ostwald ripening present in the coupled Navier-Stokes/Cahn-Hilliard system below? How strong is it in dependence of the parameters?

16 Overview 1 Phase Separation and Cahn-Hilliard Equation 2 Analysis of the Navier-Stokes Equations 3 Diffuse Interface Model for Matched Densities 4 Other Diffuse Interface Models Outlook Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

17 Well-Posedness of Nonlinear Evolution Equations Typical results: 1 Local strong solutions: Usually short-time existence or global in time existence close to equilibria Solutions are unique and regular. Basis: Linearization, contraction argument 2 Global weak solutions: Solutions exists globally, but are not necessarily unique and regular. Basis: A priori/energy estimates, approximation and compactness General question: Gap between weak and strong solutions? Famous example: (Incompressible) Navier-Stokes equations in R 3 Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

18 Well-Posedness for the Navier-Stokes Equations t v + v v ν v + p = f div v = 0 v t=0 = v 0 Leray 34: Weak solutions exist for x R d, d = 2, 3, 0 < t <. d = 2: Uniqueness/smoothness for 0 < t <. d = 3: Uniqueness/smoothness for 0 < t < T 0. Possible singularities in time have measure zero. Serrin 62: Uniqueness/Smoothness for d = 3 if T ( ) r v(x, t) q q 2 dx dt < with 0 R 3 r + 3 < 1, 3 < q <. q Scheffer 77, Cafferelli/Kohn/Nirenberg 82: Singularities in space-time have Hausdorff-Dimension 5 3, 1, resp. Since 2000: Uniqueness/smoothness in d = 3 is a Millenium Problem (1 Mio. $ prize). Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

19 Overview 1 Phase Separation and Cahn-Hilliard Equation 2 Analysis of the Navier-Stokes Equations 3 Diffuse Interface Model for Matched Densities 4 Other Diffuse Interface Models Outlook Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

20 Basic Modeling Ansatz: Assume that the interfacial energy is given by E mix (c) = ε c(x) 2 dx + ε 1 f (c(x)) dx, 2 Ω where the free energy density f is a suitable double well potential. Diffusion: Assume that t c + v c = div J J = m µ µ := δe mix δc Ω (Fick s law) = ε c + ε 1 f (c) (chemical potential) where µ = δe mix δc is the chemical potential, m > 0. Classical models: Pure transport of the interface (m=0). Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

21 Diffuse Interface Model in the Case of Matched Densities If the densities of the fluids are the same, then one can derive: t v + v v div(ν(c)dv) + p = ε div( c c) (5) } {{ } } {{ } inner friction surface tension div v = 0 (6) where Dv = 1 2 ( v + v T ) together with t c + v c = m µ (7) µ = ε c + ε 1 f (c) (8) v Ω = n c Ω = n µ Ω = 0 on Ω (0, ), (9) (v, c) t=0 = (v 0, c 0 ) in Ω. (10) Derivation: Hohenberg & Halperin 74, Gurtin et al. 96 Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

22 Diffuse Interface Model in the Case of Matched Densities If the densities of the fluids are the same, then one can derive: t v + v v div(ν(c)dv) + p = ε div( c c) (5) } {{ } } {{ } inner friction surface tension div v = 0 (6) t c + v c = m µ (7) µ = ε c + ε 1 f (c) (8) where Dv = 1 2 ( v + v T ) Analytic results on well-posedness: Starovoitov 93, Boyer 03, X.Feng 06 Energy dissipation: For sufficiently smooth solutions we have d E(c(t), v(t)) = ν(c) Dv 2 dx m µ 2 dx with dt Ω Ω v(t) 2 E(c(t), v(t)) = E mix (c(t)) + dx 2 Ω Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

23 Diffuse Interface Model in the Case of Matched Densities If the densities of the fluids are the same, then one can derive: t v + v v div(ν(c)dv) + p = ε div( c c) (5) } {{ } } {{ } inner friction surface tension div v = 0 (6) where Dv = 1 2 ( v + v T ) Remark: (5) can be replaced by: t c + v c = m µ (7) µ = ε c + ε 1 f (c) (8) t v + v v div(ν(c)dv) + g = µ c where g = p + ε 1 f (c) + ε 2 c 2. Use (8) multiplied by c and ε div( c c) = ε c c ε c 2 2 Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

24 Theorem (Existence, Regularity, Uniqueness, A. 07/ 09) Let d = 2, 3. For every v 0 L 2 σ(ω), c 0 H 1 (Ω) with E mix (c 0 ) < there is a weak solution (v, c, µ) of (5)-(8), which satisfies (v, c) L (0, ; L 2 (Ω)), 2 c, f (c) L 2 loc ([0, ); L6 (Ω)). ( v, µ) L 2 (0, ; L 2 (Ω)), Moreover, c BUC([0, ); W 1 q (Ω)) with q > d. For (v 0, c 0 ) sufficiently smooth: 1 If d = 2, then the weak solution is unique and regular. 2 If d = 3, there are some 0 < T 0 < T 1 < such that the weak solution is regular and (locally) unique on (0, T 0 ) and [T 1, ). 3 There is a critical point c of E mix s.t. (v(t), c(t)) t (0, c ). Remarks: ε > 0 and m > 0 are essential! For f one can choose e.g.: { θ((1 c) log(1 c) + (1 + c) log(1 + c))c θ c c 2, c [ 1, 1], f (c) = + else. Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

25 Structure of the Proof First study the separate systems: 1 Cahn-Hilliard equation with convection and singular potential (based on E mix (c) = E 0 (c) θ 2 c 2 2 with E 0 convex) 2 (Navier-)Stokes system with variable viscosity Existence of weak solutions: Approximation and compactness argument Higher Regularity: Use regularity results for separate systems Uniqueness: Gronwall s inequality once c L (0, T ; C 1 (Ω)) and v L (0, T ; W 1 s (Ω)), s > d. Crucial ingredient for higher regularity: A priori estimate for c BUC([0, ); W 1 q (Ω)), q > d! Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

26 A priori Estimates for c W 2 r -estimate for c: Formally multiply µ(x, t) = c(x, t) + f θ (c(x, t)) by f (c(x, t)) = f 0 (c(x, t)) θ cc(x, t) to obtain f (c(t)) 2 dx + f (c(t)) c(t) 2 dx C µ(t) Ω Ω } {{ } 2. θ c Similarly, for 2 r < where f (c(t)) r + c(t) W 2 r C r ( µ(t) r + c(t) 2 ). c L 2 uloc ([0, ); W 2 6 (Ω)) c L 2 uloc ([0, );X ) = sup c L 2 (t,t+1;x ). t 0 Modifications: Higher regularity in time in Besov spaces. Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

27 Higher Time Regularity for c L (0, ; H 1 (0) )-estimate of tc: Multiplying by 1 N tc yields t 2 c + ( c f (c) } {{ } t c) = t (v c) θ c t c L (0, ;H 1 (0) ) + tc L 2 (Q) C(c 0 ) where V n (Ω) = {ϕ H 1 (Ω) n : n ϕ Ω = 0}. µ L (0, ; H 1 (Ω)) ( ) 1 + t v L 3 4 uloc (0, ;V n) c L (0, ; Wr 2 (Ω)), r = 6 if d = 3 and 1 < r < if d = 2. Problem: In general t v L 4 3 uloc (0, ; H 1 (Ω) n ) L 4 3 uloc (0, ; V n)! Solution: Replace t c by h τ h c. Use v B τ 4 ;uloc([0, ); H s (Ω)) 3 with 0 < s < 1 2, τ > 2 3 as well as Hs 0 (Ω) = Hs (Ω) and H s (Ω) = H s (Ω).... c BUC([0, ); Wq 1 (Ω)), q > 3. Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

28 Maximal Regularity for the Stokes Equation We consider the Stokes equation with variable viscosity t v div(ν(x, t)dv) + p = f in Ω (0, T ), (9) div v = 0 in Ω (0, T ), (10) v Ω = 0 on Ω (0, T ), (11) v t=0 = 0 in Ω (12) where Dv = 1 2 ( v + v T ) in a suitable domain Ω R d with Ω W 2 1 r r, ν BUC([0, T ]; W 1 r (Ω)), where 2 d < r. Theorem (A. & Terasawa 09, A 10) Let 1 < q < with q, q r, ν(x) ν 0 > 0, and 0 < T <. Then for every f L q (Ω (0, T )) d there is a unique solution of v of (9)-(12) s.t. ( t v, 2 v, p) L q (Ω (0,T )) C T f L q (Ω (0,T )). NB: fg W 1 q (Ω) falls f W 1 q (Ω), g W 1 r (Ω), 1 < q r, und r > d. Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

29 Overview 1 Phase Separation and Cahn-Hilliard Equation 2 Analysis of the Navier-Stokes Equations 3 Diffuse Interface Model for Matched Densities 4 Other Diffuse Interface Models Outlook Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

30 Diffuse Interface Model without Diffusion In the case without diffusion (m = 0) we have t v + v v div(ν(c)dv) + p = ε div( c c) (13) div v = 0 (14) t c + v c = 0 (15) where Dv = 1 2 ( v + v T ) plus initial and boundary conditions. Energy-dissipation: For smooth solutions we have d E(c(t), v(t)) = ν(c) Dv 2 dx with dt Ω v(t) 2 E(c(t), v(t)) = E mix (c(t)) + dx 2 Remark: Existence of weak, global solution is open! There is no control of the curvature µ = ε c + ε 1 f (c) of the diffuse interface! Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33 Ω

31 Diffuse Interface Model without Diffusion In the case without diffusion (m = 0) we have t v + v v div(ν(c)dv) + p = ε div( c c) (13) div v = 0 (14) t c + v c = 0 (15) where Dv = 1 2 ( v + v T ) plus initial and boundary conditions. Theorem (A. & Terasawa 09) Let q > d 2, q 3, Ω R d be bounded and v 0, c 0, Ω be sufficiently smooth. Then there is some T > 0 such that (13)-(15) has a unique solution (v, p, c) with t v, 2 v, p L q (Ω (0, T )), c C([0, T ]; W 2 q (Ω)) W 1 q (0, T ; W 1 q (Ω)) Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

32 Methods of Characteristics/Lagrangian-Coordinates The solution of t c + v c = 0, c t=0 = c 0 is c(x, t) = c 0 (Xt 1 (x)) where X t : Ω Ω is the trajectory of mass particles ξ Ω, i.e. d dt X t(ξ) = v(x t (ξ), t) for t (0, T ), ξ Ω, X 0 (ξ) = ξ for ξ Ω. u(ξ, t) = v(x t (ξ), t) solves the transformed quasi-linear system: t u div u (ν(c 0 )D u u) + u p = div u ( u c 0 u c 0 ) in Ω (0, T ) div u u = 0 in Ω (0, T ) where the coefficients of u, D u, div u depend on X t I for T 1. Linearization + contraction mapping principle Theorem if the linearized system possesses maximal regularity! Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

33 Quasi-Incompressible Model Lowengrub &Truskinovsky 98 derived: ρ t v + ρv v div(ν(c)dv) + p = ε div( c c) (16) } {{ } surface tension t ρ + div(ρv) = 0 (17) ρ t c + ρv c = m µ (18) ) 2 ρ µ = ρ (p + c 2 + ε 1 f (c) ερ 1 c (19) c 2 in Ω (0, T ), where Dv = 1 2 ( v + v T ), together with suitable initial and boundary conditions. v, p are the velocity and pressure of the fluid mixture. ρ = ˆρ(c) is the density given as a constitutive function. c = c 1 c 2 is the difference of the (mass) concentrations of the fluids. µ is the chemical potential and m > 0 the (constant) mobility. f : R [0, ) is a (homogeneous) free energy density Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

34 Quasi-Incompressible Model Lowengrub &Truskinovsky 98 derived: ρ t v + ρv v div(ν(c)dv) + p = ε div( c c) (16) } {{ } surface tension t ρ + div(ρv) = 0 (17) ρ t c + ρv c = m µ (18) ) 2 ρ µ = ρ (p + c 2 + ε 1 f (c) ερ 1 c (19) c 2 New difficulties: div v 0. p enters equation for chemical potential (19). (16)-(17) and (18)-(19) are coupled in highest order if ρ const.! Analytic results: A. 09: Existence of weak solutions for slightly modified free energy/system A. forthcoming: Existence of strong solutions locally in time. Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

35 Workshop on Phase Field Models in Fluid Mechanics February 14-16, 2011, University of Regensburg Main Speakers Franck Boyer (Marseille) Charlie Elliott (Warwick) Chun Liu (Penn State) John Lowengrub (Irvine) Christian Rohde (Stuttgart) Helmut Abels (Regensburg) Organizers: Helmut Abels (Regensburg), Harald Garcke (Regensburg), Günther Grün (Erlangen) Webpage: Supported by: Johannes-Kepler-Forschungszentrum Regensburg, SPP 1506 Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

36 References (I) H. Abels. Diffuse interface models for two-phase flows of viscous incompressible fluids. Lecture Notes, Max Planck Institute for Mathematics in the Sciences, No. 36/2007, H. Abels. Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Comm. Math. Phys., 289(1):45 73, H. Abels. On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Rat. Mech. Anal., 194(2): , H. Abels. Nonstationary Stokes system with variable viscosity in bounded and unbounded domains. Discrete Contin. Dyn. Syst. Ser. S, 3(2): , H. Abels and E. Feireisl. On a diffuse interface model for a two-phase flow of compressible viscous fluids. Indiana Univ. Math. J., 57(2): , H. Abels and Y. Terasawa. Non-homogeneous Navier-Stokes systems with order-parameter dependent stresses. Preprint, Max Planck Institute for Mathematics in the Sciences, No. 19/2009, to appear in Math. Math. Methods Appl. Sci., H. Abels and Y. Terasawa. On Stokes Operators with variable viscosity in bounded and unbounded domains. Math. Ann., 344(2): , H. Abels and M. Wilke. Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal., 67(11): , Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

37 References (II) J. F. Blowey and C. M. Elliott. The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis. European J. Appl. Math., 2(3): , J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys., 28, No. 2: , A. Debussche and L. Dettori. On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal., 24(10): , C. M. Elliott and H. Garcke. On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal., 27(2): , C. M. Elliott and S. Luckhaus. A generalized equation for phase separation of a multi-component mixture with interfacial free energy. Preprint SFB 256 Bonn No. 195, C. M. Elliott and S. Zheng. On the Cahn Hilliard equation. Arch. Rational Mech. Anal., 96: , X. Feng. Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal., 44(3): (electronic), K.-H. Hoffmann and P. Rybka. Convergence of solutions to Cahn-Hilliard equation. Comm. Part. Diff. Eq., 24: , Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33

38 References (III) P.C. Hohenberg and B.I. Halperin. Theory of dynamic critical phenomena. Rev. Mod. Phys., 49: , N. Kenmochi, M. Niezgódka, and I. Paw low. Subdifferential operator approach to the Cahn-Hilliard equation with constraint. J. Differential Equations, 117(2): , J. Lowengrub and L. Truskinovsky. Quasi-incompressible Cahn-Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454(1978): , L. Modica. The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal., 98(2): , L. Modica and S. Mortola. Un esempio di Γ -convergenza. Boll. Un. Mat. Ital. B (5), 14(1): , P. Sternberg and K. Zumbrun. Connectivity of phase boundaries in strictly convex domains. Arch. Rational Mech. Anal., 141(4): , Helmut Abels (U Regensburg) Diffuse Interface Models Juli 8, / 33