Nonlocal Cahn-Hilliard equations with fractional dynamic boundary conditions

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1 Nonlocal Cahn-Hilliard equations with fractional dynamic boundary conditions Ciprian G. Gal To cite this version: Ciprian G. Gal. Nonlocal Cahn-Hilliard equations with fractional dynamic boundary conditions. 39 pages <hal v> HAL Id: hal Submitted on 1 Nov 016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents whether they are published or not. The documents may come from teaching and research institutions in France or abroad or from public or private research centers. L archive ouverte pluridisciplinaire HAL est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche publiés ou non émanant des établissements d enseignement et de recherche français ou étrangers des laboratoires publics ou privés.

2 NONLOCAL CAHN-HILLIARD EQUATIONS WITH FRACTIONAL DYNAMIC BOUNDARY CONDITIONS CIPRIAN G. GAL Abstract. We consider a nonlocal version of the Cahn-Hilliard equation characterized by the presence of a fractional diffusion operator and which is subject to fractional dynamic boundary conditions. Our system generalizes the classical system in which the dynamic boundary condition was used to describe any relaxation dynamics of the orderparameter at the walls. The proposed fractional dynamic boundary condition appears to be more general in the sense that it incorporates nonlocal effects which were completely ignored in the classical approach. We aim to deduce well-posedness and regularity results as well as to establish the existence of finite dimensional attractors for this system. Contents 1. Introduction 1. Functional framework and well-posedness 4.1. A backward Euler scheme 4.. The fractional dynamic boundary condition Weak and strong solutions Finite dimensional attractors 7 5. Appendix: the Ising spin exchange model Final remarks 38 References Introduction The classical form of the Cahn-Hilliard equation [6] developed in the later 1950 s has been successfully applied in a variety of phenomena in science see also [9]. In particular it is frequently used as a model for phase separation phenomena in two-component materials which occupy a given domain R d d 3. The standard form of the classical Cahn-Hilliard equation is given by 1.1 t ϕ = µ µ = ϕ + F ϕ in 0 after we have scaled up all the relevant physical constants to one and is usually subjected to no-flux boundary conditions ν µ = 0 on 0 ν ϕ = 0 on 0. In this context one has ±1 corresponding to the pure phases of the material while ϕ 1 1 corresponds to the transition in the interface between the two material phases. A physically relevant choice for F is a singular logarithmic potential which is often approximated by a regular polynomial potential typically the double-well potential F r = r 1. We point out that 1. implies the conservation of mass for 1.1 while 1.3 implies that the diffused interface which separates the two phases of the material intersects the solid wall at a perfect contact angle of π/. This unfortunately is a strong assumption for many materials and binary systems. We refer the reader for further discussion to [7 17] and the references therein. In order to remedy this situation one needs 1991 Mathematics Subject Classification. 35R09 37L30 8C4. Key words and phrases. nonlocal Cahn-Hilliard equation fractional dynamic boundary conditions regional fractional Laplacian fractional Wentzell Laplacian well-posedness regularity global attractor. 1

3 CIPRIAN G. GAL to allow a dynamic contact angle to deviate from the stationary one of π/ and so the contact line can move also relative to the solid boundary of the domain. From this point of view it was proposed in the 1990 s see [5 13] that the dynamics near the solid wall is better described by dynamic boundary conditions of the form 1.4 t ψ ψ + ν ϕ + βψ + G ψ = 0 on 0 where ψ = ϕ in the trace sense whenever the latter exists. Generally β > 0 is a constant denotes the Laplace-Beltrami operator on and the nonlinear function G characterizes the possible preferential attraction or repulsion of one of the components by the walls. Thus in more general binary systems 1.4 must replace the condition 1.3 to account for relaxation dynamics on the boundary. We recall that various other modifications of 1.4 and 1. have also been proposed recently to account for the interaction of the two-phase material with the solid wall. See for instance [7 Remark 1.4] and the corresponding references therein. We note that an important feature of and 1.4 is that any sufficiently smooth solution ϕ ψ satisfies the energy identity d dt E totϕ t ψ t + µ x t dx + t ψ x t ds = 0 for all t 0 where the free bulk plus surface energy is given by 1.5 E tot ϕ ψ = ϕ + F ϕ dx + ψ + βψ + Gψ ds. As far as theoretical results are concerned for the classical Cahn-Hilliard systems and together with 1.4 we now have a rather complete picture on wellposedness regularity and longtime behaviour in terms of global attractors and the global asymptotic stability as time goes to infinity of trajectories associated with these systems. We refer the reader to the survey paper [7] for an extensive list of references. Although the derivation of the Cahn-Hilliard system comprising and 1.4 may appear physically sound it was only introduced as an approximation where mass currents/fluxes in and on were driven by local chemical potential functionals containing derivatives of the density up to second order; this is in agreement with the presence of gradient terms in 1.5. This derivation was performed in [5 13] starting out from a nonlocal microscopic description of the kinetics of phase changes for a lattice gas of interacting particles by ignoring any nonlocal contribution to the total free energy in the continuum limit and by restricting oneself only to the nearest-neighbour interactions. Thus the interaction between any two nearest neighbours was effectively assumed to be short-ranged. Interestingly this was already pointed out by the authors of [5 13]. Note that such approximations of first order in the free energy functional which eventually lead to the classical system were already assumed by Cahn and Hilliard in their influential paper [6] but there the interactions with the boundary walls were also ignored. Besides the interaction was once again assumed to be short ranged. Hence a rigorous derivation was required in order to develop a phase-field model which can account for both short ranged interactions as well as any long range interactions between any two distant particles and without making any kind of approximations. Such rigorous analysis was performed only recently by Giacomin and Lebowitz [1] starting out from a microscopic model for a lattice gas with long-range potentials of the form J x = J x with J C R d. It turns out that in place of the approximation 1.1 one obtains instead the following nonlocal version of the Cahn-Hilliard equation: 1.6 t ϕ = µ µ = Jx y ϕ x ϕy dy + F ϕ in 0. More precisely in this formulation the radially symmetric kernel J encodes both the local and nonlocal information in a manner in which any microscopic particle interaction takes place throughout the domain. It is interesting to point out that for 1.6 the

4 NONLOCAL CHE WITH DYNAMIC BC 3 corresponding free energy takes on the form 1.7 E nonloc ϕ = Jx y ϕx ϕy dxdy + F ϕdx. Rigorous mathematical results for the nonlocal Cahn-Hilliard equation 1.6 together with 1. have been established only recently. We refer the reader to [ ] for wellposedness results to [ ] for regularity and longtime behaviour in terms of global attractors and to [16 6 7] for issues concerning the global asymptotic stability of trajectories as time goes to infinity. However in the general formulation of Giacomin and Lebowitz [1] the interaction between particles inside the bulk domain and particles on was ignored and so any stationary boundary condition for ϕ was somewhat encoded in the properties of the interaction kernel J. This is in particular true for a strongly singular kernel J / L 1 loc R d see [1 18 3] and it covers the case when a regional fractional Laplace operator s s 0 1 appears in the formulation of the chemical potential in 1.6 of course in that case the integral in the second equation of 1.6 must be understood in a principal valued sense. We recall that the use of fractional operators is quite natural for systems that describe long ranged interactions when particles are allowed to take arbitrarily large steps up to the system size with a small finite probability for each such step in addition to any short-ranged interactions between close neighbours. In this sense normal classical diffusion in the Cahn-Hilliard equation 1.1 describes phase-transitions in systems close to equilibrium whereas anomalous diffusion in the presence of fractional Laplace operators in 1.6 seems to be inherent in phase-transitions phenomena that is far from equilibrium. Returning to our first point of this paragraph our goal is to consider the case of phase-separation phenomena that also accounts for the presence of particle interactions with the wall. We follow an idea consistent with the point of view of [13] except that now our dynamic boundary condition incorporates any of the aforementioned nonlocal effects see Appendix and may simply reduce to the classical description 1.4 under certain conditions. To give a precise formulation of the nonlocal value problem we wish to investigate we first recall the definition for the regional fractional Laplace operator as given by for u L s ux uy ux = C ds P.V. dy x y d+s = C ds lim ε 0 provided that the limit exists. Here {y y x >ε} L 1 := {u : R measurable and 1.9 C ds = ss Γ d+s π d Γ1 s ux uy dy x x y d+s ux dx < } 1 + x d+s where Γ denotes the usual Gamma function. We refer to [ 3 33] cf. also Section. below for a class of functions for which the limit in 1.8 exists. Our interest lies with the nonlocal boundary value problem: 1.10 t ϕ = µ µ = s ϕ + F ϕ in 0 subject to the boundary conditions: such that ψ = ϕ and ν µ = 0 on 0 t ψ + l ψ + C sn s ϕ + βψ + G ψ = 0 on ϕ t=0 = ϕ 0 in ψ t=0 = ψ 0 on.

5 4 CIPRIAN G. GAL In 1.1 C s N s s 1/ 1 denotes the so-called fractional normal derivative and l l 0 1 corresponds to the fractional surface diffusion operator on see Section. We observe that at least formally when s = l = 1 the boundary condition 1.1 translates to the classical boundary condition 1.4 and 1.10 is formally the classical local Cahn-Hilliard equation 1.1. A further physical justification for using the simplest model of an interacting particle systems is given in the Appendix see Section 5. It is our goal to give a comprehensive treatment of in terms of existence regularity and stability with respect to the initial data of properly defined solutions. We will also discuss and derive sufficient conditions for the nonlinearities F G so that possesses finite dimensional global and exponential attractors. Our main goal is to relax the usual framework see [7] associated with the classical Cahn-Hilliard equations subject to classical dynamic boundary conditions 1.4 in the sense that we wish to consider the more general case when is a bounded domain with Lipschitz continuous boundary. This is in particular important when the domains of the operators involved in the weak and strong formulations associated with the problem no longer enjoy explicit characterizations in terms of standard Sobolev spaces. Indeed this is not at all expected and in fact impossible in the nonlocal framework that includes fractional diffusion operators like the regional fractional Laplace s and the fractional Laplace-Beltrami boundary operator l. These features make the usual constructive arguments based on the Galerkin approximation scheme for the classical local Cahn-Hilliard equation very difficult to apply as well as place additional regularity constraints on the parameters involved in the nonlocal counter-part. To resolve these issues in the nonlocal framework we develop a time-implicit discretization scheme for an abstract initial value problem in a generic setting that includes all the problems considered in this article and much more. In fact such abstract schemes have already proved their usefulness in the treatment of other nonlocal problems see [15]. We then take advantage of several recent results of Gal and Warma [19 0] suited for nonlocal operators associated with the fractional dynamic boundary condition 1.1 to develop a suitable solution theory for problem Such results will also become crucial in the last part of this contribution where we will show that our problem admits an exponential attractor as well as a finite dimensional global attractor. The article is organized as follows. In Section we define the functional setting needed for our approach: we first consider in subsection.1 an abstract boundary value for two generic self-adjoint operators A B and solve this problem by appealing to a backward Euler scheme and by employing fixed point arguments. Then in subsection. we give a proper definition for the fractional dynamic boundary condition in 1.1 as well as define all the corresponding operators involved in the subsequent analysis. In Section 3 we define what we mean by a weak and strong solution of and then provide and prove appropriate well-posedness results. Section 4 is dedicated to the regularity of weak solutions as well as the existence of finite dimensional attractors for Section 6 contains some additional remarks and some open questions.. Functional framework and well-posedness.1. A backward Euler scheme. Let us first introduce an abstract initial-boundary value problem:.1. subject to the condition t u t + Aµ t = 0 in H A 0 T t U t + BU t + f U t = µ t in H B 0 T.3 U t=0 = U 0 in H B. Let X be a locally compact metric space and m a Radon measure on X such that suppm = X. We let H A = L X m be a given Hilbert space with inner product HA and assume that A 0 is a self-adjoint operator on H A such that 0 is an eigenvalue of A. Similarly let H = L Y m Y for some Radon measure m Y that is supported on a locally

6 NONLOCAL CHE WITH DYNAMIC BC 5 compact metric space Y. Then consider the product Hilbert space H B = H A H endowed with the natural product norm U H B = u H A + u H for all vector-valued functions U = u u H B. In. we further identify µ H A with its pair µ 0 H B. On H B we define a densely defined operator B that is positive and self-adjoint. The matrix-valued function f U = f 1 u f u is assumed to belong to Cb 1 R R. Furthermore we let V A = DA + I 1/ and V B = D B 1/ and endow them with the natural inner products of V A and V B respectively. Finally we also assume that I + A 1 and B 1 are compact operators on H A and H B respectively. Without loss of generality we also assume that the constant function 1 D B = {U V B : BU H B } since this is our case of interest. The application we have mind for the abstract problem.1-.3 in Sections 3 4 is as follows: X = is a bounded domain in R d d 3 with Lipschitz continuous boundary Y = where m is the usual Lebesgue measure on X and m Y is the standard surface measure on. Furthermore A and B are the weak Neumann Laplacian and fractional Wentzell Laplacian respectively cf. Section. below. Definition.1. Let T > 0 be given but otherwise arbitrary. Our notion of strong solution for.1-.3 is as follows: the pair U µ satisfies U L 0 T ; D B L 0 T ; V B t U L 0 T ; H B f U L 0 T ; H B µ L 0 T ; D A such that t u t + Aµ t = 0 in H A -sense t U t + BU t + f U t = µ t in the H B -sense for a.e. t 0 T. The initial condition U t=0 = U 0 is also satisfied in a strong H B -sense. The proof of the following result is based on a time-discretization scheme and compactness arguments. Theorem.. Let f 1 f be globally Lipschitz functions over R and assume there exist c f C f > 0 such that f U U c f U C f with c f C B where U H B C B U V B. Then for every U 0 V B the abstract nonlinear problem.1-.3 possesses at least one strong solution in the sense of Definition.1. Remark.1. Note that. is a system of equations for the vector-valued function U = u u. Theorem. can be also modified for the case of a single equation when one merely identifies U with u and lets H A = H B. Remark.. We also observe that as a consequence of Definition.1 we have U C w [0 T ] ; V B where the latter space stands for the space of weakly continuous functions from [0 T ] with values in V B. We point out that in fact more can be said about the solution of Definition.1. Indeed U C [0 T ] ; V B owing to the fact that by. we have d.4 dt U t V B = BU H B f U BU HB + µ BU HB which follows by testing. with BU H B. Since all the terms on the right-hand side of.4 belong to L 1 0 T we find the absolute continuity of U t V B on [0 T ]. This together with weak continuity gives the desired strong continuity. Theorem. will become crucial in the subsequent Section 3 in the existence of solutions for when we shall specialize the operators A B. Our goal in this subsection is to provide a complete proof of Theorem.. We shall employ a technique based on semi-discretization in time which can be used to construct explicit approximate solutions that will converge to a solution of the original problem under natural conditions on the physical parameters. We will divide this programme into several key steps. We first construct our timedifference scheme. To this end let n be an integer and P be a partition of the time interval [0 T ] such that P = {t 0 t 1... t n } with 0 = t 0 < t 1 <... < t m <... < t n = T where t m = mh for 0 m n and h = T/n is the step size. In applications one may also

7 6 CIPRIAN G. GAL choose variable step sizes for every subinterval t m 1 t m but in our case a fixed step size suffices for the sake of simplicity. Then for 1 m n we consider the time difference equations associated with.1-.3:.5.6 such that δ t u m = Aµ m in H A δ t U m + BU m + f U m = µ m in H B.7 U 0 = U 0 V B. Here we have set δ t u m = um u m 1 δ t U m = δ t u m δ t u m h δ t u m is defined in a similar way and we think of U m and µ m as approximations of the functions U x t m and µ x t m respectively. By analogy with Definition.1 we say that.5-.7 has a strong solution in the following sense. Definition.3. We say U m µ m is a strong solution if U m D B µ m D A for every 1 m n the initial condition U 0 = U 0 is satisfied in the strong H B -sense and the relations in.5-.6 are satisfied in the strong sense of H A and H B respectively. Step 1. Our first goal is to prove the existence of at least one solution U m µ m to the system.5-.7 for a fixed m and assuming that U m 1 is already known. To show this claim we apply the Leray-Schauder degree theory to prove the existence of a solution to the following nonlinear system:.8.9 u + haµ = g in H A hbu + U + hf U = hµ + l in H B where we have set U µ = U m µ m and g = u m 1 l = U m 1. We wish to formulate problem.8-.9 in terms of a fixed point for a nonlinear map S 1 U µ = U µ where S λ is a compact homotopy depending on the parameter λ [0 1] defined as follows: S λ : [0 1] V B V A V B V A such that S λ V w = U µ is a solution of v + haµ = λg in H A hbu + V = hw + λl hλf V in H B. In this step we may also assume without loss of generality that A = A is positive. Indeed if A 0 we could replace equation.8 by u + h A + I µ = g + hµ and then.10 by v + h A + I µ = λg + hw. Next we observe that if g H A and v H A then µ D A is the unique solution of.10. Moreover if l H B and H A w = w 0 H B and V H B which yields that f V H B owing to f Cb 1 then.11 has a unique solution U D B. Here we have endowed both D A and D B with their corresponding graph norms. Since the injections D A V A and D B V B are continuous the mapping S λ is well-defined. Lemma.4. S is continuous as a mapping from [0 1] V B V A V B V A. Proof. Let λ n λ [0 1] and V n w n V w strongly in V B V A. Write Un λn µ λn n = S λ n V n w n Un λ µ λ n = S λ Vn w n and U λ µ λ = S λ V w respectively. From it follows that Un λn Un λ µ λn n µ λ n satisfies the system.1.13 ha hb µ λn n Un λn Un λ µ λ n = λ n λ g in H A = λ n λ l h λ n λ f V n in H B. Taking the corresponding inner products of in H A H B with µ λn n µ λ n and Un λn Un λ respectively we easily deduce the estimates: µ λn µ λ.14 n C λ n λ g HA VA n

8 .15 U λn n NONLOCAL CHE WITH DYNAMIC BC 7 Un λ C λ n λ VB l HB + V n HB for some constant C > 0 independent of n λ n λ. Note that since g H A and l H B and {V n } H B is bounded we have as n that µ λn n µ λ n 0 strongly in V A as well as Un λn Un λ 0 strongly in V B. On the other hand from we see that U λ n U λ µ λ n µ λ satisfies.16 ha µ λ n µ λ = v n v in H A.17 hb Un λ U λ + V n V = h w n w hλ f V n f V in H B. As before taking the inner products of in H A H B with µ λ n µ λ and Un λ U λ respectively we immediately get.18 µ λ n µ λ VA C v n v HA C V n V HB and.19 Un λ U λ VB C V n V HB + w n w HA owing to the fact that f Cb 1 for some constant C > 0 independent of n. Thus if the right-hand sides of go to zero as n then µ λ n µ λ 0 strongly in V A - sense and Un λ U λ 0 strongly in V B. This completes the proof in view of and Lemma.5. S λ is also compact. Proof. Let {λ n V n w n } be any bounded sequence in [0 1] V B V A and consider U n µ n = S λ n V n w n as a solution of Exploiting the same arguments as in and we easily see that {U n µ n } is bounded in V B V A uniformly with respect to n N. Then taking the inner products of the equations for U n µ n with Aµ n and BU n in H A and H B respectively and using the aforementioned fact we also arrive at the fact that {U n µ n } is bounded in D B D A uniformly in n owing to f Cb 1. Since I + A 1 and B 1 are compact self-adjoint operators the injections D B V B D A V A are dense and compact. This yields in particular the compactness of S λ in V B V A. Lemma.6. The Leray Schauder degree for the mapping S 1 is for some open ball B 0 M = D Id S 1 B 0 M 0 = 1 { } U µ V B V A : U µ VB V A < M for some M > 0. In particular problem.8-.9 has at least one strong solution. Proof. For a definition of the Leray-Schauder degree please refer to [9 Chapter Section 8.3]. In order to prove the claim it suffices to show that any possible fixed point of S λ can be estimated independently of λ [0 1]. More precisely we claim that if U µ V B V A is such that S λ U µ = U µ for some λ [0 1] then there exists a constant M > 0 such that.0 U µ VB V A < M. First let us consider the linear system of equations given by u + haµ = 0 in H A and hbu + U = hµ in H B. Observing that S 0 U µ = U µ is the unique solution of such a system our first requirement is that M > 0 is large enough such that the ball B 0 M contains this unique solution. On the other hand we recall that any fixed point U µ of solves the system.1. u + haµ = λg in H A hbu + U = hµ + λl hλf U in H B.

9 8 CIPRIAN G. GAL Let us now take the inner product in H A of.1 with hµ the inner product in H B of. with U and then add the resulting relations. We deduce h U V B + U H B + h A 1/.3 µ H A = λh g µ HA + λ l U HB λh f U U HB. Since by assumption f U U c f U C f we obtain from.3 for every λ [0 1] that CB h c f U H B + h U V B + U H B + h A 1/ µ H A µ HA h g HA µ 1HA + g 1 HA µ 1 HA + ε U + C HB ε l HB C εh g H A + εh A 1/ µ µ + C g 1HA HA H A + ε U + C HB ε l HB for any ε > 0 for some constants C C ε > 0 independent of λ [0 1]. Therefore choosing a sufficiently small ε 1/ we derive the estimate:.4 U V B + h A 1/ µ C h g H H A + l µ H B + g 1HA HA. A Next taking the inner product in H B of. with V 1 we find µ 1 HA = h B 1/ U B 1/ 1 + U 1 HB λ l x 1 HB + hλ f U 1 HB. H B By virtue of.4 and the fact that f is globally Lipschitzian on R we have for every λ [0 1].5 µ 1 HA C 1 + U VB + l HB for some C > 0 independent of λ. Estimate.5 together with.4 then yields by token of elementary Young inequalities.6 U V B + µ V A C 1 + g + l HA HB for C = C h A B c f C f > 0 independent of λ. Thus if we take M > 0 large enough such that M > C1 + g H A + l H B.0 ensures that S λ U µ U µ for any U µ B 0 M for all λ [0 1]. Finally the homotopy invariance of the degree D Id S λ B 0 M 0 for λ [0 1] implies that D Id S 1 B 0 M 0 = D Id S 0 B 0 M 0 = 1 which is enough to conclude that problem.8-.9 has at least one solution U µ V B V A. The latter also immediately yields the regularity U µ D B D A owing once again to the facts that g H A l H B and f is globally Lipschitz. Remark.3. It turns out that each solution U µ of.1-.3 is also unique a fact which can be easily checked. We will not need this property here and so we leave the details to be verified by the interested reader. Step. In this step we are interested in deducing a priori estimates for solutions of.5-.7 which are uniform with respect to the step size h. In what follows C > 0 will denote a constant that is independent of h but which only depends on f and the operators A B. Since A = A 0 and 1 NullA we may assume that δ t u m 1 HA = 0; this implies that u m 1 HA = u m 1 1 = u H 0 1 A HA for every 1 m n. Next we take the inner

10 NONLOCAL CHE WITH DYNAMIC BC 9 product in H A of.5 with µ m µ m 1 HA and the inner product in H B of.6 with δ t U m. Adding the relations that we obtain we find δ t U m H B + A 1/ µ m BU m U m U m 1 H A h H B = f U m δ t U m HB. The basic identity x x y = x y + x y gives 1 BU m U m U m 1 h H B = 1 B 1/ U m B 1/ U m 1 + B 1/ U m U m 1 h H B H B while for f Cb 1 we get f U m δ t U m HB ε δ t U m H B + C ε U m H B + 1. Therefore for a sufficiently small ε > 0 from.7 we obtain after we add the relations for m = 1... r r h δ t U m H B + A 1/ µ m r + U r.8 V B + U m U m 1 V B C h H A r U m H B U 0 V B for 1 r n and h 1. We can estimate µ m 1 HA by taking the inner product in H B of.6 with V 1. Indeed we have µ m 1 HA = f U m 1 HB + B 1/ U m B 1/ 1 + δ t U m 1 HB. H B A repeated application of Young s inequality gives r.9 h µ m r 1 HA C h U m V B U 0 V B recalling once again that.8 is satisfied. We can combine the two estimates of.8 and.9. We find r r h δ t U m + µ m + U r.30 HB VA V B + U m U m 1 V B C 1 + U 0 V B + h r U m V B. Now let us recall Gronwall s lemma in discrete form. Proposition.7. Let {y r } and {a r } be two nonnegative sequences and c 0 a positive constant. If y r c 0 + r a my m for r 1 then r.31 y r c 0 exp a m for r 1. Apply now.31 to.30 to conclude that U r V B e CT 1 + U 0 V B for every 1 r n. Also going back to.30 we also obtain the following estimates that are uniform with respect to h: r r.3 h δ t U m + µ m C HB VA 0 U m U m 1 C V 0 B H B

11 10 CIPRIAN G. GAL.33 sup U r V B C 0 1 r n sup f U r V B C 0 1 r n for some constant C 0 > 0 which depends only on C T > 0 and the initial datum U 0 V B. The last inequality in.33 is immediate on account of the first one and the fact that f is globally Lipschitz. Next we need higher-order estimates for U m µ m. We take the inner product in H A of.5 with Aµ m and use Young and Cauchy-Schwarz inequalities to observe that Aµ m H A C δ t u m H A C δ t U m H B. This estimate together with.3 yields r.34 h µ m DA C 0 for all 1 r n. Analogously we take the inner product in H B of equation.6 with BU m use the fact that f Cb 1 and exploit once more to easily arrive at the estimate r.35 h U m DB C 0 for all 1 r n. Step 3. We can now complete the proof of Theorem.. To this end we need to introduce the corresponding piecewise constant interpolating functions U h µ h and also the corresponding linear interpolate functions Ũh as follows. Let P = {t 0 t 1... t n } be a partition with 0 = t 0 < t 1 <... < t m <... < t n = T where t m = mh for 0 m n and h = T/n is the step size. For the vectors {µ m } n m=0 H A {U m } n m=0 H B = H A H consider the following interpolating functions.36 µ h t = µ m U h t = U m for t [t m 1 t m ] and.37 Ũ h t = U m + t t m U m U m 1 t [t m 1 t m ]. h With these definitions we may now rewrite the entire problem.5-.7 in terms of U h µ h and Ũh as follows: such that t ũ h + Aµ h = 0 in H A 0 T t Ũ h + BU h + fu h = µ h in H B 0 T.40 Ũ h 0 = U 0 in H B. By rewriting the estimates obtained in Step in terms of the interpolations functions U h µ h and Ũh we deduce the following result. Lemma.8. We have the estimates: and U h L 0T ;V B C 0 µ h L 0T ;DA C 0 U h L 0T ;DB C 0 f U h L 0T ;H B C 0 Ũh L 0T ;V B C 0 Ũh L 0T ;DB C 0.44 t Ũ h L 0T ;H B C 0. Proof. The first of.41 and the last of.4 are immediate consequences of.33. To prove the second of.41 we observe n mh µ h L 0T ;DA A = + I µ m dt HA h m 1h n µ m DA C 0

12 NONLOCAL CHE WITH DYNAMIC BC 11 by virtue of.34. In a similar fashion one can deduce the first estimate of.4 owing to.35. Also by token of the first of.3 we have n mh n t Ũ h L 0T ;H B = δ t U m H B dt h δ t U m H B C 0. m 1h By a similar argument using.33 and.35 we obtain the other estimates in.43. We now pass to the limit in as the step size h 0 +. According to the uniform estimates there exist subsequences still labelled by U h µ h Ũ h such that as h 0 + we have the following convergence properties: U h U weakly * in L 0 T ; V B U h U weakly in L 0 T ; D B µ h µ weakly in L 0 T ; D A t Ũ h t U weakly in L 0 T ; H B Ũ h Ũ weakly * in L 0 T ; V B Ũ h Ũ weakly in L 0 T ; D B. These convergence relations and the compact embedding also allow us to conclude L 0 T ; D B H 1 0 T ; H B L 0 T ; V B.51 U h U strongly in L 0 T ; V B L 0 T ; H B. By definition we further observe Ũh U h L 0T ;H B = n = h 3 mh m 1h h t t m δ t U m H B dt n δ t U m H B = h 3 tũh L 0T ;H B. Therefore by.44 we may infer that Ũh U h L 0T ;H B C 0 h. Thus passing to the limit as h 0 and recalling we find that Ũ U in H B a.e. on 0 T. Finally also with help from.51 we can then pass to the limit in the nonlinear term of.39 to find that f U h f U strongly in L 0 T ; H B. Subsequently one may employ once again to pass to the limit in the remaining linear terms of in a standard fashion. Thus we have proved that the limit function U µ yields a strong solution in the sense of Definition.1. The proof of Theorem. is now complete. We can prove further regularity properties for the strong solution of Theorem.. Theorem.9. Let the assumptions of Theorem. be satisfied and assume U 0 D B. Then the strong solution U µ of.1-.3 has the following regularity:.5 U L 0 T ; D B W 1 0 T ; H B µ L 0 T ; D A. Proof. In order to deduce.5 we need to formally differentiate the equations of.1-.3 with respect to t > 0 and then set V := t U w = t µ. In this case V w solves the following system { t v + Aw = 0 in H.53 A 0 T t V + BV + f U V = w in H B 0 T subject to the initial condition.54 V t=0 := V 0 = µ0 BU 0 f U 0. 0

13 1 CIPRIAN G. GAL Here in we have once again identified H A w with w 0 H B and let µ 0 be the solution of the following abstract boundary value problem.55 A + I µ 0 = BU f 1 u 0 in H A where U 0 = u 0 u 0 D B f 1 is globally Lipschitz in R and there is the operator representation BU1 BU = for U D B. BU Note that for U 0 D B we have µ 0 D A H A by.55 and so in.54 V 0 H B. Moreover the matrix-valued function f U = f 1 u f u belongs to C b R R by assumption. A bound for V L 0 T ; H B is equivalent to a bound for U W 1 0 T ; H B in.5. In order to derive this bound rigorously we shall instead work with a timediscretization of.53 by connecting the problem to the one employed earlier in To this end for the same approximations U m µ m of problem.5-.6 we further define for 1 m n V m = U m U m 1 /h and w m = µ m µ m 1 /h and then set δ t V m := V m V m 1 i.e. δ t V m = δ t v m δ t v m. h For 1 m n consider the system of time-difference equations such that δ t v m = Aw m in H A δ t V m + BV m + f U m V m = w m in H B.58 V 0 = V 0 H B. Recall first that u m 1 HA = u m 1 1 = u H 0 1 A HA for every 1 m n; this yields v m 1 HA = 0. We are interested in obtaining the required uniform bound for V m L 0 T ; H B with respect to the step size h. Thus we take the inner product in H A of.56 with A 1 v m and the inner product in H B of.57 with V m. Adding the relations that we obtain we find.59 1 h + 1 h V m HB V m 1 HB + V m V m 1 HB v m VA v m 1 + v m v m 1 VA VA + BV m V m HB = f U m V m V m H B owing to the identity x x y = x y + x y. Since f Cb 1 we derive further from.59 that.60 V m H B V m 1 H B + V m V m 1 H B + h V m V B C f h V m H B. Now sum.60 over all m = 1... r; it follows that r V r H B + V m V m h V m H B V B C h r V m H B + V 0. H B In view of the Gronwall s inequality in discrete form see Proposition.7 we infer that.6 max 1 r n V r H B C 0

14 NONLOCAL CHE WITH DYNAMIC BC 13 for some constant C 0 > 0 which depends only on C T > 0 and the initial datum V 0 H B. Now returning to.61 by virtue of.6 we also derive r.63 V m V m 1 + h V m H B V B C 0 uniformly with respect to h. As usual the a posteriori analysis of the time-discretization relies on an appropriate system of equations. For a partition P = {t 0... t n } and interpolating functions of it suffices then to define new functions Ṽh t w h t which are continuous affine on each interval [t m 1 t m ] and equal to V m = U m U m 1 /h and w m = µ m µ m 1 /h respectively at t = t m also Ṽh 0 = V 0 and µ h 0 = µ 0. In particular we can define new functions V h t = t Ũ h t w h t = t µ h t where µ h t = µ m + t t m µ m µ m 1 t [t m 1 t m ] h as well as Ṽ h t = V m + t t m V m V m 1 t [t m 1 t m ]. h With these definitions we can rewrite in terms of V h w h and Ṽh as follows: such that t ṽ h + Aw h = 0 in H A 0 T t Ṽ h + BV h + f U h V h = w h in H B 0 T.66 Ṽ h 0 = V 0 in H B. Exactly as in Lemma.8 by rewriting the estimates obtained in in terms of the function V h we deduce the following estimate V h L 0T ;H B L 0T ;V B C 0. Passing to a suitable subsequence as h 0 + we can then infer t Ũ h t U weakly-* in L 0 T ; H B t Ũ h t U weakly in L 0 T ; V B. Recalling now that f U h L 0 T ; H B see.4 we can use.39 to write t Ũ h 1 + B 1/ U h B 1/ 1 + fu h 1 HB = µ h 1 HA H B H B and observe that µ h 1 HA L 0 T on account of.67 and.45. Moreover by note that µ h solves the following abstract boundary value problem.69 µ h + Aµ h = BU h 1 + f 1 u h in H A 0 T. Also by virtue of.38 if we set µ h := µ h µ h 1 HA / 1 H A we have µ h µ h HA = A 1 t ũ h µ h owing to the fact that t ũ h 1 HA = 0. Therefore µ h L 0 T ; H A provided that t ũ h L 0 T ; H A L 0 T ; VA which actually is satisfied by.67. It follows that the bound µ h L 0 T ; H A is also uniform with respect to the step size h. We can exploit once again.57 and.67 to deduce by comparison in.39 that BU h L 0 T ; H B uniformly in h > 0. Owing to the global Lipschitz continuity of f 1 from.69 we also get the uniform bound µ h L 0 T ; D A. These uniform estimates entail that as we pass along a subsequence as h 0 we additionally have the convergence properties: { Uh U weakly-* in L 0 T ; D B µ h µ weakly-* in L 0 T ; D A. Thus the final claim in the statement of the theorem easily follows. H A

15 14 CIPRIAN G. GAL Remark.4. As in Remark. one can show by similar arguments that any strong solution satisfying the assumptions of Theorem.9 also satisfies U C [0 T ] ; D B and t U C [0 T ] ; H B... The fractional dynamic boundary condition. We need to introduce some notation and preliminary results related to some auxiliary problems. We denote by L p and L p the norms on Lp and L p respectively. In the case p = and stand for the usual scalar products in L and L respectively. The norms on H s and H s are indicated by H s and H s respectively for any s R. Let now R d d 3 be a bounded domain with Lipschitz continuous boundary i.e. is of class C 01. For s 0 1 we denote by H s := {ϕ L : the fractional order Sobolev space endowed with the norm Cds ϕx ϕy u H s = x y d+s dxdy + ϕx ϕy x y d+s dxdy < } 1/ β ϕ ds having defined C ds as in 1.9 for some β > 0. It is well-known that for 0 < s 1/ the spaces H s and H s 0 = D H s coincide with equivalent norms see [ Corollary.8 and Remark.3]. We also let H s = H s H s C c and note that Hs contains H0 s as a closed subspace. By definition Hs 0 is also the smallest closed subspace of H s containing D. In order to talk about traces for H0 s in the case when s 0 1/] we need to exploit a different potential-theoretic description of H0 s using the notion of relative capacity see e.g. [33 Definition 3.]; cf. also []. Without going into too much detail let us mention that the most common property of the relative capacity is that it measures small sets more precisely than the usual Lebesgue measure on. Thus the potential description gives H0 s = {u H s ũ = 0 r.q.e. on } where ũ denotes the relative quasi-continuous version of u see e.g. [33 Theorem 4.5]. Here the property ũ = 0 r.q.e. on means that there exists a polar set P i.e whose relative capacity is zero such that ũ = 0 almost everywhere on \P see [33 Definition 3.3]. Also every u H s admits a unique up to a relatively polar set P relatively quasi-continuous version ũ : R such that ũ = u a.e. in see for instance [33 Theorem 3.7]. Therefore every ϕ H0 s is zero S-a.e. on i.e. more precisely its relative quasi-continuous version ϕ = 0 on up to a relatively polar set where S denotes the usual Lebesgue surface measure on. Thus we can more than formally say that the trace for ϕ H0 s is actually null when 0 < s 1/ so that there are no nontrivial boundary conditions for ψ in that case. Therefore from this point on we shall simply assume that 1/ < s < 1 for ϕ to possess a nontrivial trace ψ = ϕ but which is well-defined in the usual sense. Indeed when 1/ < s < 1 the trace ψ = ϕ is well defined as a bounded operator from H s L q for every ϕ H s C where q = d 1 / d s if d and s 1/ 1 and q = if d = 1 and s 1/ 1. Furthermore by standard Sobolev function theory the injection H s into L d/d s is continuous if d and s 1/ 1 and it is continuous in C when d = 1 and s 1/ 1. Clearly the continuous injection H s L is also compact see [30]. Next for l 0 1 we let H l := {ψ L ψx ψy : x y d 1+l ds x ds y < } be the fractional order Sobolev space endowed with the norm β ψ H l := ψ ds + C d 1l ψx ψy 1/ x y d 1+l ds x ds y. Similarly we have H l L p is continuous with p := d 1 d 1 l > when d > 1+l H l L is continuous provided that d < 1 + l and H l L r for any

16 NONLOCAL CHE WITH DYNAMIC BC 15 r [ when d = 1 + l. Clearly the continuous injection H l L is also compact. To this end for 1/ < s < 1 and l 0 1 we define another scale of Hilbert spaces V sl := {U = ϕ ψ : ϕ H s ψ = ϕ H l } endowed with norm U V sl = ϕ H s + ψ H l. We shall now recall an integration by parts formula that was derived in [ Theorem 3.3] for the regional fractional Laplacian 1.8 in smooth domains. For this consider the following constant C 1s t 1 1 s t 1 1 s C s := ss 1 0 t s dt where t 1 = max {t 1}. Theorem.10. Let R N be a bounded domain of class C 11 and recall that 1/ < s < 1. Define the set C s := {u : ux = fxρx s 1 + gx x for some f g C } where ρx :=distx x. For u Cs and z we define the fractional normal derivative N s u of the function u by.70 N s duz νzt uz = lim t s for z. t 0 dt Then for every u Cs and v Hs one has s u L N s u L and v s udx = C ds vx vyux uy.71 x y d+s dxdy C s vn s uds. One can also introduce a weak version for the fractional normal derivative on nonsmooth domains that are only of class C 01. Definition.11. Recall that 1/ < s < 1. i Let u W s. We say that s u L if there exists w L such that C ds vx vyux uy x y N+s dxdy = wvdx for all v Cc and so for all v H0 s by density. In that case one writes s u = w. ii Let u H s such that s u L. We say that u has a weak fractional normal derivative in L if there exists g L such that.7 v s udx = C ds vx vyux uy x y d+s dxdy gvds for all v H s. In this case g is uniquely determined by.7; we may simply write g = C s N s u and refer to g as the fractional normal derivative of u. iii Consequently the following Green identity holds:.73 v s udx = C ds C s vx vyux uy x y d+s dxdy vn s uds for all v H s whenever u W s s u L and N s u exists as an element of L. Remark.5. If is a bounded domain of class C 11 and if u C s then N s u coincides exactly with function given by definition.70. In that case N s u L and s u L ; see for instance [].

17 16 CIPRIAN G. GAL To give a proper definition of a fractional power for the Laplace-Beltrami operator acting on we let H l = H l denote the dual of the reflexive Hilbert space H l and consider the operator D l : H l H l by the following definition:.74 D l u v := C d 1l ux uyvx vy x y d 1+l ds x ds y where denotes the duality map between H l and H l. It turns out that D l is bounded and the quadratic form associated with the right-hand side of.74 defines a bilinear symmetric continuous form for u v H l. Next we can define a fractional power of the Laplace-Beltrami operator as a principal-valued integral l u = C d 1lP.V. = C d 1l lim ε 0 + ux uy x y d 1+l ds y : x y >ε ux uy x y d 1+l ds y whenever the latter exists and observe that D l u v = l u v for all u v Hl. Viewing L as a subspace of H l and denoting the part of the operator D l in L by D l again we then have the following characterization:.75 DD l = {u H l l u L } D l u = l u. We also observe that C d 1l.76 ux uyvx vy x y d 1+l ds x ds y = D l u x v x ds x for u D D l and v H l. We refer the reader for more details about this operator to [19 Section.4] the proofs of Propositions Finally we can now give the definition for the linear fractional Wentzell dynamic boundary condition associated with condition 1.1. Theorem.1. [19 Theorem.1]Assume that is of class C 01 and let s 1/ 1 and l 0 1. We define a bilinear symmetric form as follows:.77 a W U V = C ds + C d 1l a W : V sl V sl R + ux uyvx vy x y d+s dxdy + ux uyvx vy x y d 1+l ds x ds y βuvds for all U = u u V = v v V sl. Let A sl W be the closed linear self-adjoint operator on L L associated with this form a W. Then the following are satisfied: i The domain DA sl W of Asl W consists of functions U = u u V sl such that s u L D l u L N s u exists as an element of L and In this case A sl W ii Also A sl W C s N s u + D l u + βu L. has the following operator representation: A sl u s W = u C s N s u + D l u + βu u has compact resolvent and the first eigenvalue λsl W1 is positive. Finally the injection DA sl W L L is continuous provided that d = 1 and if d = 3: { s > d sd 1 4 if l d or s > d+1 4 if l s 1.. or l > d 1 4 if s 1 l < sd 1 d

18 NONLOCAL CHE WITH DYNAMIC BC 17 We call A sl W the realization of the regional fractional Laplace operator s with the fractional Wentzell type boundary condition. We conclude this subsection with a regularity result for a nonlinear elliptic boundary value problem associated with the fractional Wentzell operator A sl W. Theorem.13. [0 Section 3]Assume s 1/ 1 d/4 1 and l d 1 /4 1 if d 3. Consider the following nonlocal boundary value problem: { s.78 u + θ u = h 1 in D l u + C s N s u + βu + ϖ u = h on where h 1 h L L. i Assume that θ ϖ C R satisfy the following conditions:.79 θ x x α 0 x α 1 ϖ x x β 0 x β 1 for all x R with x x 0 for some x 0 > 0 and some positive constants on the right-hand side of.79. Then every bounded solution U = u u V sl L L of.78 satisfies the explicit estimate: u L + u L 1 C + h 1 L + h L for some constant C > 0 independent of U h 1 and h. ii Consequently any bounded solution of.78 is also a strong solution i.e. s u L N s u L and D l u L and the following estimate holds: s u dx + D l u + N s u ds Q1 + h 1 L + h L for some positive function Q independent of U h 1 and h. Remark.6. For further clarification we say that U is a bounded solution of.78 if U V sl L L and a W U V + θ u vdx + ϖ u vds = h 1 vdx + h vds for all V V sl. Finally for a domain with Lipschitz continuous boundary we define the weak Neumann Laplacian A N µ = µ with domain given by.80 D A N = { µ H 1 : µ L ν µ = 0 on }. The boundary condition in.80 is understood in the following variational sense:.81 µωdx = µ ω for all ω H 1 whenever µ L and µ H 1. By results in spectral theory it is well known that A = A 0 on H A = L and I + A 1 is compact in L. 3. Weak and strong solutions Now let us give the assumptions on the potential functions F G. H1 F G C R and there exists c F c G > 0 such that F x c F G x c G x R. H There exist c f c g > 0 and p q 1 ] such that F x p c f F x + 1 G x q c g Gx + 1 x R. H3 There exist C i > 0 i = such that F x C 1 x p/p 1 C G x C 3 x q/q 1 C 4 x R for some p q 1 ].

19 18 CIPRIAN G. GAL Remark 3.1. We observe that H1 implies that both potentials are quadratic perturbations of some strictly convex functions. Assumption H is satisfied by potentials of arbitrary polynomial growth of orders p = p/ p 1 and q = q/ q 1 respectively. In particular for the double-well potential F x = x 1 one can take p = 4/3. Let us define the weak energy space { } Y m = U = ϕ ψ V sl : F ϕ L 1 G ψ L 1 ϕ m for some given m > 0 where we have also set ϕ = 1 ϕ x dx. We equip Y m with the following metric d U 1 U = U 1 U V sl + F ϕ 1 F ϕ dx 1/ + G ψ 1 G ψ ds. Our definition of a weak solution for the boundary value problem is as follows. Definition 3.1. Let ϕ 0 ψ 0 Y m and 0 < T < + be given. We say U = ϕ ψ is a weak solution of if ψ t = ϕ t for all 0 < t < T and the functions ϕ ψ and µ satisfy ϕ ψ L 0 T ; Y m t ϕ L 0 T ; H µ L 0 T ; H 1 t ψ L 0 T ; L F ϕ L 0 T ; L 1 G ψ L 0 T ; L 1 F ϕ L 0 T ; L p G ψ L 0 T ; L q. For every V = v ṽ V sl L p L q ω H 1 a.e. t 0 T we have t ϕ t ω + µ t ω = 0 a W U t V + t ψ t ṽ + = µ v. We have ϕ0 = ϕ 0 in and ψ 0 = ψ 0 on. 1/ F ϕ t v + G ψ t ṽ Remark 3.. Setting ω 1 in 3.5 we deduce the conservation of mass ϕt 1 = ϕ 0 1 for all t 0. The initial conditions ϕ 0 = ϕ 0 ψ 0 = ψ 0 also make sense in a strong L -sense since by we have ϕ ψ C[0 T ]; L L. In 3.6 each is understood as the duality map between L p and L p and between L q and L q respectively. Here we recall once again that 1/p+1/p = 1 and 1/q +1/q = 1. We also define what we mean by a strong solution. To this end we define a strong energy space { } Z m = ϕ 0 ψ 0 DA sl W : µ 0 H 1 ϕ m and endow it with the norm 3.7 ϕ 0 ψ 0 Z m = ϕ 0 ψ 0 DA sl W + µ 0 H 1 where µ 0 is computed via the equation µ 0 = s ϕ 0 + F ϕ 0 in. In 3.7 the norm in Z m = DA sl W H1 is meant for the triplet ϕ 0 ψ 0 µ 0 but we kept the notion ϕ 0 ψ 0 Zm instead of ϕ 0 ψ 0 µ 0 Zm in order to simplify the exposition.

20 NONLOCAL CHE WITH DYNAMIC BC 19 Definition 3.. Let 0 < T < + be given. We say U = ϕ ψ is a strong solution of if it is a weak solution in the sense of Definition 3.1 and ϕ ψ and µ satisfy 3.8 ϕ ψ L 0 T ; L L L 0 T ; DA sl W 3.9 t ϕ t ψ L 0 T ; V sl µ L 0 T ; H 1 L 0 T ; D A N. In particular for the strong solution we have t ϕ = µ a.e. in 0 T while the boundary condition ν µ = 0 is understood in the sense of.81 a.e. on 0 T. Finally we have 3.10 µ = s ϕ + F ϕ a.e. in 0 T and 3.11 t ψ + D l ψ + C s N s ϕ + βψ + G ψ = 0 a.e. on 0 T. The first goal of this subsection is to prove the existence of at least one strong energy solution to problem by passing to the limit as ɛ α 0 in a regularized version of the system admitting a sufficiently smooth approximate solution ϕ ɛα ψ ɛα. The existence of such smooth solutions will be ensured by the abstract result of Theorem. which was proven by means of a backward Euler scheme. Then we will derive additional uniform estimates for the solutions ϕ ɛα ψ ɛα as ɛ α 0. Theorem 3.3. Let ϕ 0 ψ 0 Z m for s 1/ 1 d/4 1 and l d 1 /4 1 if d 3. Assume that F G satisfy the assumption H1. Then there exists at least one strong solution U = ϕ ψ L 0 T ; Z m to problem in the sense of Definition 3.. Remark 3.3. In the range provided for s 1/ 1 d/4 1 and l d 1 /4 1 we have by part ii of Theorem.1 that DA sl W L L and therefore for any ϕ ψ L 0 T ; Z m the nonlinearities F ϕ L 0 T and G ψ L 0 T are well-defined. Proof. Step 1. An approximating problem P ɛα associated with can be formulated abstractly in For the sake of notation we let f = F and g = G. We take for ɛ 0 1 smooth functions f ɛ f g ɛ g uniformly on compact intervals of R with the following properties: 3.1 f ɛ j x c fɛ g ɛ j x c gɛ for all x R for j = 0 1 and some positive constants c fɛ c gɛ which may behave like ɛ m as ɛ 0 for some m = m l N. Since F ɛ s = s 0 f ɛζdζ and G ɛ s = s 0 g ɛζdζ we may also assume 3.13 lim sup F ɛ x = F x lim sup G ɛ x = G x for all x R ɛ 0 ɛ 0 such that 3.14 F ɛ x c F and G ɛ x c G for all x R owing to assumption H1. Such constructions can be easily realized in a standard way as follows: one lets F ɛ x = F x for x ɛ 1 F ɛ x = F i ɛ 1 x ɛ 1 i for x > ɛ 1 and F ɛ x = i=1 F i ɛ 1 x + ɛ 1 i for x < ɛ 1. i=1 A similar construction applies to G ɛ see for instance [7] and the references therein. In particular we observe that 3.1 and 3.14 imply that both f ɛ g ɛ are

21 0 CIPRIAN G. GAL globally Lipschitz functions on R and that f ɛ x g ɛ x x x C 1 x + x C for all x x R for some sufficiently large constants 0 < C 1 λ sl W1 1 C > 0. We also preliminarily observe that due to H1 F and G as well as F ɛ G ɛ are merely quadratic perturbations of the convex functions F x = F x + c F x G x = G x + c G x respectively F ɛ = F ɛ + c F x G ɛ = G ɛ + c G x. We insert an artificial viscosity α 0 1 into the problem by adding the viscous term α t ϕ in the definition of the chemical potential in 1.10 i.e. we let µ = s ϕ + α t ϕ + F ϕ. More precisely our approximated problem P ɛα ɛ α 0 1 consists of finding a function ϕ ɛα ψ ɛα which has the regularity stated in such that and t ϕ ɛσ + A N µ ɛα = 0 α t ϕ ɛσ + A sl W t ψ ɛα subject to the initial conditions t ϕ ɛα L 0 T ; L ϕɛα ψ ɛα fɛ ϕɛα + = g ɛ ψɛα 3.17 ϕ ɛα 0 = ϕ 0 ψ ɛα 0 = ψ 0. In we have let B = A sl W µɛα 0 with D B = DAsl W and A = A N with domain D A = D A N. We recall that A = A 0 on H A = L and I + A 1 is compact in L provided that is of class C 01 while B = B > 0 on H B = L L and B 1 is compact by Theorem.1. We also prepare a fixed initial datum µ ɛα 0 L that solves 3.18 µ ɛα 0 + αa N µ ɛα 0 = µ 0 := s ϕ 0 + F ϕ 0 in L whenever it is assumed that ϕ 0 ψ 0 Z m DA sl W. Observe that the latter also yields that µ ɛα 0 H 1 uniformly with respect to ɛ α > 0. Indeed from 3.18 we immediately see that µ ɛα 0 + α A L N µ ɛα 0 = µ L 0 µ ɛα 0 so that µ ɛα 0 L d and αa N µ ɛα 0 L uniformly in ɛ α provided that µ 0 H 1 the latter is clearly satisfied by assumption cf. the definition of Z m. In particular 3.19 µ ɛα 0 + α A H 1 N µ ɛα 0 C ϕ L 0 ψ 0 Z m for some C > 0 independent of α ɛ. By taking a sufficiently small ɛ 1/M 0 if necessary where ϕ 0 ψ 0 L L M 0 since in that case g ɛ ψ 0 g ψ 0 we also specify a fixed set of data v ɛα 0 L such that 3.0 v ɛα 0 = D l ψ 0 + C s N s ϕ 0 + βψ 0 + g ψ 0. Then it is clear that 3.1 v ɛα 0 L Q ϕ 0 ψ 0 Z m for some function Q > 0 independent of ɛ α. Therefore the abstract results of Theorem. and Theorem.9 yield the existence of at least one smooth solution U ɛα = ϕ ɛα ψ ɛα to problem Pɛα in the sense of Definition.1 by setting V B = V sl and V A = H 1 we also note that the proof of Theorem. was given in the case α = 1 but clearly it can be easily adapted to accommodate the case α > 0. In particular we deduce the existence of a smooth function { Uɛα L 0 T ; DA sl W W 1 0 T ; L L µ ɛα L 0 T ; D A N