From nonlocal to local Cahn-Hilliard equation. Stefano Melchionna Helene Ranetbauer Lara Trussardi. Uni Wien (Austria) September 18, 2018

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1 From nonlocal to local Cahn-Hilliard equation Stefano Melchionna Helene Ranetbauer Lara Trussardi Uni Wien (Austria) September 18, 2018 SFB P D ME S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 1 / 26

2 Index 1 Introduction 2 Local CH 3 Nonlocal CH 4 Main result S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 2 / 26

3 Cahn-Hilliard equation It has been proposed in 1958 by [Cahn, Hilliard 1958, Cahn 1961] It describes the process of phase separation (spinodal decomposition) in binary alloys (iron-nickel) Phase field model u [0, 1] (vs sharp interface model u {0, 1}) It has a variety of applications: image processing [Capuzzo Dolcetta, Finzi Vita, March 2002] population dynamics [Cohen, Murray 1981] formation of Saturn rings [Tremaine 2003] tumour growth [Colli, Garcke, Gilardi, Lam, Rocca, Sprekels, Scala... ] S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 3 / 26

4 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

5 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

6 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

7 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

8 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

9 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

10 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

11 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

12 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

13 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

14 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

15 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

16 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

17 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

18 Settings u R: real valued function representing the local concentration of one of the two components u = 0, u = 1: pure phases Two species A, B with concentrations c A and c B = 1 c A at each point. Then u(x, t) = 1 c A (x, t) and: u 1 pure phase B; u 0 pure phase A. Source of the picture: Wikipedia S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 4 / 26

19 Local Cahn-Hilliard Free energy functional E CH (u) = where ( τ 2 ) 2 u 2 + F(u) dx τ: small positive parameter related to the transition region thickness F: double well potential with two global minima in the pure phases u 2 : reflects intermolecular interactions (penalising the creation of interfaces) S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 5 / 26

20 Local Cahn-Hilliard Corresponding evolution problem (4 th order PDE): where u t + J CH = 0, µ: mobility (constant = 1) J CH = µ(u) v CH, v CH = δe CH(u) δu v = δe δu : chemical potential = τ 2 u + F (u) S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 6 / 26

21 Nonlocal Cahn-Hilliard Proposed by [Giacomin, Lebowitz 1997] Free energy functional E NL (u) = 1 K (x, y)(u(x) u(y)) 2 dxdy + 4 F(u(x))dx, where K (x, y): positive and symmetric convolution kernel F: double well potential with two global minima in the pure phases S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 7 / 26

22 Nonlocal Cahn-Hilliard Corresponding evolution problem (2 nd order PDE): u t + J NL = 0, J NL = µ(u) v NL, v NL = δe NL(u) δu = (K 1)u K u + F (u) S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 8 / 26

23 Nonlocal Cahn-Hilliard Corresponding evolution problem (2 nd order PDE): u t + J NL = 0, J NL = µ(u) v NL, v NL = δe NL(u) δu = (K 1)u K u + F (u) Goal: prove the convergence of solutions of the nonlocal Cahn-Hilliard equation to solutions of the local version in a periodic setting. S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 8 / 26

24 Local vs nonlocal They share fundamental features: underlying gradient flow structure, lack of comparison principle, separation from the pure phases,... Both energy functionals allow the same Γ-limit for vanishing interface thickness [Gal, Grasselli, Miranville, Rocca,... ] Pointwise convergence is of little use due to non convexity and lack of coercivity of the nonlocal energy functional E NL in H 1. There exists Γ-convergence for the energy functionals [Ponce 2004] but it is not trivial to prove convergence for solutions of the corresponding dynamic problems using [Sandier, Serfaty 2011] S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 9 / 26

25 Overview of known results and properties Eq F µ Existence Separation Long time pol non-deg Garcke 00, Temam probably false Temam 88 deg Elliott Garcke 96?? CH log non-deg Elliott Luckhaus 91 Miranville Zelik 04 Cherfils Zelik Miranville 11 deg Elliott Garcke 96? Debussche Dettori 95 pol non-deg Bates Han 04 probably false Gal Grasselli 17 deg??? NLCH log non-deg Gal Giorgini Grasselli 17 Gal Giorgini Grasselli 17 Gal Giorgini Grasselli 17 deg Gajewski Zacharias 03 Londen Petzeltova 11 Londen Petzeltova 11 S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 10 / 26

26 Hypothesis H1 d-dimensional flat torus with d 3 S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 11 / 26

27 Hypothesis H1 d-dimensional flat torus with d 3 H2 Family of convolution kernels parametrised by a parameter ε: ( K ε (x, y) = ε d 2 x y J 2 ) ε with J : R R sufficiently smooth nonnegative function with compact support and 1 J( z 2 ) z 2 dz = 1 d S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 11 / 26

28 Hypothesis H1 d-dimensional flat torus with d 3 H2 Family of convolution kernels parametrised by a parameter ε: ( K ε (x, y) = ε d 2 x y J 2 ) ε with J : R R sufficiently smooth nonnegative function with compact support and 1 J( z 2 ) z 2 dz = 1 d H3 F C 2 (R) double well potential with two global minima at 0 and 1 such that F (s) 0 for s (, a] [a, + ) with a nonnegative, and C l ( u 3 + 1) F (u) C u ( u 3 + 1) for C l, C u > 0 S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 11 / 26

29 Hypothesis H1 d-dimensional flat torus with d 3 H2 Family of convolution kernels parametrised by a parameter ε: ( K ε (x, y) = ε d 2 x y J 2 ) ε with J : R R sufficiently smooth nonnegative function with compact support and 1 J( z 2 ) z 2 dz = 1 d H3 F C 2 (R) double well potential with two global minima at 0 and 1 such that F (s) 0 for s (, a] [a, + ) with a nonnegative, and C l ( u 3 + 1) F (u) C u ( u 3 + 1) for C l, C u > 0 H4 u 0,ε L 2 () converges strongly in L 2 () to the limit u 0 H 1 () and satisfies E ε (u 0,ε ), E(u 0 ) C 0 for some constant C 0 > 0 independent of ε S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 11 / 26

30 Definition of solutions Definition (Weak solution to the nonlocal Cahn-Hilliard equation) Let ε > 0 and T > 0 be fixed. We define u ε to be a weak solution to the nonlocal Cahn-Hilliard equation on [0, T ] associated with the initial datum u 0,ε L 2 () if u ε H 1 (0, T ; (H 1 ()) ) L 2 (0, T ; H 1 ()), satisfies t u ε, ϕ (H 1 ()),H 1 () + for all ϕ H 1 (), and u ε (0) = u 0,ε. [(K ε 1)u ε K ε u ε + F (u ε )] ϕ dx = 0 S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 12 / 26

31 Definition of solutions Definition (Weak solution to the local Cahn-Hilliard equation) Let T > 0 be fixed. We define u to be a weak solution to the Cahn-Hilliard equation on [0, T ] associated with the initial datum u 0 H 1 () if u H 1 (0, T ; (H 1 ()) ) L 2 (0, T ; H 2 ()), satisfies t u, ϕ (H 1 ()),H 1 () + for all ϕ H 2 (), and u(0) = u 0. u ϕ dx F (u) ϕ dx = 0 S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 13 / 26

32 Main result Remark: existence and uniqueness of weak solutions to both problems are well known with different choices for the boundary conditions both systems have been largely studied (qualitative properties, numerical aspects, long-time behaviour, asymptotics with different kinds of boundary conditions and different potentials) S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 14 / 26

33 Main result Remark: existence and uniqueness of weak solutions to both problems are well known with different choices for the boundary conditions both systems have been largely studied (qualitative properties, numerical aspects, long-time behaviour, asymptotics with different kinds of boundary conditions and different potentials) Theorem Let u ε be a solution of the nonlocal CH with periodic boundary conditions and kernel K ε (x, y). Then u ε u in L 2 (0, T ; H 1 ()) H 1 (0, T ; H 1 ()) where u is a solution of the local CH. S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 14 / 26

34 Some comments Note that in the Neumann case CH has two boundary conditions (one for u and one for the chemical potential), while NLCH has just one (for the chemical potential v). The mobility can be annoying for the estimates on the chemical potential. S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 15 / 26

35 Some comments Note that in the Neumann case CH has two boundary conditions (one for u and one for the chemical potential), while NLCH has just one (for the chemical potential v). The mobility can be annoying for the estimates on the chemical potential. Idea: use advantage of the dynamic structure: for every fixed ε, u ε H 1 S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 15 / 26

36 Proof: uniform estimates t u ε, ϕ (H 1 ()),H 1 () + [(K ε 1)u ε K ε u ε + F (u ε )] ϕ dx = 0 Test function: ϕ = u ε 0 = 1 d 2 dt u ε 2 L 2 () + [ (K ε 1)u ε K ε u ε + F (u ε ) ] u ε dx S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 16 / 26

37 Proof: uniform estimates t u ε, ϕ (H 1 ()),H 1 () + [(K ε 1)u ε K ε u ε + F (u ε )] ϕ dx = 0 Test function: ϕ = u ε 0 = 1 d 2 dt u ε 2 L 2 () + 0 = 1 d 2 dt u ε 2 L 2 () + [ (K ε 1)u ε K ε u ε + F (u ε ) ] u ε dx [(K ε 1) u ε 2 (K ε u ε ) u ε +F (u ε ) u ε 2 ] dx }{{} 1 2 Kε(x,y) uε(x) uε(y) 2 dx dy S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 16 / 26

38 Proof: uniform estimates Change of variable x y ε =: z 1 d 2 dt u ε 2 L 2 () + 1 J( z 2 ) u ε (y + εz) u ε (y) 2 ε = F (u ε ) u ε 2 dx B 1 u ε 2 L 2 (). 2 dy dz S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 17 / 26

39 Proof: uniform estimates Change of variable x y ε =: z 1 d 2 dt u ε 2 L 2 () + 1 J( z 2 ) u ε (y + εz) u ε (y) 2 ε = F (u ε ) u ε 2 dx B 1 u ε 2 L 2 (). 2 dy dz Aim: estimate the blue term with u ε 2 Key idea: [Ponce 2004], Poincaré type inequality S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 17 / 26

40 Key idea R N with N 1, bounded domain with Lipschitz boundary, 1 p < Poincaré It exist C p > 0 s.t. f f p C p Df p, f W 1,p () Let (ρ n ) L 1 (R N ) be a sequence of radial functions satisfying: ρ n 0 a.e. in R N ρ n = 1 n 1 R N ρ n (h)dh = 0 δ > 0 lim n h >δ S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 18 / 26

41 Theorem 1 Let (ρ n ) L 1 (R N ) be a sequence of radial functions as defined before. Given δ > 0, there exists n 0 1 sufficiently large, such that ( f f p Cp ) f (x) f (y) p + δ K p,n x y p ρ n ( x y ) dx dy for every f L p () and n n 0. S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 19 / 26

42 Theorem 1 Let (ρ n ) L 1 (R N ) be a sequence of radial functions as defined before. Given δ > 0, there exists n 0 1 sufficiently large, such that ( f f p Cp ) f (x) f (y) p + δ K p,n x y p ρ n ( x y ) dx dy for every f L p () and n n 0. Observe: this formulation is stronger than Poincaré. It is easy to see that f (x) f (y) p x y p ρ n ( x y ) dx dy Df p C R N Df p S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 19 / 26

43 Theorem 2 compactness If (f n ) L p (R n ) is a bounded sequence such that f n (x) f n (y) p x y p ρ n ( x y )dxdy B, n 1 then (f n ) is relatively compact in L p. Assume that f n f in L p () then f W 1,p () if 1 < p < f BV () if p = 1 S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 20 / 26

44 Proof: uniform estimates t u ε, ϕ (H 1 ()),H 1 () + [(K ε 1)u ε K ε u ε + F (u ε )] ϕ dx = 0 Test function: ϕ = ( ) 1 U ε where ( ) 1 : (H 1 ()) H 1 () is the map assigning to every v (H 1 ()) the unique solution w of the equation w = v such that the mean value is zero, i.e. w = 0. We define U ε = u ε u ε. S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 21 / 26

45 Proof: uniform estimates t u ε, ϕ (H 1 ()),H 1 () + [(K ε 1)u ε K ε u ε + F (u ε )] ϕ dx = 0 Test function: ϕ = ( ) 1 U ε where ( ) 1 : (H 1 ()) H 1 () is the map assigning to every v (H 1 ()) the unique solution w of the equation w = v such that the mean value is zero, i.e. w = 0. We define U ε = u ε u ε. u ε u ε 2 L 2 () C p and u ε u ε 2 L 2 () C p ( ) 2 J( z 2 ) z 2 Uε (y + εz) U ε (y) dy dz B 2 ε z J( z 2 ) z 2 u ε (y + εz) u ε (y) ε z 2 dy dz, S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 21 / 26

46 Uniform estimates 1 d 2 dt u ε 2 L 2 () J( z 2 ) u ε (y + εz) u ε (y) ε 2 dy dz B 1 u ε 2 L 2 () S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 22 / 26

47 Uniform estimates 1 d 2 dt u ε 2 L 2 () J( z 2 ) u ε (y + εz) u ε (y) ε 2 dy dz B 1 u ε 2 L 2 () ( ) u ε(t ) 2 L 2 () + B 1 u ε 2 2C L 2 (0,T ;L 2 ()) 1 p 2 u ε(0) 2 L 2 () S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 22 / 26

48 Uniform estimates 1 d 2 dt u ε 2 L 2 () J( z 2 ) u ε (y + εz) u ε (y) ε 2 dy dz B 1 u ε 2 L 2 () ( ) u ε(t ) 2 L 2 () + B 1 u ε 2 2C L 2 (0,T ;L 2 ()) 1 p 2 u ε(0) 2 L 2 () T 0 J( z 2 ) u ε (y + εz) u ε (y) ε 2 dy dz dt C S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 22 / 26

49 Proof: convergence Goal: prove the limit u to be a weak solution of the local Cahn-Hilliard equation u ε u weakly in L 2 (0, T ; H 1 ()) t u ε t u weakly in L 2 (0, T ; (H 1 ()) ) u ε u strongly in C([0, T ]; L 2 ()) for some limit u L 2 (0, T ; H 1 ()) H 1 (0, T ; (H 1 ()) ) S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 23 / 26

50 Proof: convergence Goal: prove the limit u to be a weak solution of the local Cahn-Hilliard equation u ε u weakly in L 2 (0, T ; H 1 ()) t u ε t u weakly in L 2 (0, T ; (H 1 ()) ) u ε u strongly in C([0, T ]; L 2 ()) for some limit u L 2 (0, T ; H 1 ()) H 1 (0, T ; (H 1 ()) ) [Ponce 2004] u L 2 (0, T ; H 2 ()) S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 23 / 26

51 Proof: convergence Test function: ϕ C () T 0 = ( t u ε )ϕ dx dt 0 }{{} I T F (u ε ) ϕ dx dt 0 } {{ } II 1 T K ε (x, y)(u ε (x) u ε (y))( ϕ(x) ϕ(y)) dy dx dt 2 0 }{{} III (I) (II): the growth conditions and continuity on F suffice to pass to the limit S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 24 / 26

52 Proof: convergence Thanks to (H2): 1 d Hd 1 (S d 1 ) 0 J(r 2 )r d+1 dr = 1 and by using the weak convergence: 1 T J( z 2 ) z (u ε (y + εz) u ε (y)) ( ϕ(y + εz) ϕ(y)) dy dzdt ε z ε z 1 T u(y) ϕ(y)dy dt 2 Thus, the limit u satisfies T T ( t u)ϕ dx dt T u ϕ dx dt F (u) ϕ dx dt = 0. 0 S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 25 / 26

53 Outlook and open questions Proved the convergence of weak solutions of the NLCH equation to the weak solutions to the local one as the convolution kernel approximates a Dirac delta (case with periodic boundary conditions) using a compactness argument. Dirichlet / Neumann boundary conditions? S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 26 / 26

54 Outlook and open questions Proved the convergence of weak solutions of the NLCH equation to the weak solutions to the local one as the convolution kernel approximates a Dirac delta (case with periodic boundary conditions) using a compactness argument. Dirichlet / Neumann boundary conditions? Thanks for your attention SFB P D ME Melchionna S., Ranetbauer H. and T. L., From nonlocal to local Cahn-Hilliard equation, submitted (2018) S. Melchionna, H. Ranetbauer, L.Trussardi From nonlocal to local Cahn-Hilliard equation 26 / 26