Cahn-Hilliard equations on an evolving surface
aa r X i v : . [ m a t h . A P ] J a n CAHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE
DIOGO CAETANO, CHARLES M. ELLIOTT
Abstract.
We describe a functional framework suitable to the analysis of the Cahn-Hilliard equationon an evolving surface whose evolution is assumed to be given a priori . The model is derived frombalance laws for an order parameter with an associated Cahn-Hilliard energy functional and we establishwell-posedness for general regular potentials, satisfying some prescribed growth conditions, and for twosingular nonlinearities – the thermodynamically relevant logarithmic potential and a double obstaclepotential. We identify, for the singular potentials, necessary conditions on the initial data and theevolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integralsof solutions are preserved over time, and prove well-posedness for initial data on a suitable set ofadmissible initial conditions. We then briefly describe an alternative derivation leading to a model thatinstead preserves a weighted integral of the solution, and explain how our arguments can be adapted inorder to obtain global-in-time existence without restrictions on the initial conditions. Some illustrativeexamples and further research directions are given in the final sections. Introduction
In this paper, we study well-posedness of Cahn-Hilliard equations on a prescribed moving surfacein R , for the different cases of a smooth, logarithmic and double obstacle nonlinearities and with aconstant mobility. Even though there already exists some work concerning this problem on evolvingdomains, mostly from the point of view of numerical analysis, different models have been proposedand rigorous well-posedness results are still missing. We start this article by introducing these modelsand by discussing our main results. The models and main results.
In what follows, we fix a parametrised evolving surface { Γ( t ) } t ∈ [0 ,T ] with velocity field V = V ν + V τ , with V ν being the normal velocity of the evolution and V τ thetangential component of the velocity of the parametrization. We denote by u a scalar order parametersatisfying, for all subregions Σ( t ) ⊂ Γ( t ) , the following balance law ddt Z Σ( t ) u = − Z ∂ Σ( t ) q · µ = − Z Σ( t ) ∇ Γ · q, where µ denotes the outer unit conormal and q = q d + q a is a flux consisting of an advective term, q a = u V a with V a a tangential vector field, and a diffusive term q d = − M ( u ) ∇ Γ w , where M ( · ) is amobility function and w = − ∆ Γ u + F ′ ( u ) denotes the chemical potential of the system, defined as the functional derivative of the Cahn-Hilliardfunctional E CH ( u ) = Z Γ( t ) |∇ Γ u | F ( u ) . (1.1)The functional (1.1) measures the total free energy with the gradient term accounting for the surfaceenergy of the interface and the nonlinearity F representing the homogeneous free energy, which willbe: (i) a smooth potential generalising the frequently considered quartic polynomial F ( r ) = ( r − , r ∈ R ; (F s )(ii) the thermodynamically relevant logarithmic potential defined by F ( r ) = (1 − r ) log(1 − r ) + (1 + r ) log(1 + r ) + 1 − r , r ∈ [ − , (F log ) (iii) a double obstacle potential F ( r ) = ( (1 − r ) / if | r | ≤ ∞ otherwise . (F obs )We hence obtain, for the cases (F s ), (F log ), the system ∂ • u + u ∇ Γ · V − ∇ Γ · (cid:0) u ( V τ − V a ) (cid:1) = ∇ Γ · ( M ( u ) ∇ Γ w ) − ∆ Γ u + F ′ ( u ) = w, (CH )and for (F obs ) the system reads as ∂ • u + u ∇ Γ · V − ∇ Γ · (cid:0) u ( V τ − V a ) (cid:1) = ∇ Γ · ( M ( u ) ∇ Γ w ) − ∆ Γ u + ∂I ( u ) − u ∋ w, (CH ′ )where ∂I denotes the subdifferential of the indicator function of the set [ − , . For the system with(F s ), we establish in Section 4 well-posedness results analogous to those known for the equation on afixed domain. However, it turns out that, for the singular potentials (F log ), (F obs ), some conditionsrelating the boundedness of the solutions, the evolution of the domains and the effect of the Cahn-Hilliard dynamics are necessary in order to obtain global in time results. These are the results ofSections 5.1, 5.2. This same derivation is considered in [ER15], [OS16].More recently, in [OXY20, YQO19, ZTL +
19] different derivations of the model have been proposedby considering instead a conservation law for the quantity ρc , where ρ denotes the total density of thesystem and c is a scaled difference of concentrations, which can loosely be seen as the analogue of u inthe previous model. We do not intend to make any physical considerations about the models, but it isnonetheless interesting to collect the equations these authors consider and to see how our techniquescould be adapted to analyse them.The derivation in [ZTL +
19] starts from the balance laws ddt Z Σ( t ) ρ = 0 (1.2)for the total density ρ and ddt Z Σ( t ) ρc = − Z ∂ Σ( t ) q · µ = − Z Σ( t ) ∇ Γ · q, (1.3)where again q is a flux for the conserved quantity ρc . From (1.2) we thus obtain ∂ • ρ + ρ ∇ Γ · V = 0 , (1.4)which can be viewed for each x ∈ Γ( t ) as an ODE for ρ . The free energy functional and the associatedchemical potential are now defined as ˜E CH ( c ) = Z Γ( t ) |∇ Γ c | ρF ( c ) dΓ and w = 1 ρ δ ˜E CH δc = − ρ ∆ Γ c + F ′ ( c ) . As a consequence we are led to the Cahn-Hilliard system ρ∂ • c − ∇ Γ · (cid:0) ρc ( V τ − V a ) (cid:1) = ∇ Γ · ( ρM ( c ) ∇ Γ w ) − ∆ Γ c + ρF ′ ( c ) = ρw, (CH )and for (F obs ) the system reads as ρ∂ • c − ∇ Γ · (cid:0) ρc ( V τ − V a ) (cid:1) = ∇ Γ · ( ρM ( c ) ∇ Γ w ) − ∆ Γ c + ρ∂I ( c ) − ρc ∋ ρw. (CH ′ )In Section 6 we briefly explain how to adapt the arguments in Sections 4, 5.1, 5.2 to establish well-posedness for (CH ), (CH ′ ), by formally obtaining the necessary a priori bounds. This system isparticularly interesting to us not only because it can be dealt with by essentially the same techniques,but also because we can now prove well-posedness without any extra conditions on the surfaces or theinitial data, as opposed to the first model with the singular potentials. The reason for this is that (1.4)determines a relation between the weight function ρ and the Jacobian determinant of the flow map, sothat the evolution of the surfaces is now encoded into the equation via the presence of ρ . As we willsee, this allows for global well-posedness for any initial data in H (Γ ) not identically equal to ± . AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 3
In [YQO19], [OXY20] yet another derivation is proposed (the former for a Cahn-Hilliard system andthe latter for an Allen-Cahn equation). In [YQO19], a conservation law as in (1.3) is considered for ρc and the chemical potential is defined as the functional derivative of the energy (1.1), and a derivationas above leads, for instance for the smooth potential (F s ) and with a constant mobility M ≡ , to thesystem ρ∂ • c − ∇ Γ · (cid:0) ρc ( V τ − V a ) (cid:1) = ∆ Γ w − ∆ Γ c + F ′ ( c ) = w. (CH )In [OXY20], the authors consider a third different energy E( c ) = Z Γ( t ) ρ (cid:18) |∇ Γ c | F ( c ) (cid:19) dΓ , and define the chemical potential as the functional derivative of E with respect to c . Again, proceedingin the same way as above, this leads to the system ρ∂ • c − ∇ Γ · (cid:0) ρc ( V τ − V a ) (cid:1) = ∆ Γ w −∇ Γ · ( ρ ∇ Γ c ) + ρF ′ ( c ) = w. (CH )Although these are similar to the models we consider in this article, the a priori estimates do not quitework out in the same way as for (CH ), (CH ), and as such we shall leave a detailed analysis of (CH ),(CH ) for future work. In spite of this, we observe that, for the case of a smooth potential, local in time(or even global in time for the case of potentials with quadratic growth) existence is easy to establishusing our techniques.In summary, in this work we present a rigorous derivation for (CH ), (CH ′ ) and establish existence,uniqueness and stability of weak solutions. We also address extra regularity results for all the potentials.We then briefly consider (CH ), (CH ′ ) and explain how our arguments can be adapted to obtain thesame type of results, including global in time existence for the singular potentials without restrictingthe set of admissible initial conditions. The study of (CH ), (CH ) seems to require a new approachfor the a priori estimates, so we leave it for future work. Background and motivation.
Our interest in this equation is part of the more general problem ofunderstanding equations on non-static domains. The study of partial differential equations on movingdomains, of which evolving hypersurfaces are an example, has been a very active area of researchrecently and many applications to physics, materials science, biology, among other sciences, have beenconsidered. Generalising systems that are usually considered in stationary domains to spaces thatevolve with time has been seen to lead to more accurate and realistic models, for instance for the studyof surface dissolution of binary alloys, for phenomena of cell motility, and also for the modelling of elasticmembranes (such as lipid bilayer membranes or cell tissues). Some references for these applicationsare [BEM11, EE08, ESV12, EAK +
01, GLS13, VSG + R n was introduced in [CH58] tostudy spinodal decomposition in binary alloys; more precisely, it models the phase separation of analloy consisting of two components when the temperature of the system has been quenched to atemperature below the critical temperature, leading to a spatially separated two-phase structure asopposed to the uniform mixed state of equilibrium. See also [Cah61]. It has more recently beenunderstood to provide a good model to describe later stages of the evolution of phase transitionphenomena. The introduction of the equation in [CH58] and subsequent studies in natural sciences,see for example [NCS84] and included references, generated also interest on the equation from themathematics community. The basis for the analytic and numerical studies of the equation is thearticle [EZ86], where the authors consider the problem with a polynomial nonlinearity. From thenon, many generalizations of the equation were considered and several applications to different areas ofmathematics were explored; some examples are [Ell89, EF89, EFM89, EL92] and [EG96, DD14, DD16] DIOGO CAETANO, CHARLES M. ELLIOTT for the degenerate version of the equation. See also [GK18, Hei15] where the gradient flow structureof the problem is exploited, [CENC96, Gar13] for relations with geometric flows and [PD96] for astochastic version. It is also worthwhile mentioning the articles [Peg86, CENC96, OS16] where anasymptotic analysis is performed for the parameter ε → ; the latter does so for the equation on anevolving surface. We mention particularly the work in [DD95, EL91], where the logarithmic potentialis considered (see also the survey [CMZ11]), and the article [BE91, BE92], where the authors analysethe double obstacle problem. These served as motivation for a significant part of this paper. For ageneral overview we refer also to the survey article [Ell89] or the recent book [Mir19].The results we present in this article are mostly motivated by those in [BE91, EL91, ER15, GK18],which we generalise to the case of moving surfaces in R . As mentioned above, the biggest difficultiesarise for (CH ), (CH ′ ) with the singular potentials, due to an interplay between the boundedness ofsolutions, the evolution of the surfaces and the initial data. As we shall see, different regimes need tobe studied for these cases. Structure of the paper.
Finally, let us describe the structure of the paper. We begin by introducingthe necessary analytic background and notation for posing PDEs on evolving spaces in Section 2, and weproceed to a derivation of our model and the statement of our results in Section 3. Section 4 is devotedto the study of (CH ) with a smooth nonlinearity satisfying some polynomial growth conditions, andfor this case we establish well-posedness of the equation by using on the Galerkin method. This isthe starting point for the approximation method we use to tackle the singular potentials in Section 5,for which it turns out that the moving nature of the domains has an impact on existence of (globalin time) solutions. We identify some necessary conditions for well-posedness, and prove existence anduniqueness of solutions in these regimes by approximation with more regular nonlinearities. In Section6 we analyse (CH ), (CH ′ ), which does not preserve the integral of solutions, and Section 7 containssome simple examples. We finish with a discussion of our results and some open questions we proposeto address in future work. Our results are, to the best of our knowledge, new in the literature, andgeneralise the classical results for the Cahn-Hilliard equation on a fixed domain or surface. Notation.
For simplicity of notation we will frequently omit the differentials d t , dΓ on the integrals.It should be clear that, in time, we always integrate with respect to Lebesgue measure and over thesurfaces we use the surface measure. For a function u ∈ L (Γ) , we denote its mean value over thesurface Γ by ( u ) Γ := 1 | Γ | Z Γ u, where | Γ | denotes the measure of Γ . As for the constants appearing in estimates, we use C , C , . . . positive constants which might be different in different equations. More precisely, inside each equationthe constants appear with indices in increasing order, but every time we start a new estimate the firstconstant will again be denoted C . If only one constant is involved, then we denote it simply by C .We believe this keeps the notation clear and helps keep track, inside each estimate, of when a newconstants appear. 2. Preliminaries
Evolving surfaces.
Fix
T > and a C -evolving surface { Γ( t ) } t ∈ [0 ,T ] in R . More precisely, thismeans we have a regular, closed, connected, orientable C -surface Γ in R and a smooth flow map Φ : [0 , T ] × Γ → R such that(i) denoting Γ( t ) := Φ t (Γ ) , the map Φ t := Φ( t, · ) : Γ → Γ( t ) is a C -diffeomorphism, with inverse map Φ t : Γ( t ) → Γ ; (ii) Φ = id Γ .It follows from the definition above that, for each t ∈ [0 , T ] , Γ( t ) is also a regular, closed, connected,orientable C -surface. We can also naturally define diffeomorphisms between the surfaces at different AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 5 instants s, t ∈ [0 , T ] by Φ st := Φ t ◦ Φ s : Γ( s ) → Γ( t ) . Now in order to analytically treat problems in an evolving surface it is convenient to assume that Φ isthe flow of some prescribed vector field in R , which we will do in this article: Assumption ( A Φ ) : There exists a velocity field V : [0 , T ] × R → R with regularity V ∈ C (cid:0) [0 , T ]; C ( R , R ) (cid:1) such that, for any t ∈ [0 , T ] and every x ∈ Γ , ddt Φ t ( x ) = V (cid:0) t, Φ t ( x ) (cid:1) , in [0 , T ]Φ ( x ) = x. We denote the tangential and normal components of V by V τ , V ν , respectively. Note that, due tocompactness, there exists C V > independent of t such that, k V τ ( t ) k C (Γ( t )) , k V ν ( t ) k C (Γ( t )) ≤ k V ( t ) k C (Γ( t )) ≤ C V , for t ∈ [0 , T ] . (2.1.1)In this setting we can define the normal material derivative of a scalar quantity u on Γ( t ) by ∂ ◦ u := ˜ u t + ∇ ˜ u · V ν , and its material time derivative as ∂ • u := ∂ ◦ u + ∇ ˜ u · V τ = ˜ u t + ∇ ˜ u · V , (2.1.2)where ˜ u denotes any extension of u to a neighbourhood of Γ( t ) . This last definition takes into accountnot only the evolution of the domains but also the movement of points in the surface. The followingtransport formula will be useful throughout the article: Theorem 2.1.
Let Σ( t ) ⊂ Γ( t ) be evolving with velocity V = V ν + V τ , where V ν , V τ denote, respec-tively, the normal and tangential components of V . Define V ∂M := V τ · µ, where µ denotes the outer unit conormal to ∂ Σ( t ) . Then ddt Z Σ( t ) u = Z Σ( t ) ∂ • u + u ∇ Γ · V = Z Σ( t ) ∂ ◦ u + u ∇ Γ · V ν + Z ∂ Σ( t ) u V ∂M , for every function u for which the quantities above make sense. In our context, by taking Σ( t ) = Γ( t ) and recallng that Γ( t ) is a closed surface we obtain ddt Z Γ( t ) u = Z Γ( t ) ∂ • u + u ∇ Γ · V = Z Γ( t ) ∂ ◦ u + u ∇ Γ · V ν , which shows that the time evolution of integral quantities depends only on the normal component ofthe velocity.Denoting now by J t , respectively J t , the change of area element from Γ to Γ( t ) , respectively from Γ( t ) to Γ , satisfying Z Γ( t ) η = Z Γ ˜ η J t , Z Γ ˜ η = Z Γ( t ) η J t (2.1.3)where, given η : Γ( t ) → R , we are writing ˜ η : Γ → R for the function ˜ η ( p ) = η (cid:0) Φ t ( p ) (cid:1) , ∀ p ∈ Γ . Combining (2.1.3) with the transport theorem above, it follows that, for each p ∈ Γ , ddt J t ( p ) = J ( t, p ) ∇ Γ · V ( t, Φ( t, p )) , from where J t ( p ) = exp (cid:26)Z t ∇ Γ · V ( s, Φ( s, p )) (cid:27) . (2.1.4) DIOGO CAETANO, CHARLES M. ELLIOTT
Due to (2.1.1), this can then be used to prove that there exists a constant C A > , depending only onthe bound for V ν , such that < | Γ | C A ≤ | Γ( t ) | ≤ C A | Γ | , ∀ t ∈ [0 , T ] . Under the stronger assumption ( A Φ ) we can prove the following additional results: Lemma 2.2.
Suppose ( A Φ ) holds. (a) Given t ∈ [0 , T ] and u ∈ L (Γ( t )) or u ∈ H (Γ( t )) , define φ − t u = u ◦ Φ t . Then φ − t : L (Γ( t )) → L (Γ ) , φ − t : H (Γ( t )) → H (Γ ) , and these maps are isomorphisms between the two spaces with constants of continuity indepen-dent of t , and the same result is true for φ t := ( φ − t ) − ; (b) Given u ∈ L (Γ ) (resp. H (Γ ) ), the map t
7→ k φ t u k L (Γ( t )) (cid:0) resp. t
7→ k φ t u k H (Γ( t )) (cid:1) is continuous; (c) [Poincaré inequality] For any t ∈ [0 , T ] , there exists a constant C P > , independent of t , suchthat, for any u ∈ H (Γ( t )) , k u − ( u ) Γ( t ) k L (Γ( t )) ≤ C P k∇ Γ( t ) u k L (Γ( t )) ; (d) [Sobolev embedding theorems] For any t ∈ [0 , T ] , the following continuous embeddings holdwith continuity constants independent of t : (d1) for all p ∈ [1 , + ∞ ) , H (Γ( t )) ֒ → L p (Γ( t )) ; (d2) for k, r ≥ integers with k − r > , H k (Γ( t )) ֒ → C r (Γ( t )) ; (d3) for k, r ≥ integers and α ∈ (0 , with k − r − α ≥ , H k (Γ( t )) ֒ → C r + α (Γ( t )) . The proofs follow from calculations involving the change of variables and integration by parts for-mulas (see, for example, [Vie11, Section 3]). We emphasize that, in parts (c) and (d), the constants canbe chosen independently of t , and the statements in (d) make use of the fact that Γ( t ) is -dimensional.2.2. Time-dependent Bochner spaces.
Observe that, in part (b) of the result above, we used theflow map to define a pullback operator for functions defined on the hypersurface by φ − t u = u ◦ Φ t , t ∈ [0 , T ] , u ∈ L (Γ( t )) or u ∈ H (Γ( t )) (2.2.1)Now if we denote, for t ∈ [0 , T ] , H ( t ) = L (Γ( t )) and V ( t ) = H (Γ( t )) , then the results in (b), (c)above show that the pairs ( H, φ ( · ) ) and ( V, φ ( · ) | V ) are compatible in the sense of [AES15a]. We canthus consider, for t ∈ [0 , T ] , the evolving Hilbert spaces L (Γ( t )) and H (Γ( t )) , in which case the workin [AES15a] gives rise to the following time-dependent Bochner spaces:(i) The separable Hilbert spaces L L and L H consisting of (equivalence classes) of functions such u : [0 , T ] → [ t ∈ [0 ,T ] L (Γ( t )) × { t } t (¯ u ( t ) , t ) or u : [0 , T ] → [ t ∈ [0 ,T ] H (Γ( t )) × { t } t (¯ u ( t ) , t ) that φ − ( · ) ¯ u ( · ) ∈ L (0 , T ; L (Γ )) or φ − ( · ) ¯ u ( · ) ∈ L (0 , T ; H (Γ )) ; we identify u ≡ ¯ u . Thesespaces are endowed with the inner products ( u, v ) L L := Z T ( u ( t ) , v ( t )) L (Γ( t )) , u, v ∈ L L , ( u, v ) L H := Z T ( u ( t ) , v ( t )) H (Γ( t )) , u, v ∈ L H . We also identify ( L L ) ∗ ≡ L L via the Riesz map, and set L H − := ( L H ) ∗ .(ii) For X ( t ) = L (Γ( t )) or X ( t ) = H (Γ( t )) , the Banach space L ∞ X , of those functions u ∈ L X such that t φ − t u ( t ) ∈ L ∞ (0 , T ; X (0)) , with the norm k u k L ∞ X := ess sup t ∈ [0 ,T ] k u ( t ) k X ( t ) ; AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 7 (iii) For X ( t ) = L (Γ( t )) or X ( t ) = H (Γ( t )) , the Banach space C X of functions u ∈ L X such that t φ − t u ( t ) ∈ C ([0 , T ]; X (0)) , with the norm k u k C X := sup t ∈ [0 ,T ] k u ( t ) k X ( t ) ; (iv) More generally, for X ( t ) = L (Γ( t )) or X ( t ) = H (Γ( t )) and k ∈ N , the space of k timesdifferentiable functions C kX of functions u ∈ L X such that t φ − t u ( t ) ∈ C k ([0 , T ]; X (0)) ;(v) For X ( t ) = L (Γ( t )) or X ( t ) = H (Γ( t )) , the space of test functions D X (0 , T ) consisting of u ∈ L X such that t φ − t u ( t ) ∈ D ((0 , T ); X (0)) := C ∞ c ((0 , T ); X (0)) . It remains to define an appropriate notion of a time derivative for functions in the above time-dependent spaces. For regular functions, we do it in the natural way, by pulling back to the referencedomain X (0) , differentiating in time, and pushing forward to return to X ( t ) . For X ( t ) = L (Γ( t )) or X ( t ) = H (Γ( t )) , the (strong) time derivative of a function u ∈ C X is defined to be ∂ • u ( t ) := φ t ddt φ − t u ( t ) ∈ C X . (2.2.2)By considering values along curves that follow the evolution, it can be seen that, whenever bothdefinitions (2.2.2) and (2.1.2) make sense, they coincide.The definition (2.1.2) can now be abstracted to a weaker sense as follows. Let u ∈ L H . A functional v ∈ L H − is said to be the weak time derivative of u , and we write v = ∂ • u , if, for any η ∈ D H (0 , T ) ,we have Z T h v ( t ) , η ( t ) i H − × H = − Z T ( u ( t ) , ∂ • η ( t )) L − Z T Z Γ( t ) u ( t ) η ( t ) ∇ Γ( t ) · V ( t ) . Observe that ∂ • η is the strong material derivative of η . Of course, for a regular function, the strongtime derivative (2.1.2) coincides with the weak derivative defined above. We are then interested inlooking for solutions to parabolic PDEs lying in the space H H − := (cid:8) u ∈ L H : ∂ • u ∈ L H − (cid:9) . Often the following criterion for weak differentiability is easier to apply (see [AES15a, Lemma 3.5]):
Lemma 2.3.
Let u ∈ L H and g ∈ L H − . Then u is weakly differentiable with ∂ • u = g if and only if,for all η ∈ H (Γ ) , ddt ( u ( t ) , φ t η ) L (Γ( t )) = h g ( t ) , φ t η i H − (Γ( t )) × H (Γ( t )) + Z Γ( t ) u ( t ) φ t η ∇ Γ( t ) · V ( t ) . We conclude this section with two results which will be of use later in the text. The first one relatessome of the spaces defined above.
Proposition 2.4.
Suppose assumption ( A Φ ) holds. (a) L H ֒ → L L ∼ = ( L L ) ∗ ֒ → L H − with each inclusion continuous and dense; (b) The space H H − is continuously embedded in C L ; (c) [Aubin–Lions Lemma] The space H H − is compactly embedded in L L . For details on the above, see [AES15a] (or [AES15b, Sections 3, 4] for a more summarized version),and [AET17, Lemma B.3] for the Aubin–Lions-type result. In the next proposition, we collect twoformulas for the time-derivatives of the L -inner product, and a proof can be found e.g. in [DE07,Lemma 5.2]. Proposition 2.5.
Let u, v ∈ H H − . Then the function t ( u ( t ) , v ( t )) L (Γ( t )) is absolutely continuous,and we have ddt ( u, v ) L = h ∂ • u, v i H − × H + h ∂ • v, u i H − × H + Z Γ( t ) uv ∇ Γ · V . If additionally ∇ Γ ∂ • u, ∇ Γ ∂ • v ∈ L L , then also t ( ∇ Γ( t ) u ( t ) , ∇ Γ( t ) v ( t )) L (Γ( t )) is absolutely continu-ous, and ddt ( ∇ Γ u, ∇ Γ v ) L = ( ∇ Γ ∂ • u, ∇ Γ v ) L + ( ∇ Γ u, ∇ Γ ∂ • v ) L + Z Γ( t ) B( V ) ∇ Γ u · ∇ Γ v, DIOGO CAETANO, CHARLES M. ELLIOTT where, for a vector field v = ( v , v , v ) , B( v ) = ( ∇ Γ · v ) I − D ( v ) and the matrix D ( v ) = ( D ij ( v )) i,j =1 is defined by D ij ( v ) = 12 X k =1 δ ik D k v j + δ jk D k v i = D i v j + D j v i . We now have all the background needed to state our problem.
Remark 2.6.
We presented the precise framework necessary to treat the Cahn-Hilliard equation, butthe setting described can be made more general.(i) On the one hand, it can be extended with the natural changes to compact hypersurfaces Γ( t ) ⊂ R n +1 , with any n ∈ N , see for instance [DE07]. Restriction to dimension n = 2 plays arole only in the statement of the Sobolev embeddings in Lemma 2.2(d).(ii) On the other hand, the work in [AES15a] allows to define L pX for any p ∈ [1 , + ∞ ] and a familyof time-dependent Banach spaces { X ( t ) } t ∈ [0 ,T ] with maps φ − t : X ( t ) → X satisfying the sameproperties as the pullback map in (2.2.1).3. Problem setup
For the rest of the text, fix
T > and an evolving C -surface { Γ( t ) } t ∈ [0 ,T ] in R satisfying assumption ( A Φ ) .3.1. Derivation of the equation.
We start by deriving the Cahn-Hilliard equation on the evolvingsurface { Γ( t ) } t ∈ [0 ,T ] from a conservation law. Fix t ∈ [0 , T ] , let u represent a scalar quantity on Γ( t ) and let q be a surface flux for u . Consider the following balance law for an arbitrary portion Σ( t ) ⊆ Γ( t ) evolving with the given purely normal velocity V ν ddt Z Σ( t ) u = − Z ∂ Σ( t ) q · µ, where ∂ Σ( t ) is the boundary of Σ( t ) and µ is the conormal on ∂ Σ( t ) , i.e. the unit normal to ∂ Σ( t ) which is tangential to Σ( t ) . Note that the normal component of q does not contribute to the flux, andhence we can assume that q is tangential. Using integration by parts we have Z ∂ Σ( t ) q · µ = Z Σ( t ) ∇ Γ · q − Z Σ( t ) q · νH = Z Σ( t ) ∇ Γ · q, where H denotes the mean curvature. The second term vanishes because q is tangential.On the other hand, using the transport formula in Theorem 2.1 gives ddt Z Σ( t ) u = Z Σ( t ) ∂ ◦ u + u ∇ Γ · V ν , so combining the previous equations yields Z Σ( t ) ∂ ◦ u + u ∇ Γ · V ν + ∇ Γ · q = 0 , for any portion Σ( t ) ⊆ Γ( t ) . This implies that ∂ ◦ u + u ∇ Γ · V ν + ∇ Γ · q = 0 on Γ( t ) . (3.1.1)We are interested in considering the surface flux q to be the sum of a diffusive term of the form q d = − M ( u ) ∇ Γ w and an advection term q a = u V a , with an advective tangential velocity V a . In thediffusive flux, M ( u ) is a mobility term and w , the chemical potential , is defined by w = − ∆ Γ u + F ′ ( u ) , where the homogeneous free energy F will be taken of different types in the following sections. Thechemical potential w is the functional derivative of the free energy E CH [ u ] = Z Γ( t ) |∇ Γ u | F ( u ) , which we will refer to throughout the text as the Cahn-Hilliard energy functional . Substituting q = q d + q a in (3.1.1) leads to ∂ ◦ u + u ∇ Γ · V ν + ∇ Γ · ( u V a ) = ∇ Γ · (cid:0) M ( u ) ∇ Γ (cid:0) − ∆ Γ u + F ′ ( u ) (cid:1)(cid:1) , AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 9 which we rewrite using the definition of the material time derivative to obtain the 4th order advectiveCahn-Hilliard equation ∂ • u + u ∇ Γ · V − ∇ Γ · ( u ( V τ − V a )) = ∇ Γ · (cid:0) M ( u ) ∇ Γ (cid:0) − ∆ Γ u + F ′ ( u ) (cid:1)(cid:1) . We will treat the problem above as the system of two second-order equations ∂ • u + u ∇ Γ · V − ∇ Γ · ( u V τa ) = ∇ Γ · ( M ( u ) ∇ Γ w ) − ∆ Γ u + F ′ ( u ) = w together with some initial condition for the function u , where we denote V τa := V τ − V a . There areno boundary conditions because the hypersurfaces are closed.We shall make some simplifying assumptions on the system above. The thermodynamically relevantmobility term M ( u ) should vanish at the pure phases ± , effectively restricting diffusion to the interfacebetween the two components. It is generally taken to be M ( u ) = 1 − u , and (3.1.2) becomes adegenerate system. Even in the classical setting, there are not many results available for the degenerateCahn-Hilliard equation (see [EG96, DD14, DD16]), and it is common to consider instead the constantmobility Cahn-Hilliard problem. We will do so in this text. Taking M ≡ , we are thus led to thesystem ∂ • u + u ∇ Γ · V − ∇ Γ · ( u V τa ) = ∆ Γ w − ∆ Γ u + F ′ ( u ) = w , (3.1.2)and this is the problem we analyse over the subsequent sections. We also make some assumptions onthe different velocity fields involved in the derivation:(i) (2.1.1) holds;(ii) for the advective velocity V a , we assume a uniform bound k V a k L ∞ ≤ C a .We now have all the conditions to formulate the weak form of (3.1.2).3.2. Problem setup.
We start by introducing some notation. Define, for t ∈ [0 , T ] , the bilinear forms:(i) for η, ϕ ∈ L (Γ( t )) , the terms of order m ( t ; η, ϕ ) := Z Γ( t ) η ϕg ( t ; η, ϕ ) := Z Γ( t ) η ϕ ∇ Γ( t ) · V ( t ); (ii) for η ∈ L (Γ( t )) , ϕ ∈ H (Γ( t )) , the first order term a N ( t ; η, ϕ ) := Z Γ( t ) η V τa ( t ) · ∇ Γ( t ) ϕ ; (iii) for η, ϕ ∈ H (Γ( t )) , the second order terms a S ( t ; η, ϕ ) := Z Γ( t ) ∇ Γ( t ) η · ∇ Γ( t ) ϕb ( t ; η, ϕ ) := Z Γ( t ) B( V ( t )) ∇ Γ( t ) η · ∇ Γ( t ) ϕ ; (iv) for η ∈ H − (Γ( t )) , ϕ ∈ H (Γ( t )) , the duality pairing m ∗ ( t ; η, ϕ ) := h η, ϕ i H − (Γ( t )) , H (Γ( t )) For simplicity of notation, we will from now on omit the dependence on t ∈ [0 , T ] . The weak form ofthe equations in (3.1.2) then reads as m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 ,a S ( u, η ) + m ( F ′ ( u ) , η ) − m ( w, η ) = 0 , (3.2.1)where the equations are satisfied for almost all t ∈ [0 , T ] and all test functions η ∈ L H . As usual,this is obtained by assuming u, w are sufficiently smooth functions satisfying (3.1.2), multiplying theequations by an arbitrary η ∈ L H , integrating over Γ( t ) and using integration by parts.As for the nonlinear term F ′ ( u ) , as we explained in the Introduction, it will be taken of smooth,logarithmic and double obstacle type, see (F s ), (F log ), (F obs ), and the second equation in (3.2.1) is to be interpreted as a variational inequality for the third case. Regardless of the choice of the potential,we observe that, if u solves the first equation of (3.2.1), then its integral is preserved over time; indeed,by testing the first equation with η = 1 , we obtain from Theorem 2.1 that ddt Z Γ( t ) u ( t ) = h ∂ • u ( t ) , i H − , H + Z Γ( t ) u ( t ) ∇ Γ · V = 0 . (3.2.2)This is an important feature of the Cahn-Hilliard dynamics, and it will play an important role in theanalysis of the equation, particularly for the study of the singular potentials.4. Smooth potentials
In this section, we establish well-posedness of the Cahn-Hilliard system for the case of a potential F satisfying, for some positive constants α , α , α , α > and real numbers β , β , β , β , β ∈ R :(A1) F ( ϕ ) ≥ β ;(A2) F = F + F , where:(A2.1) F and F are C ( R ) ;(A2.2) F ≥ is convex and satisfies, for some q ∈ [1 , ∞ ) , | F ′ ( ϕ ) | ≤ α | ϕ | q + β ;(A2.3) | F ′ ( ϕ ) | ≤ α F ( ϕ ) + β and | ϕF ′ ( ϕ ) | ≤ α F ( ϕ ) + β ;(A2.4) | F ′ ( ϕ ) | ≤ α | ϕ | + β .The growth conditions above prescribe the behaviour of the potential at infinity, and are motivatedby the ones given in [GK18]. They require the nonlinearity to behave polynomially at infinity, and tobe the sum of a convex term with a non-convex part which is essentially quadratic. These encapsulatethe idea expressed in the introduction that we consider potentials whose graphs have a W-shape, andare satisfied in particular by the double-well polynomial potentials F ( r ) = r − r , F ( r ) = ( r − , which are frequently considered in the literature. In Section 5, we will establish well-posedness for thesingular potentials by regularisation of these nonlinearities, and the conditions above are also generalenough to be applied to these approximating problems.The goal of this section is to prove well-posedness for the following problem. Problem 4.1.
Given u ∈ H (Γ ) and a potential F satisfying the assumptions (A1)-(A2) above, the Cahn-Hilliard system with a smooth potential is the following problem: find a pair ( u, w ) such that:(a) u ∈ H H − ∩ L ∞ H and w ∈ L H ;(b) the equations m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 ,a S ( u, η ) + m ( F ′ ( u ) , η ) − m ( w, η ) = 0 . (CH s )hold, for all η ∈ L H and almost every t ∈ [0 , T ] ;(c) u (0) = u almost everywhere in Γ .The pair ( u, w ) is called a weak solution of (CH s ). Remark 4.2.
Part (a) in the definition above ensures that our notion of a weak solution makes sense.Indeed:(i) the nonlinear term in (CH s ) is well-defined; since u ( t ) ∈ H (Γ( t )) for a.a. t ∈ [0 , T ] , the growthconditions (A1)-(A2) for F imply that, for such t , k F ′ ( u ( t )) k L (Γ( t )) ≤ C k | u ( t ) | q k L (Γ( t )) + C k | u | k L (Γ( t )) ≤ C k u k qH (Γ( t )) + C k u k L (Γ( t )) , (4.0.1)due to the Sobolev embedding theorem in Lemma 2.2d), so that F ′ ( u ( t )) ∈ L (Γ( t )) ;(ii) the initial condition in (c) is also well-defined because u ∈ H H − ֒ → C L , see Lemma 2.4b).We prove existence of a solution by employing an evolving space Galerkin method. AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 11
Galerkin approximation.
In order to define the approximating spaces, we pick a basis { χ j : j ∈ N } ⊂ H (Γ ) consisting of smooth functions such that χ is constant, which we transport using theflow map to { χ tj := φ t ( χ j ) : j ∈ N } ⊂ H (Γ( t )) basis for H (Γ( t )) . This definition implies the followingtransport formula for the basis functions, ∂ • χ tj ≡ , ∀ j ∈ N , which will be useful in setting up the approximating problem. We then define the approximation spacesas V M ( t ) = span { χ t , . . . , χ tM } and L V M := (cid:8) η ∈ L H : η ( t ) ∈ V M ( t ) , t ∈ [0 , T ] (cid:9) . It follows that L V M is dense in L H (and hence so it is in L L ).We consider also the L -projection operator P tM : L (Γ( t )) → V M ( t ) ⊂ H (Γ( t )) determined by the formula ( P tM η − η, ϕ ) L (Γ( t )) = 0 , for η ∈ L (Γ( t )) and ϕ ∈ V M ( t ) . It follows that P tM satisfies, for all η ∈ L (Γ( t )) , k P tM η k L (Γ( t )) ≤ k η k L (Γ( t )) and P tM η → η in L (Γ( t )) . We make the following additional assumption:
Assumption ( A P ) : We suppose that the projections P tM satisfy:(i) At time t = 0 , if η ∈ H (Γ ) then k P M η k H (Γ ) ≤ k η k H (Γ ) and P M η → η in H (Γ ) . (ii) For any ε > , t ∈ [0 , T ] , and η ∈ H (Γ( t )) , there exists ˜ M ∈ N such that we have theapproximation estimate k P tM η − η k L (Γ( t )) ≤ ε k η k H (Γ( t )) , ∀ M ≥ ˜ M. The assumption above is reasonable, and satisfied by the usual choices of a Galerkin scheme; thiscan be seen with an easy calculation for the Fourier expansion, and in [ER17] for a finite elementapproximation. We now set up the Galerkin approximation for (CH s ) in these spaces L V M as follows. Problem 4.3.
The
Galerkin approximation for (CH s ) is the following problem: for each M ∈ N , findfunctions u M , w M ∈ L V M with ∂ • u M ∈ L V M such that, for any η ∈ L V M and all t ∈ [0 , T ] , m ∗ ( ∂ • u M , η ) + g ( u M , η ) + a N ( u M , η ) + a S ( w M , η ) = 0 ,a S ( u M , η ) + m ( F ′ ( u M ) , η ) − m ( w M , η ) = 0 , (CH M s )and u M (0) = P M u almost everywhere in Γ .We now proceed to find a local solution for the problem. Proposition 4.4 (Well-posedness for (CH M s )) . There exists a unique local solution pair to (CH M s ) .More precisely, there exist functions ( u M , w M ) satisfying (CH M s ) on an interval [0 , t ∗ ) , < t ∗ ≤ T ,together with the initial condition u M (0) = P M u . The functions are of the form u M ( t ) = M X i =1 u Mi ( t ) χ ti , w M ( t ) = M X i =1 w Mi ( t ) χ ti , t ∈ [0 , t ∗ ) with coefficient functions u Mi ∈ C ([0 , t ∗ )) and w Mi ∈ C ([0 , t ∗ )) , for every i ∈ { , . . . , M } .Proof. We write u M ( t ) = P Mi =1 u Mi ( t ) χ ti and w M ( t ) = P Mi =1 w Mi ( t ) χ ti and test (CH M s ) with the basisfunction χ tj to write the problem above as M X i =1 ˙ u Mi ( t ) m ij ( t ) + u Mi ( t ) g ij ( t ) + u Mi ( t ) a Nij ( t ) + w Mi ( t ) a Sij ( t ) = 0 M X i =1 u Mi ( t ) a Sij ( t ) + f j ( u M ( t )) − w Mi ( t ) m ij ( t ) = 0 , which in matrix form reads as M ( t ) ˙ u M ( t ) + G ( t )( u M ( t )) + A N ( t ) u M ( t ) + A S ( t ) w M ( t ) = 0 A S ( t ) u M ( t ) + F ( u M ( t )) − M ( t ) w M ( t ) = 0 , where we denote, for i, j = 1 , . . . , M and any t , the solution vectors u M ( t ) = ( u M ( t ) , . . . , u MM ( t )) , w M ( t ) = ( w M ( t ) , . . . , w MM ( t )) , the coefficient matrices (cid:0) M ( t ) (cid:1) ij = m ij ( t ) := m ( t ; χ ti , χ tj ) , (cid:0) G ( t ) (cid:1) ij = g ij ( t ) := g ( t ; χ ti , χ tj ) , (cid:0) A S ( t ) (cid:1) ij = a Sij ( t ) := a S ( t ; χ ti , χ tj ) , (cid:0) A N ( t ) (cid:1) ij = a Nij ( t ) := a N ( t ; χ ti , χ tj ) and the nonlinear term by F ( u M ( t )) j = F j ( u M ( t )) := m ( t ; F ( u M ( t )) , χ tj ) . Solving the first equation for w M ( t ) and substituting on the second we obtain a semilinear first-orderODE for u M together with an initial condition u M (0) = P M u . Since F ′ is C , the result follows fromthe general theory of ODEs. (cid:3) We are now ready to establish a priori bounds for the equation. Before we state the next result, letus recall the definition of the Cahn-Hilliard energy functional: E CH [ u ] = Z Γ( t ) |∇ Γ( t ) u | F ( u ) . Proposition 4.5.
There exists ˜ M ∈ N such that the local solution ( u M , w M ) of (CH M s ) satisfies theenergy estimate sup [0 ,t ∗ ) E CH [ u M ] + 12 Z t ∗ k∇ Γ w M k L ≤ C, for M ≥ ˜ M, (4.1.1) where the constant C > depends only on the final time T , the constants in (A1)-(A2) for the potential F and the initial condition u . In particular, solutions ( u M , w M ) are defined on [0 , T ] , and we have sup [0 ,T ] E CH [ u M ] + 12 Z T k∇ Γ w M k L ≤ C, for M ≥ ˜ M. (4.1.2) Proof.
Differentiate the energy functional and use the equations (CH M s ) to obtain ddt E CH [ u M ] = a S (cid:0) ∂ • u M , u M (cid:1) + b (cid:0) u M , u M (cid:1) + m (cid:0) F ′ ( u M ) , ∂ • u M (cid:1) + g (cid:0) F ( u M ) , (cid:1) = m (cid:0) w M , ∂ • u M (cid:1) + b (cid:0) u M , u M (cid:1) + g (cid:0) F ( u M ) , (cid:1) = − g (cid:0) u M , w M (cid:1) − a N (cid:0) u M , w M (cid:1) − a S ( w M , w M ) + b (cid:0) u M , u M (cid:1) + g (cid:0) F ( u M ) , (cid:1) , which gives ddt E CH [ u M ] + k∇ Γ w M k L = − g ( u M , w M ) − a N (cid:0) u M , w M (cid:1) + b (cid:0) u M , u M (cid:1) + g (cid:0) F ( u M ) , (cid:1) To estimate the terms on the right hand side, we use the assumptions on the velocity field to obtain,for the last two terms, b (cid:0) u M , u M (cid:1) + g (cid:0) F ( u M ) , (cid:1) ≤ C V k∇ Γ u M k L + C V m (cid:0) F ( u M ) , (cid:1) + C ≤ C + C V E CH [ u M ] . Using the assumption on the tangential velocities and Young’s inequality yields | a N ( u M , w M ) | ≤ C k u M k L + 12 k∇ Γ w M k L . Observe that testing the first equation with η = 1 shows that the integral of u M is preserved, exactlyas in (3.2.2), and thus there exist C , C > such that k u M k L ≤ C k∇ Γ u M k L + C , (4.1.3) AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 13 from where | a N ( u M , w M ) | ≤ C k∇ Γ u M k L + C + 12 k∇ Γ w M k L . Combining the above leads to ddt E CH [ u M ] + 12 k∇ Γ w M k L ≤ − g ( u M , w M ) + C + C E CH [ u M ] . Now we note that by definition of the projections P tM we have g ( u M , w M ) = m (cid:0) w M , P M ( u M ∇ Γ · V ) (cid:1) , and so testing the second equation with η = P M ( u M ∇ Γ · V ) leads to | g ( u M , w M ) | ≤ (cid:12)(cid:12) a S (cid:0) u M , u M ∇ Γ · V (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) m (cid:0) F ′ ( u M ) , P M ( u M ∇ Γ · V ) (cid:1)(cid:12)(cid:12) ≤ C + C k∇ Γ u M k L + (cid:12)(cid:12) m (cid:0) F ′ ( u M ) , P M ( u M ∇ Γ · V ) (cid:1)(cid:12)(cid:12) ≤ C + C E CH [ u M ] + (cid:12)(cid:12) m (cid:0) F ′ ( u M ) , P M ( u M ∇ Γ · V ) (cid:1)(cid:12)(cid:12) (4.1.4)As for the remaining term, we note that (cid:12)(cid:12) m (cid:0) F ′ ( u M ) , P M ( u M ∇ Γ · V ) (cid:1)(cid:12)(cid:12) ≤ (cid:12)(cid:12) m (cid:0) F ′ ( u M ) , u M ∇ Γ · V ) (cid:12)(cid:12) + (cid:12)(cid:12) m (cid:0) F ′ ( u M ) , (cid:0) P M ( u M ∇ Γ · V ) − u M ∇ Γ · V (cid:1) (cid:1)(cid:12)(cid:12) ≤ C m ( F ( u M ) ,
1) + k F ′ ( u M ) k L k P M ( u M ∇ Γ · V ) − u M ∇ Γ · V k L . Using Assumption ( A P ) , for any ε > we can choose ˜ M ∈ N sufficiently large so that k P M ( u M ∇ Γ · V ) − u M ∇ Γ · V k L ≤ ε k u M ∇ Γ · V k H ≤ C ε k u M k H , which leads to, combining the estimate above with Remark (4.0.1), (cid:12)(cid:12) m (cid:0) F ′ ( u M ) , P M ( u M ∇ Γ · V ) (cid:1)(cid:12)(cid:12) ≤ C E CH ( u M ) + C ε k u M k q +1 H ≤ C + C E CH ( u M ) + C ε (cid:0) E CH [ u M ] (cid:1) q +1 . All in all, we have proved that ddt E CH [ u M ] + 12 k∇ Γ w M k L ≤ C (cid:16) C + E CH [ u M ] + ε (cid:0) E CH [ u M ] (cid:1) q +1 (cid:17) , (4.1.5)for some constants C, C > independent of both M and t . By picking ε > small enough, we canapply the generalised Gronwall inequality in Lemma A.1 to obtain a uniform bound sup [0 ,t ∗ ] E CH [ u M ] + Z t ∗ k∇ Γ w M k L ≤ C + C E CH ( P M u ) ≤ C , for M ≥ ˜ M, (4.1.6)where C is independent of M and t ∗ due to Assumption ( A P ) (i). (cid:3) Remark 4.6.
Observe that Assumption ( A P ) (ii) was used in order to obtain a small coefficient infront of the higher order term in (4.1.6). However, in the case that the potential has quadratic growthat infinity (i.e. q = 1 in (A2.2)), we can obtain the global energy estimate without resorting toAssumption ( A P ) . Indeed, in this case we would have k F ′ ( u M ) k L ≤ C + C k u M k L , and so wedirectly replace (4.1.4) with | g ( u M , w M ) | ≤ C + C k∇ Γ u M k L + C k u M k L ≤ C + C E CH [ u M ] , which then leads, instead of (4.1.6), to ddt E CH [ u M ] + 12 k∇ Γ w M k L ≤ C + C E CH [ u M ] . The same conclusion as above now follows from the classical Gronwall inequality, valid for all M .For the rest of the section, we always consider M ≥ ˜ M so that the energy estimate above issatisfied. Observe that, unlike the case of a stationary domain, in our setting the energy (4.1.2) doesnot necessarily decrease along a solution to the Cahn-Hilliard equation. It is also important to notethat the constants involved in the previous estimates are independent of M but rely strongly on thoseconstants appearing in assumptions (A1)-(A2) for the potential.As a consequence of the result above: Corollary 4.7 (A priori bounds) . We have uniform bounds for u M , w M in L ∞ H , L H , respectively.More precisely, sup t ∈ [0 ,T ] k u M ( t ) k H (Γ( t )) + Z T k w M ( t ) k H (Γ( t )) ≤ C where C > is a constant depending only on the initial condition, on the potential and on the finaltime T .In particular, we also have that u M is bounded in L ∞ L p , for all p ∈ [1 , + ∞ ) , and F ′ ( u M ) is boundedin L ∞ L .Proof. The energy estimate (4.1.2) immediately yields a uniform bound for ∇ Γ u M in L ∞ L , and so(4.1.3) implies that u M is uniformly bounded in L ∞ H .To show the bound for the H -norm of w M , we first note that testing the second equation with η = 1 leads to m (cid:0) w M , (cid:1) = m (cid:0) F ′ ( u M ) , (cid:1) , and this is uniformly bounded due to (4.0.1) and the uniform estimate for u M in H . Combined withthe uniform bound for ∇ Γ w M in L L given by the previous result, a uniform bound for w M in L H follows again from an application of Poincaré’s inequality.Finally, the fact that u M is bounded in L ∞ L p follows from the Sobolev embedding result in Lemma2.2d), and this combined with the growth assumptions on F imply that F ′ ( u M ) is bounded in L ∞ L . (cid:3) Passage to the limit.
The a priori bounds in the previous result allow us to obtain limit functions u ∈ L ∞ H and w ∈ L H such that, as M → ∞ and up to taking subsequences,(i) u M ∗ ⇀ u in L ∞ H and w M ⇀ w in L H ;(ii) the uniform bound for u M in L ∞ H also implies that u M ( t ) ⇀ u ( t ) in H (Γ( t )) for a.a. t ;(iii) by compactness, u M ( t ) → u ( t ) in L (Γ( t )) for a.a. t ;(iv) u M ( t ) → u ( t ) pointwise a.e. in Γ( t ) .The Sobolev embedding in Lemma 2.2d) additionally implies u ∈ L ∞ L p , for all p ∈ [1 , + ∞ ) . Due tocontinuity of F ′ , from point (iv) above it follows that F ′ ( u M ( t )) → F ′ ( u ( t )) pointwise a.e. in Γ( t ) . Wealso observe that, using the equation and the transport formula in (2.1), we have ddt k u M k L (Γ( t )) = m ( ∂ • u M , u M ) + 12 g ( u M , u M ) = − a N ( u M , u M ) − a S ( w M , u M ) − g ( u M , u M ) . Integrating over [0 , T ] and using the a priori bounds leads to k u M ( T ) k L (Γ( T )) ≤ C k P M u k L + C ≤ C , from where we additionally obtain z ∈ L (Γ( T )) such that u M ( T ) ⇀ z in L (Γ( T )) . Remark 4.8.
It is important to note that we have not yet obtained the existence of a weak timederivative for the limit u . In the classical setting, ∂ • u is obtained as a limit of the sequence ( ∂ • u M ) M ,but in the current time-dependent framework it seems not possible to obtain a uniform estimate for ( ∂ • u M ) M in L H − by the usual duality arguments. In particular, we do not have a uniform estimate for ( u M ) M in H H − , which precludes us from applying the Aubin-Lions compactness lemma in Proposition2.4(c) and obtaining stronger convergence results for ( u M ) M .In the next result we show the existence of ∂ • u by integrating the first equation by parts, in orderto transfer the time derivative to the test functions, and passing the obtained equations to the limit. Proposition 4.9.
Let ( u, w, z ) be the limit functions above. (i) There exists ∂ • u ∈ L H − , and u ∈ C L . (ii) We have u (0) = u and u ( T ) = z .Proof. (i). For any η ∈ L V M with ∂ • η ∈ L V M , integrating over [0 , T ] we can write the first equation ofthe system as m ( u M ( T ) , η ( T )) − m ( P M u , η (0)) + Z T a N ( u M , η ) + a S ( w M , η ) − m ( u, ∂ • η ) = 0 AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 15
For j ≤ M , take η ( t ) = ψ ( t ) χ tj with ψ ∈ C ([0 , T ]) to get m ( u M ( T ) , ψ ( T ) χ Tj ) − m ( P M u , ψ (0) χ j ) + Z T ψ ( t ) a N ( u M , χ tj ) + ψ ( t ) a S ( w M , χ tj ) − ψ ′ ( t ) m ( u M , χ tj ) = 0 . Passing to the limit M → ∞ , we obtain m ( z, ψ ( T ) χ Tj ) − m ( u , ψ (0) χ j ) + Z T ψ ( t ) a N ( u, χ tj ) + ψ ( t ) a S ( w, χ tj ) − ψ ′ ( t ) m ( u, χ tj ) = 0 . (4.2.1)Given η ∈ H (Γ ) , there exist coefficients a j ∈ R and a sequence η M = P Mj =1 a j χ j (note η M ∈ V M (0) for all M ) such that η M → η in H (Γ ) . Hence φ t η M = P Mj =1 a j χ tj converges to φ t η in H (Γ( t )) . Multiplying (4.2.1) by a j and summing over j = 1 , ..., M gives m ( z, ψ ( T ) φ T η M ) − m ( u , ψ (0) η M ) + Z T ψ ( t ) a N ( u, φ t η M ) + ψ ( t ) a S ( w, φ t η M ) − ψ ′ ( t ) m ( u, φ t η M ) = 0 and if we furthermore take ψ ∈ D (0 , T ) this simplifies to Z T ψ ( t ) a N ( u, φ t η M ) + Z T ψ ( t ) a S ( w, φ t η M ) = Z T ψ ′ ( t ) m ( u, φ t η M ) Letting M → ∞ yields Z T ψ ( t ) a N ( u, φ t η ) + Z T ψ ( t ) a S ( w, φ t η ) = Z T ψ ′ ( t ) m ( u, φ t η ) which means that t m ( u ( t ) , φ t η ) is weakly differentiable with ddt m ( u ( t ) , φ t η ) = − a N ( u, φ t η ) − a S ( w, φ t η ) It then follows from Lemma 2.2 that u ∈ H H − and that ∂ • u ∈ L H − satisfies Z T m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 , (4.2.2)as desired. The extra regularity u ∈ C L is a consequence of the continuous embedding H H − ֒ → C L ,see Proposition 2.4.(ii). Let η ∈ H H − . Using the transport formula in Theorem 2.1 and the equation for u , we have m ( u ( T ) , η ( T )) − m ( u (0) , η (0)) = Z T m ∗ ( ∂ • u, η ) + g ( u, η ) + m ∗ ( ∂ • η, u )= Z T m ∗ ( ∂ • η, u ) − a N ( u, η ) − a S ( w, η ) Taking η ( t ) = ψ ( t ) χ tj for any j ∈ N and ψ ∈ C ([0 , T ]) , we obtain m ( u ( T ) , ψ ( T ) χ Tj ) − m ( u (0) , ψ (0) χ j ) = Z T ψ ′ ( t ) m ( u, χ tj ) − Z T ψ ( t ) a N ( u, χ tj ) − Z T ψ ( t ) a S ( u, χ tj ) Comparing this with (4.2.1), we see that m ( u ( T ) , ψ ( T ) χ Tj ) − m ( u (0) , ψ (0) χ j ) = m ( z, ψ ( T ) χ Tj ) − m ( u , ψ (0) χ j ) . Picking ψ with ψ (0) = 1 , ψ ( T ) = 0 (resp. ψ (0) = 0 , ψ ( T ) = 1 ) we obtain u (0) = u (resp. u ( T ) = z ). (cid:3) We can now show that indeed the limit pair ( u, w ) solves (CH s ). Proposition 4.10.
The limit pair ( u, w ) is a solution to (CH s ) .Proof. We have already proved that u, w lie in the desired spaces, that u (0) = u , and it also followsfrom the previous proof that Z T m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 , ∀ η ∈ L H . (4.2.3) To obtain the second equation, let η ∈ L H and take η M ∈ L V M such that η M → η in L H . Weimmediately obtain for the linear terms that Z T a S ( u M , η M ) −→ Z T a S ( u, η ) , Z T m ( w M , η M ) −→ Z T m ( w, η ) , (4.2.4)For the nonlinear term, we note that F ′ ( u M ) ∈ L L converges pointwise a.e. to F ′ ( u ) ∈ L L andsatisfies k F ′ ( u M ) k L L ≤ C due to the a priori bounds, so that the generalised Dominated ConvergenceTheorem B.2 implies F ′ ( u M ) ⇀ F ′ ( u ) in L L , from where we also obtain Z T m (cid:0) F ′ ( u M ) , η M (cid:1) → Z T m (cid:0) F ′ ( u ) , η (cid:1) . (4.2.5)Combining (4.2.4) and (4.2.5) then gives Z T a S ( u, η ) + m ( F ′ ( u ) , η ) − m ( w, η ) dt = 0 . (4.2.6)Considering now test functions of the form ψ ( t ) η ( t, x ) with ψ ∈ C ∞ c (0 , T ) and η ∈ L H , equations(4.2.3) and (4.2.6) read as Z T ψ (cid:0) m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) (cid:1) dt = 0 Z T ψ (cid:0) a S ( u, η ) + m ( F ′ ( u ) , η ) − m ( w, η ) (cid:1) dt = 0 , and hence, for almost all t ∈ [0 , T ] , by the fundamental theorem of calculus of variations we obtain m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 ,a S ( u, η ) + m ( F ′ ( u ) , η ) − m ( w, η ) = 0 . In other words, ( u, w ) is a solution to (CH s ), as desired. (cid:3) We finally establish stability of solutions with respect to initial data under an additional assumptionon the non-convex part F of the potential. See Appendix C for the definition of k · k − . Proposition 4.11 (Stability) . Suppose F ′ is Lipschitz. For any u , , u , ∈ H (Γ ) with ( u , ) Γ =( u , ) Γ , if u , u denote the solutions of (CH s ) with u (0) = u , and u (0) = u , , then there exist C > , independent of t , such that, for almost all t ∈ [0 , T ] , k u ( t ) − u ( t ) k − ≤ e Ct k u , − u , k − . (4.2.7) In particular, if F ′ is Lipschitz, there exists at most one weak solution to the Cahn-Hilliard system (CH s ) .Proof. Suppose that we have two solution pairs ( u , w ) and ( u , w ) , and let us denote ξ u = u − u and ξ w = w − w . Subtracting the corresponding equations yields, for any η ∈ L H , m ∗ ( ∂ • ξ u , η ) + g ( ξ u , η ) + a N ( ξ u , η ) + a S ( ξ w , η ) = 0 , (4.2.8) a S ( ξ u , η ) + m ( F ′ ( u ) − F ′ ( u ) , η ) − m ( ξ w , η ) = 0 . (4.2.9)Recall that the mean value of both u and u must be preserved, and thus, for any t ∈ [0 , T ] , ξ u ( t ) has zero mean value over Γ( t ) . Thus the inverse Laplacian G ξ u of ξ u is well defined and an element of H H − (see Appendix C), and we can test (4.2.8) with G ξ u to get m ∗ ( ∂ • ξ u , G ξ u ) + g ( ξ u , G ξ u ) + a N ( ξ u , G ξ u ) + a S ( ξ w , G ξ u ) = 0 which is equivalent to ddt k ξ u k − + a N ( ξ u , G ξ u ) + m ( ξ w , ξ u ) = m ( ξ u , ∂ • G ξ u ) = a S ( G ξ u , ∂ • G ξ u ) . (4.2.10)Testing now (4.2.9) with ξ u yields k∇ Γ ξ u k L + m ( F ′ ( u ) − F ′ ( u ) , ξ u ) + m ( F ′ ( u ) − F ′ ( u ) , ξ u ) = m ( ξ w , ξ u ) . (4.2.11) AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 17
We estimate the nonlinear terms as follows. For the first term, observe that convexity of F impliesthat F ′ is monotone, and thus m ( F ′ ( u ) − F ′ ( u ) , ξ u ) ≥ . As for the second term, we use Lipschitz continuity of F ′ to obtain | m ( F ′ ( u ) − F ′ ( u ) , ξ u ) L | ≤ L k ξ u k L where L > is some positive constant. Hence, from (4.2.11) we obtain k∇ Γ ξ u k L ≤ m ( ξ w , ξ u ) + C k ξ u k L . (4.2.12)Adding (4.2.10) and (4.2.12) we get ddt k ξ u k − + k∇ Γ ξ u k L ≤ C k ξ u k L + a S ( G ξ u , ∂ • G ξ u ) − a N ( ξ u , G ξ u ) ≤ C k ξ u k L + a S ( G ξ u , ∂ • G ξ u ) + C k ξ u k − We now estimate the terms on the right hand side. For the first one, we can use Young’s inequalityto get C k ξ u k L = C a S ( ξ u , G ξ u ) ≤ k∇ Γ ξ u k L + C k ξ u k − , and for the second one we have a S ( G ξ u , ∂ • G ξ u ) L = 12 ddt a S ( G ξ u , G ξ u ) − b ( G ξ u , G ξ u ) ≤ ddt k ξ u k − + C k ξ u k − . In conclusion, ddt k ξ u k − + k∇ Γ ξ u k L ≤ C k ξ u k − , (4.2.13)and an application of Gronwall’s inequality implies (4.2.7).If u (0) = u (0) , then it follows from (4.2.13) that ξ u is constant, and since it has zero mean valueit must be u = u . From (4.2.9) we obtain w = w , giving uniqueness. (cid:3) We summarize our findings of this section in the following result.
Theorem 4.12.
Let u ∈ H (Γ ) and F : R → R be a potential satisfying assumptions (A1)-(A2).Then, there exists a pair ( u, w ) with u ∈ H H − ∩ L ∞ H and w ∈ L H satisfying, for all η ∈ L H and a.a. t ∈ [0 , T ] , m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 ,a S ( u, η ) + m ( F ′ ( u ) , η ) − m ( w, η ) = 0 , (CH s ) and u (0) = u almost everywhere in Γ . The solution u satisfies the additional regularity u ∈ C L ∩ L ∞ L p , for all p ∈ [1 , + ∞ ) . Furthermore, provided F ′ is Lipschitz, then u , , u , ∈ H (Γ ) with ( u , ) Γ = ( u , ) Γ , if u , u denote the solutions of (CH s ) with u (0) = u , and u (0) = u , , then there exist C > , independentof t , such that, for almost all t ∈ [0 , T ] , k u ( t ) − u ( t ) k − ≤ e Ct k u , − u , k − . (4.2.14) In particular, if F ′ is Lipschitz, the pair ( u, w ) is unique. The stability estimate (4.2.14) above follows immediately from (4.2.13).4.3.
Extra regularity.
To conclude, we study additional regularity properties of the solutions.
Theorem 4.13.
Denote by ( u, w ) a solution pair of the Cahn-Hilliard system with a smooth potentialgiven by Theorem 4.12. (i) We have the regularity u ∈ L H ; (ii) If Γ is a C -surface, the diffeomorphisms Φ t are C and F ′′ ( u ( t )) ∈ L (Γ( t )) for a.a. t ∈ [0 , T ] ,then u ∈ L H ; (iii) If u ∈ H (Γ ) and | F ′′ ( r ) | ≤ C | r | q − + C , then we have u ∈ L ∞ H , ∂ • u ∈ L L and w ∈ L ∞ L ∩ L H . (iv) If Γ is a C -surface, the diffeomorphisms Φ t are C , ∆ Γ F ′ ( u ( t )) ∈ L (Γ( t )) for a.a. t ∈ [0 , T ] ,and u ∈ H (Γ ) , then u ∈ L H , and u is a strong solution of the system, i.e. it satisfies ∂ • u + u ∇ Γ · V − ∇ Γ · (cid:0) u ( V τ − V a ) (cid:1) = ∆ Γ (cid:0) − ∆ Γ u + F ′ ( u ) (cid:1) in L (Γ( t )) . (4.3.1) for almost all t ∈ [0 , T ] .Proof. The second equation gives, for almost all t ∈ [0 , T ] , − ∆ Γ( t ) u ( t ) = w ( t ) − F ′ ( u ( t )) ∈ L (Γ( t )) , with the estimate k ∆ Γ( t ) u ( t ) k L (Γ( t )) ≤ k w ( t ) k L (Γ( t )) + k F ′ ( u ( t )) k L (Γ( t )) ≤ k w ( t ) k L (Γ( t )) , and elliptic regularity (see e.g. [DE13, Lemma 3.2]) then implies u ( t ) ∈ H (Γ( t )) , satisfying k u ( t ) k H (Γ( t )) ≤ k ∆ Γ( t ) u ( t ) k L (Γ( t )) + C k u ( t ) k H (Γ( t )) ≤ k w ( t ) k L (Γ( t )) + C k u ( t ) k H (Γ( t )) , (4.3.2)where C > can be taken to be independent of t . Integrating the above over [0 , T ] gives u ∈ L H , asdesired.In the second case, we have instead − ∆ Γ( t ) u ( t ) ∈ H (Γ( t )) , and we can use the extra regularity ofthe surfaces and the nonlinear term to show that u ∈ L H .Now suppose u ∈ H (Γ ) . Integrating by parts we have a S ( P M u , w M (0)) = a S ( u , w M (0)) = − m (∆ Γ u , w M (0)) , and thus from the second equation of (CH M s ) at t = 0 we obtain k w M (0) k L = a S ( u M (0) , w M (0)) + m ( F ′ ( u M (0)) , w M (0))= − m (∆ Γ u , w M (0)) + m ( F ′ ( u M (0)) , w M (0)) ≤ (cid:0) k ∆ Γ u k L + C k u M (0) k qH (cid:1) k w M (0) k L ≤ (cid:0) k ∆ Γ u k L + C k u k qH (cid:1) k w M (0) k L , (4.3.3)which implies that k w M (0) k L is uniformly bounded. To obtain ∂ • u ∈ L L , we differentiate the secondequation to obtain, for all η ∈ L V M with ∂ • η ∈ L V M , a S ( ∂ • u M , η ) + b ( u M , η ) + m ( F ′′ ( u M ) ∂ • u M , η ) + g ( F ′ ( u M ) , η ) = m ( ∂ • w M , η ) + g ( w M , η ) . The terms involving ∂ • η vanish because ∂ • η is still an admissible test function and ( u M , w M ) is thesolution pair to (CH M s ). Testing the above with η = w M gives a S ( ∂ • u M , w M ) + b ( u M , w M ) + m ( F ′′ ( u M ) ∂ • u M , w M ) + g ( F ′ ( u M ) , w M )= 12 ddt k w M k L + 12 g ( w M , w M ) . (4.3.4)Now taking η = ∂ • u M in the first equation of (CH M s ) gives k ∂ • u M k L + g ( u M , ∂ • u M ) + a N ( u M , ∂ • u M ) + a S ( w M , ∂ • u M ) = 0 , (4.3.5)and combining (4.3.4) with (4.3.5) we obtain ddt k w M k L + k ∂ • u M k L = − g ( u M , ∂ • u M ) − a N ( u M , ∂ • u M ) − g ( w M , w M ) + b ( u M , w M ) + m ( F ′′ ( u M ) ∂ • u M , w M ) + g ( F ′ ( u M ) , w M ) . Using the uniform bounds for u M , the first four terms on the right are estimated as − g ( u M , ∂ • u M ) − a N ( u M , ∂ • u M ) − g ( w M , w M ) + b ( u M , w M ) ≤ k ∂ • u M k L + C + C k w M k H , and the last two terms we estimate using the uniform bounds for u M and the Sobolev inequality: m ( F ′′ ( u M ) ∂ • u M , w M ) + g ( F ′ ( u M ) , w M ) ≤ k ∂ • u M k L + C k F ′′ ( u M ) k L k ( w M ) k L AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 19 + C k F ′ ( u M ) k L k w M k L ≤ k ∂ • u M k L + C k w M k H . Putting these two estimates together we obtain k ∂ • u M k L + ddt k w M k L ≤ C + C k w M k H , and integrating yields sup [0 ,T ] k w M k L + Z T k ∂ • u M k L ≤ k w M (0) k L + C ≤ C , using (4.3.3). Letting M → ∞ yields w ∈ L ∞ L and ∂ • u ∈ L L , as desired. But then we note that thefirst equation actually gives − ∆ Γ w ( t ) ∈ L (Γ( t )) , and using elliptic regularity as in the first part ofthis proof implies w ∈ L H . The inequality in (4.3.2) gives u ∈ L ∞ H .Finally, in the conditions of (iv) we can deduce u ∈ L H from the fact that w ∈ L H , and it followsthat u satisfies, for almost all t ∈ [0 , T ] , the equation ∂ • u + u ∇ Γ · V − ∇ Γ · (cid:0) u ( V τ − V a ) (cid:1) = ∆ Γ (cid:0) − ∆ Γ u + F ′ ( u ) (cid:1) in L (Γ( t )) , finishing the proof. (cid:3) Remark 4.14.
Combining the results above with the Sobolev embedding theorems in Lemma 2.2d)and classical Schauder theory, one should obtain, for a solution pair ( u, w ) a.a. t ∈ [0 , T ] and every α ∈ (0 , ,(i) u ( t ) ∈ C α (Γ( t )) ;(ii) in the conditions of Theorem 4.13(ii), u ( t ) ∈ C α (Γ( t )) ;(iii) in the conditions of Theorem 4.13(iii), w ( t ) ∈ C α (Γ( t )) , and if additionally F ′ ( u ( t )) ∈ C α (Γ( t )) then u ( t ) ∈ C α (Γ( t )) ;(iv) in the conditions of Theorem 4.13(iv), u ( t ) ∈ C α (Γ( t )) , and if additionally F ′ ( u ( t )) ∈ C α (Γ( t )) then u ( t ) ∈ C α (Γ( t )) and u is a classical solution, i.e. (4.3.1) holds for all t ∈ [0 , T ] .We leave the study of classical solutions and extra regularity for future work.5. Non smooth potentials
In this section, we study the same problem with a logarithmic and a double obstacle potentials.These are non smooth potentials, and in both cases an appropriate weak formulation needs to beconsidered. For the former, the derivative is singular at ± , and so we need to impose the constraint | u | < for a solution. In the latter case, the energy is not differentiable, and the second equationmust be reinterpreted as a variational inequality. We shall see that, in both cases, a general statementobtained as above is not possible to obtain, and the choice of admissible initial conditions is related tothe evolution of the surfaces.5.1. Logarithmic potential.
In this section we define, for r ∈ [ − , , F θ ( r ) = θ θ c ((1 + r ) log(1 + r ) + (1 − r ) log(1 − r )) + 1 − r θ θ c F log ( r ) + 1 − r , (5.1.1)where θ represents the temperature and θ c is the critical temperature. For θ > θ c , the potential isstrictly convex with a global minimum at u = 0 . For θ < θ c , F θ is a double well potential with twoglobal minima. Let us take for simplicity θ c = 1 and assume θ < . Since in this section θ will be afixed constant, we will henceforth omit the dependence on θ of both the potential and the solution.To use the logarithmic nonlinearity it is clear that we are only interested in solutions taking valuesin the interval ( − , . As in the previous section, we expect to have a solution pair ( u, w ) solving theequations m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 a S ( u, η ) + m ( F ′ ( u ) , η ) − m ( w, η ) = 0 , (5.1.2)satisfying | u | < almost everywhere and an initial condition u (0) = u , for some u ∈ H (Γ ) with | u | ≤ . These conditions on u are enough to guarantee that the energy at the initial time makes sense. We claim that this is not possible without extra conditions on the initial data and the evolutionof the surface. Indeed, define, for t ∈ [0 , T ] and η ∈ H (Γ ) , m η ( t ) := 1 | Γ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)Z Γ η (cid:12)(cid:12)(cid:12)(cid:12) . (5.1.3)We start by proving a simple result which shows that well-posedness of the problem is related to thesize of this function m η . Proposition 5.1.
Let { Γ( t ) } t ∈ [0 ,T ] be an evolving surface in R and u ∈ H (Γ ) satisfy | u | ≤ .Suppose that there exists a subset of [0 , T ] with positive measure on which m u ( t ) ≥ . Then therecannot exist a pair ( u, w ) satisfying (5.1.2) and | u | < almost everywhere.Proof. If such a pair exists, let ˜ I ⊂ [0 , T ] be a subset of full measure in which equations (CH log ) hold, | u | < and m u ( t ) ≥ . We observe that, as before, the integral of the solution is preserved, and as aconsequence, for t ∈ ˜ I , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ( t ) u ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Γ u (cid:12)(cid:12)(cid:12)(cid:12) ≥ | Γ( t ) | , but since | u | < we also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ( t ) u ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Γ( t ) | u ( t ) | < | Γ( t ) | , which is a contradiction. (cid:3) An immediate consequence of the above and the fact that m u is continuous is the following: Corollary 5.2.
If there exists t ∈ [0 , T ] such that m u ( t ) > , then there cannot exist a pair ( u, w ) satisfying (5.1.2) and | u | < almost everywhere. The results above show that, if we are to expect existence of solutions, then it must be m u ( t ) < foralmost every t ∈ [0 , T ] . In this article, we will focus in the case where m u ( t ) < for every t ∈ [0 , T ] .In particular, note that | Γ | (cid:12)(cid:12)(cid:12)(cid:12)Z Γ u (cid:12)(cid:12)(cid:12)(cid:12) = m u (0) < . For future use, let us define the set I of admissible initial conditions by I := (cid:8) η ∈ H (Γ ) : | η | ≤ a.e. on Γ , E CH [ η ] < ∞ , and m η < (cid:9) . The first conditions ensure that the energy is defined at the initial time, and we motivated the secondcondition on the previous paragraph. Note that, since m u is continuous, there exists α ∈ [0 , suchthat ≤ m u ( t ) ≤ α < , for all t ∈ [0 , T ] .We can now state the problem we aim to solve in this section. Denote also ϕ ( r ) := (cid:16) F log (cid:17) ′ ( r ) , r ∈ ( − , . Problem 5.3.
Given an initial condition u ∈ I , the Cahn-Hilliard system with a logarithmic poten-tial is the following problem: find a pair ( u, w ) satisfying:(a) u ∈ L ∞ H ∩ H H − and w ∈ L H ;(b) for almost all t ∈ [0 , T ] , | u ( t ) | < almost everywhere in Γ( t ) ;(c) for every η ∈ L H , the equations m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 ,a S ( u, η ) + θ m ( ϕ ( u ) , η ) − m ( u, η ) − m ( w, η ) = 0 , (CH log )for almost every t ∈ [0 , T ] ;(d) u (0) = u almost everywhere in Γ .The pair ( u, w ) is called a weak solution of (CH log ). AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 21
Approximating problem.
To prove well-posedness for the problem above we will proceed byapproximation. Define, for r ∈ R and δ ∈ (0 , , F log δ ( r ) = θ (1 − r ) log( δ ) + (1 + r ) log(2 − δ ) + (1 − r ) δ + (1+ r ) − δ ) − , r ≥ − δ (1 + r ) log(1 + r ) + (1 − r ) log(1 − r ) , | r | ≤ − δ (1 + r ) log( δ ) + (1 − r ) log(2 − δ ) + (1+ r ) δ + (1 − r ) − δ ) − , r ≤ − δ . Then F log δ ∈ C ( R ) . For simplicity of notation let us also denote ϕ δ ( r ) := (cid:16) F log δ (cid:17) ′ ( r ) , r ∈ R . Consider now the following problem.
Problem 5.4.
Given δ ∈ (0 , and an initial condition u ∈ I , find ( u δ , w δ ) ∈ L ∞ H × L H , with ∂ • u δ ∈ L H − , satisfying, for all η ∈ L H and a.e. t ∈ [0 , T ] , m ∗ ( ∂ • u δ , η ) + g ( u δ , η ) + a N ( u δ , η ) + a S ( w δ , η ) = 0 ,a S ( u δ , η ) + θ m ( ϕ δ ( u δ ) , η ) − m ( u δ , η ) − m ( w δ , η ) = 0 , (CH δ log )and u δ (0) = u almost everywhere in Γ .Observe that ϕ δ is Lipschitz, so that ϕ δ ( u δ ( t )) ∈ H (Γ( t )) for almost all t ∈ [0 , T ] . It is also simpleto check that the potential F log δ satisfies all the assumptions (A1)-(A2) in the previous section, andhence it follows from Theorem 4.12 that we have existence and uniqueness of a solution pair ( u δ , w δ ) for (CH δ log ) (observe that F log δ has quadratic growth at infinity, and the observations for the Galerkinapproximation problem in Remark 4.6 hold). Note that, as we have previously mentioned, the uniformbounds we obtained in the previous section depend on the form of the potential, and consequently on δ ; this means that in order to let δ → in (CH δ log ) we need new estimates. Before doing so, we needan additional lemma. Consider the Galerkin approximation for (CH δ log ): Problem 5.5.
Given δ ∈ (0 , and an initial condition u ∈ I with E ( u ) < ∞ , find ( u Mδ , w Mδ ) ∈ L V M with ∂ • u Mδ ∈ L V M satisfying, for all η ∈ L V M and t ∈ [0 , T ] , m ∗ ( ∂ • u Mδ , η ) + g ( u Mδ , η ) + a N ( u Mδ , η ) + a S ( w Mδ , η ) = 0 ,a S ( u Mδ , η ) + θ m ( ϕ δ ( u Mδ ) , η ) − m ( u Mδ , η ) − m ( w Mδ , η ) = 0 , (CH M,δ log )and u Mδ (0) = P M u almost everywhere in Γ .Let us also introduce the following approximate Cahn-Hilliard energy: E CH δ [ u ] = Z Γ( t ) |∇ Γ u | + F δ ( u ) . We have the following uniform bound for the initial energy.
Lemma 5.6.
Let ( u Mδ , w Mδ ) be the unique solution of (CH M,δ log ) . There exists a constant C > ,independent of both M and δ , such that E CH δ [0; u Mδ (0)] ≤ C .Proof. For each M ∈ N , we have E CH δ [ P M u ] ≤ k∇ Γ P M u k L + m ( F log δ ( P M u ) ,
1) + 12 m (1 − ( P M u ) , ≤ C + C k u k H + m ( F log δ ( P M u ) , . Now convexity of F log δ implies that, for any r, s ∈ R , F log δ ( r ) ≥ F log δ ( s ) + ϕ δ ( s )( r − s ) , and combining this with F log δ ( r ) ≤ F log ( r ) , for r ≤ , leads to E CH δ [ P M u ] ≤ C + C k u k H + m ( ϕ δ ( P M u ) , P M u − u ) + m ( F log δ ( u ) , ≤ C k u k H + C + m ( ϕ δ ( P M u ) , P M u − u ) + E CH [ u ] ≤ C + m ( ϕ δ ( P M u ) , P M u − u ) , with C independent of both M and δ . Now note that ϕ ′ δ = F ′′ ,δ is bounded (for each fixed δ ), so that ϕ δ is Lipschitz continuous and thus ϕ δ ( P M u ) → ϕ δ ( u ) in L (Γ ) . Consequently m ( ϕ δ ( P M u ) , P M u − u ) → as M → ∞ , and so lim sup M →∞ E CH δ [ P M u ] ≤ C ; in particular, E CH δ [ P M u ] is uniformly bounded. (cid:3) In the next result, we obtain uniform estimates for the solution of (CH δ log ). Proposition 5.7 (A priori estimates for (CH δ log )) . Let ( u δ , w δ ) be the unique solution of (CH δ log ) . (a) There exists a constant
C > , independent of δ , such that sup t ∈ [0 ,T ] E CH δ [ t ; u δ ( t )] + Z T k∇ Γ( t ) w δ ( t ) k L (Γ( t )) dt ≤ C ; (b) u δ and w δ are uniformly bounded in L ∞ H and L H , respectively; (c) ∂ • u δ is uniformly bounded in L H − ; (d) ϕ δ ( u δ ) is uniformly bounded in L L .Proof. a) As in the proof of Proposition 4.5, we have for the solution ( u Mδ , w Mδ ) of (CH M,δ log ) ddt E CH [ u Mδ ] + 12 k∇ Γ w Mδ k L ≤ − g ( u Mδ , w Mδ ) + C + C E CH δ [ u Mδ ] , with constants C , C independent of the assumptions on the potential. Integrating between and t we obtain E CH δ [ u Mδ ] + Z t k∇ Γ w Mδ k L ≤ C + C Z t E CH δ [ u Mδ ] − Z t g ( u Mδ , w Mδ ) , where C > is the constant (independent of δ ) given by the previous result. Letting M → ∞ andusing the convergence results for the Galerkin approximation this yields E CH δ [ u δ ] + Z t k∇ Γ w δ k L ≤ C + C Z t E CH δ [ u δ ] − Z t g ( u δ , w δ ) ds. (5.1.4)Both C and C are independent of M and δ .Our aim now is to obtain an estimate on the last term on the right hand side above. Testing thesecond equation of (CH δ log ) with u δ ∇ Γ · V , we get g ( u δ , w δ ) = m ( ∇ Γ u δ , ∇ Γ ( u δ ∇ Γ · V )) + m ( ϕ δ ( u δ ) , u δ ∇ Γ · V ) − m ( u δ , u δ ∇ Γ · V ) ≤ C k∇ Γ u δ k L + C m ( ϕ δ ( u δ ) , u δ ) + C ≤ C k∇ Γ u δ k L + C k∇ Γ u δ k L + | m ( w δ , u δ ) | + C , where C , C , C are independent of δ . Taking η ≡ in the second equation of (CH δ log ) yields m ( w δ ,
1) = − m ( u δ ,
1) + m ( ϕ δ ( u δ ) , , and using the fact that | ϕ δ ( r ) | ≤ rϕ δ ( r ) + 1 and the second equation again we obtain | m ( w δ , | ≤ | m ( u δ , | + m ( ϕ δ ( u δ ) , u δ ) + C ≤ C | ( u ) Γ | + k u δ k L − k∇ Γ u δ k L + m ( w δ , u δ ) + C ≤ C | ( u ) Γ | + C k∇ Γ u δ k L + | m ( w δ , u δ ) | + C . (5.1.5)Recalling the definition of the inverse Laplacian (see Appendix C), we now have | m ( w δ , u δ ) | ≤ | m ( w δ , u δ − ( u δ ) Γ ) | + | ( u δ ) Γ | | m ( w δ , | = | a S ( w δ , G ( u δ − ( u δ ) Γ )) | + m u ( t ) | m ( w δ , |≤ | a S ( w δ , G ( u δ − ( u δ ) Γ )) | + α | m ( w δ , | (5.1.6) AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 23 and hence, using Young and Poincaré inequalities we obtain | m ( w δ , u δ ) | ≤ − α k∇ Γ w δ k L + C k∇ Γ G ( u δ − ( u δ ) Γ ) k L + α | m ( w δ , |≤ − α k∇ Γ w δ k L + C k u δ − ( u δ ) Γ k L + α | m ( w δ , |≤ − α k∇ Γ w δ k L + C k∇ Γ u δ k L + α | m ( w δ , | . (5.1.7)Combining (5.1.5) and (5.1.7) yields | m ( w δ , u δ ) | ≤ k∇ Γ w δ k L + C k∇ Γ u δ k L + C , and thus | g ( u δ , w δ ) | ≤ k∇ Γ w δ k L + C k∇ Γ u δ k L + C . Hence from (6.1.2) we get, for C , C independent of M and δ , E CH δ [ u δ ] + 12 Z t k∇ Γ w δ k L ≤ C + C Z t E CH δ [ u δ ] , (5.1.8)The conclusion now follows from Gronwall’s inequality.b) From the energy estimate in (5.1.8), we obtain a uniform bound for ∇ Γ u δ in L ∞ L , and since themean value of u δ is preserved it follows that u δ is uniformly bounded in L ∞ H . We also have a bound for ∇ Γ w δ in L L . To obtain a bound on the L L -norm of w δ , we start by noticing that, using the uniformestimates for u δ , we obtain from (5.1.5) | m ( w δ , | ≤ C + | m ( w δ , u δ ) | , (5.1.9)and now we estimate (5.1.6) without using Young’s inequality on the first term to get | m ( w δ , u δ ) | ≤ k∇ Γ w δ k L k∇ Γ G ( u δ − ( u δ ) Γ ) k L + α | m ( w δ , |≤ k∇ Γ w δ k L k u δ − ( u δ ) Γ k L + α | m ( w δ , |≤ C k∇ Γ w δ k L + α | m ( w δ , | . (5.1.10)Combining (5.1.9) and (5.1.10) yields | m ( w δ , | ≤ C + C k∇ Γ w δ k L + α | m ( w δ , | , and dividing through by − α we get | m ( w δ , | ≤ C + C k∇ Γ w δ k L . So the desired bound followsonce again by applying Poincaré’s inequality.c) This follows by duality, since, for any η ∈ L H , the previous estimates yield h ∂ • u δ , η i L H − × L H = Z T − g ( u δ , η ) − a N ( u δ , η ) − a S ( w δ , η ) ≤ C k η k L H , giving the conclusion.d) Test the second equation of (CH δ log ) with ϕ δ ( u δ ) to obtain k ϕ δ ( u δ ) k L = m ( w δ , ϕ δ ( u δ )) − a S ( u δ , ϕ δ ( u δ )) − m ( u δ , ϕ δ ( u δ ))= m ( w δ , ϕ δ ( u δ )) − m ( ϕ ′ δ ( u δ ) ∇ Γ u δ , ∇ Γ u δ ) − m ( u δ , ϕ δ ( u δ )) ≤ m ( w δ , ϕ δ ( u δ )) − m ( u δ , ϕ δ ( u δ )) , since ϕ ′ δ ≥ , and a standard application of Young’s inequality gives k ϕ δ ( u δ ) k L ≤ C k w δ k L + C . The conclusion follows from the uniform bound for w δ in L L . (cid:3) One extra challenge that we did not face in the previous section was the fact that, for the logarithmiccase, the potential is only defined on the interval [ − , and its derivative is now singular at ± . Hence,for the second equation in (CH log ) to make sense, we must prove that the solution u takes values onlyon ( − , (up to a set of measure zero). This requires the next a priori estimate for the approximation. Lemma 5.8.
There exist constants C , C > , independent of δ , such that Z Γ( t ) [ − − u δ ( t )] + dΓ + Z Γ( t ) [ u δ ( t ) − + dΓ ≤ C | log( δ ) | + C δ, where we denote, for a real-valued function f , [ f ] + = max(0 , f ) .Proof. We start by observing that the uniform bounds imply that there exists a constant
C > ,independent of δ , such that, for any δ > and t ∈ [0 , T ] , Z Γ( t ) F log δ ( u δ ) dΓ ≤ C. Now let < δ < and denote, for t ∈ [0 , T ] , A δ ( t ) = { u δ ( t ) > − δ } , A δ ( t ) = {| u δ ( t ) | ≤ − δ } , A δ ( t ) = { u δ ( t ) < − δ } . We estimate the integrals over each of these sets as follows. For the integral over A δ , observe that, for | r | < , we have (1 + r ) log(1 + r ) ≥ and (1 − r ) log(1 − r ) ≥ − e − , so that Z A δ ( t ) F log δ ( u δ ) dΓ ≥ Z A δ ( t ) (1 − u δ ( t )) log(1 − u δ ( t )) dΓ ≥ − e | Γ( t ) | ≥ − Ce , where C is a constant independent of both δ and t . For the other integral terms, we note that log( δ ) < ,and thus Z A δ ( t ) F ,δ ( u δ ( t )) dΓ ≥ Z A δ ( t ) (1 − u δ ( t )) log( δ ) + (1 + u δ ( t )) log(2 − δ ) dΓ − C ≥ log( δ ) Z A δ ( t ) − u δ ( t ) dΓ − C ≥ log( δ ) Z { u δ ( t ) > } − u δ ( t ) dΓ + δ log( δ ) | Γ( t ) | − C ≥ − log( δ ) Z Γ( t ) [ u δ ( t ) − + dΓ + Cδ log( δ ) − C, Similarly, Z A δ ( t ) F ,δ ( u δ ( t )) dΓ ≥ Z A δ ( t ) (1 + u δ ( t )) log( δ ) + (1 − u δ ( t )) log(2 − δ ) dΓ − C ≥ log( δ ) Z A δ ( t ) u δ ( t ) dΓ − C ≥ log( δ ) Z { u δ ( t ) < − } u δ ( t ) dΓ + δ log( δ ) | Γ( t ) | − C ≥ − log( δ ) Z Γ( t ) [ − − u δ ( t )] + dΓ + Cδ log( δ ) − C. Therefore | log( δ ) | Z Γ( t ) [ − − u δ ( t )] + dΓ + | log( δ ) | Z Γ( t ) [ u δ ( t ) − + dΓ ≤ C + C δ | log( δ ) | and dividing through by | log( δ ) | we obtain Z Γ( t ) [ − − u δ ( t )] + dΓ + Z Γ( t ) [ u δ ( t ) − + dΓ ≤ C | log( δ ) | + C δ. (cid:3) Passage to the limit.
As a consequence of the bounds established above, there exist u ∈ L H ,with ∂ • u ∈ L H − , and w ∈ L H , such that, as δ → (up to a subsequence) u δ ⇀ u (weakly) in L H , ∂ • u δ ⇀ ∂ • u (weakly) in L H − , w δ ⇀ w (weakly) in L H . (5.1.11)By the Aubin-Lions compactness result in Proposition 2.4(c) we additionally obtain u δ → u strongly in L L , and so also up to a subsequence (non-relabeled) we have, for almost every t ∈ [0 , T ] , u δ ( t ) → u ( t ) almost everywhere in Γ( t ) . Letting δ → in 5.8 immediately yields AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 25
Corollary 5.9.
For almost every t ∈ [0 , T ] , | u ( t ) | ≤ almost everywhere on Γ( t ) . We are not yet done since we still need to prove that the measure of the set where the solution iseither or − is zero. First, we show the following auxiliary result. Lemma 5.10.
Up to a subsequence, we have, for almost every t ∈ [0 , T ] , lim δ ց ϕ δ ( u δ ( t )) = ( ϕ ( u ( t )) , if | u ( t ) | < a.e. in Γ( t ) ∞ , otherwisealmost everywhere in Γ( t ) .Proof. Let us fix t ∈ [0 , T ] for which u δ ( t ) → u ( t ) a.e. in Γ( t ) . Such a set has full measure in [0 , T ] .Suppose first that | u ( t ) | < a.e. in Γ( t ) . Passing to a subsequence, we can further assume that | u δ ( t ) | < a.e. in Γ( t ) . Let x ∈ Γ( t ) belong to the full measure subset of Γ( t ) for which u δ ( t, x ) → u ( t, x ) , and let τ > be arbitrary. Now choose:(i) δ > such that, for δ < δ , ϕ δ ( u δ ( t, x )) = ϕ ( u δ ( t, x )) ;(ii) using continuity of ϕ at u ( x, t ) , δ > such that, if y ∈ Γ( t ) is such that | y − u ( x, t ) | < δ ,then | ϕ ( y ) − ϕ ( u ( x, t )) | < τ ;(iii) δ > such that, for δ < δ , | u δ ( x, t ) − u ( x, t ) | < δ ;Therefore, for δ < min { δ , δ , δ } , we have | ϕ δ ( u δ ( t, x )) − ϕ ( u ( t, x )) | = | ϕ δ ( u δ ( t, x )) − ϕ δ ( u ( t, x )) | < τ, proving the result for those points where | u | < .Now let t ∈ [0 , T ] and x ∈ Γ( t ) be such that | u ( t, x ) | = 1 . If u ( t, x ) = 1 , then ϕ δ ( u δ ( t, x )) ≥ min { ϕ ( u δ ( t, x )) , ϕ (1 − δ ) } → + ∞ , as δ → , and if u ( t, x ) = − , then ϕ δ ( u δ ( t, x )) ≤ max { ϕ ( u δ ( t, x )) , ϕ ( − δ ) } → −∞ , as δ → , concluding the proof. (cid:3) Combining this with the a priori bounds allows us to conclude that indeed the solution cannot takethe values ± on a set of positive measure. Lemma 5.11.
For almost every t ∈ [0 , T ] , the set { x ∈ Γ( t ) : | u ( t, x ) | = 1 } has measure zero in Γ( t ) .Proof. We have, up to a subsequence, lim δ ց ϕ δ ( u δ ( t )) u δ ( t ) = ( ϕ ( u ( t )) u ( t ) , if | u ( t ) | < a.e. in Γ( t )+ ∞ , otherwise . (5.1.12)Now observe that, testing the second equation with u δ , we obtain Z Γ( t ) ϕ δ ( u δ ) u δ ≤ C k∇ Γ u δ k L + C k w δ k L ≤ C + C k w δ k L , and therefore Z T Z Γ( t ) ϕ δ ( u δ ) u δ ≤ C T + C Z T k w δ k L ≤ C . Since ϕ δ ( r ) r ≥ , Fatou’s lemma implies that Z T Z Γ( t ) lim inf δ → ϕ δ ( u δ ) u δ ≤ lim inf δ → Z T Z Γ( t ) ϕ δ ( u δ ( t )) u δ ≤ C. But the limit of ϕ δ ( u δ ) u δ is given by (5.1.12), and thus it follows that, for almost every t ∈ [0 , T ] , itmust be | u ( t ) | < almost everywhere in Γ( t ) , proving the result. (cid:3) Remark 5.12.
The result above then implies that u remains bounded in the interval ( − , provided u is given also in that interval. This is not the case for the smooth potentials considered in Section 4,see [Mir19, Remark 4.10], which means that the problem (CH s ) does not produce physically meaningfulsolutions. It is nonetheless important to analyse it, as the quartic potentials have been seen to be goodapproximations for (5.1.1) when a shallow quench is considered (i.e. when the temperature θ is closeto the critical temperature for the system). We now have everything we need to conclude the proof of well-posedness. As in the previous section,taking the limit on the linear terms of the system is a simple consequence of the convergence results(5.1.11). As for the nonlinear term, the uniform bound for ϕ δ ( u δ ) in L L combined with ϕ δ ( u δ ) → ϕ ( u ) a.e. gives, by Theorem B.1, ϕ δ ( u δ ) ⇀ ϕ ( u ) in L L . All in all: Theorem 5.13.
The pair ( u, w ) given by (5.1.11) is the unique weak solution of the Cahn-Hilliardsystem with a logarithmic potential.Proof. The fact that ( u, w ) satisfies the equations and fulfils the initial condition follows exactly as inthe proof of Lemmas 4.9, together with what we observed in the previous paragraph. The proof ofuniqueness is also the same as in Proposition 4.11, by noting that also in this case ϕ is monotone and F ′ is Lipschitz. (cid:3) In summary, in this section we have proved:
Theorem 5.14.
Let u ∈ I and F : [ − , → R be the logarithmic potential (5.1.1) . Then, thereexists a unique pair ( u, w ) with u ∈ L ∞ H ∩ H H − and w ∈ L H such that, for a.a. t ∈ [0 , T ] , | u ( t ) | < a.e. in Γ( t ) , and satisfying, for all η ∈ L H and a.a. t ∈ [0 , T ] , m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 ,a S ( u, η ) + θ m ( ϕ ( u ) , η ) − m ( u, η ) − m ( w, η ) = 0 , and u (0) = u almost everywhere in Γ . The solution u satisfies the additional regularity u ∈ C L ∩ L ∞ L p , for all p ∈ [1 , + ∞ ) . Furthermore, if u , v ∈ I satisfy ( u ) Γ = ( v ) Γ , and u, v are the solutions of the system with u (0) = u , v (0) = v , then there exists a constant C > independent of t such that, for almost all t ∈ [0 , T ] , k u ( t ) − v ( t ) k − ≤ e Ct k u − v k − . Extra regularity.
Exactly as in the previous section, we can immediately obtain an extra degreeof regularity for u : Theorem 5.15.
For the solution ( u, w ) given by the previous theorem, we have u ∈ L H .Proof. This follows exactly as in Theorem 4.13 by noting that u solves the problem − ∆ Γ u = w − F ′ ( u ) ∈ L L , and thus elliptic regularity theory applies. (cid:3) As in the previous section, Lemma 2.2d) implies that, for almost all t ∈ [0 , T ] , u ( t ) ∈ C α (Γ( t )) forevery α ∈ (0 , .In this text we will not focus in obtaining higher regularity for the solution of the problem with alogarithmic potential, but it is interesting to analyse what challenges would arise in trying to do so.To obtain L H -regularity (or higher) as we did in Theorem 4.13, we have to prove that F ′ ( u ) ∈ L H ,which requires some integrability for F ′′ ( u ) . Calculating F ′′ ( r ) = C (cid:18) − r −
11 + r + 1 (cid:19) , r ∈ ( − , shows that F ′′ ( u ) is integrable only if we can establish a uniform estimate k u k L ∞ (Γ( t )) ≤ − ξ, (5.1.13)for some ξ > . The bound in (5.1.13) can be interpreted as a phenomenon of separation from thepure phases ; not only are the pure phases ± never reached, but there always remains at least afixed amount of the other component. For the classical Cahn-Hilliard equation on a bounded domain Ω ⊂ R n , it is well known that the solution separates from the pure phases in dimensions n = 1 , ,and the problem is still open for n ≥ (see [Mir19, Chapter 4] and references therein). Since we areworking on -dimensional manifolds, we expect (5.1.13) to be true, and consequently for u to be moreregular. This is left for future work, and we now turn to the last case of a double obstacle potential. AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 27
Double obstacle potential.
In this section, the homogeneous free energy is defined, for r ∈ R ,as F o ( r ) = ( (1 − r ) , if | r | ≤ ∞ , if | r | > I ( r ) + 1 − r , where I ( r ) = ( if r ∈ [ − , ∞ otherwise . This suggests that we will be interested in solutions lying in the set K := { η ∈ L H : | η ( t ) | ≤ a.e. on Γ( t ) for a.e. t ∈ [0 , T ] } . The energy functional for this problem is the same as in the previous sections, but since it is nownon-differentiable the chemical potential w is defined by w + ∆ Γ u + u ∈ ∂I ( u ) , where ∂I ( · ) denotesthe subdifferential of I defined above. This means that, in the double obstacle case, we will be lookingfor a pair ( u, w ) , with u ∈ K , satisfying the system m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 , for all η ∈ L H ,a S ( u, η − u ) − m ( u, η − u ) ≥ m ( w, η − u ) , for all η ∈ K, (5.2.1)and an initial condition u (0) = u for some u ∈ H (Γ ) with | u | ≤ (this ensures the energy isdefined at the initial time). Again, we show that some more assumptions on the initial condition andthe surfaces need to be made. Recall the definition in (5.1.3). Proposition 5.16.
Let { Γ( t ) } t ∈ [0 ,T ] be an evolving surface in R and u ∈ H (Γ ) be such that | u | ≤ . (a) If m u ( t ) > for t in a subset of positive measure of [0 , T ] , then there is no pair ( u, w ) satisfying u ∈ K and the system (5.2.1) ; (b) If there exists t ∈ [0 , T ] such that m u ( t ) > , then there is no pair ( u, w ) satisfying u ∈ K andthe system (5.2.1) ; (c) If m u ( t ) = 1 on a subinterval I ⊂ (0 , T ] and m u (0) < , then there is no pair ( u, w ) satisfying u ∈ K and the system (5.2.1) .Proof. For case (a), choosing t ∈ [0 , T ] for which m u ( t ) > and u ( t ) ≤ , we would have, for almostall t , | Γ( t ) | < (cid:12)(cid:12)(cid:12)(cid:12)Z Γ u (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ( t ) u ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | Γ( t ) | , which is a contradiction. Part (b) follows immediately from (a) and the fact that m u is continuous.Under the conditions in case (c), there would be a subset of [0 , T ] with positive measure in which m ( t ) = 1 and the equations (CH o ) are satisfied. Note that, for such t , m ( t ) = 1 implies that u ( t ) = 1 ,and the first equation reads as Z Γ( t ) ( ∇ Γ · V ) η + Z Γ( t ) ∇ Γ w · ∇ Γ η = 0 , for any η ∈ L H . Testing with η ≡ , we obtain Z Γ( t ) ∇ Γ · V = 0 for almost every t ∈ I . Since the function on the left hand side is continuous, actually equality musthold everywhere in I , and then in I we have ddt | Γ( t ) | = Z Γ( t ) ∇ Γ · V = 0 , and hence | Γ( t ) | = | Γ | for t ∈ I . But this is a contradiction, since for almost all t ∈ I | Γ( t ) | = Z Γ( t ) Z Γ( t ) u ( t ) = Z Γ u < | Γ | . (cid:3) For the same reasons as before, we will restrict our analysis in this section to initial conditions givenin the same set I := (cid:8) ϕ ∈ H (Γ ) : | ϕ | ≤ a.e. on Γ , and m u < (cid:9) . Note that functions in I automatically have finite energy. Our method of proof follows the same linesas the logarithmic problem. Problem 5.17.
Given u ∈ I , the double obstacle Cahn-Hilliard system is the following problem:find a pair ( u, w ) satisfying:(a) u ∈ H H − ∩ L ∞ H ∩ K and w ∈ L H ;(b) for almost all t ∈ [0 , T ] , the system m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 , for all η ∈ L H ,a S ( u, η − u ) − m ( u, η − u ) ≥ m ( w, η − u ) , for all η ∈ K, (CH o )(c) the initial condition u (0) = u almost everywhere in Γ .The pair ( u, w ) is called a weak solution of (CH o ).5.2.1. Approximating problem.
For r ∈ R and δ ∈ (0 , , define F δ ( r ) = F obs δ ( r ) + 1 − r , where F obs δ ( r ) = δ (cid:0) r − (1 + δ ) (cid:1) + δ , for r ≥ δ δ ( r − , for < r < δ , for | r | ≤ − δ ( r + 1) , for − − δ < r < − δ (cid:0) r + (1 + δ ) (cid:1) + δ , for r ≤ − − δ . We will use this function to approximate the double obstacle potential defined above. We focus firston the following problem. Let us reuse the notation of the previous section and write ϕ δ ( r ) := (cid:16) F obs δ (cid:17) ′ ( r ) , r ∈ R . Problem 5.18.
Given u ∈ I , find a pair ( u δ , w δ ) ∈ L ∞ H × L H , with ∂ • u δ ∈ L H − , satisfying, forall η ∈ L H and almost every t ∈ [0 , T ] , m ∗ ( ∂ • u δ , η ) + g ( u δ , η ) + a N ( u, η ) + a S ( w δ , η ) = 0 ,a S ( u δ , η ) + m ( ϕ δ ( u δ ) , η ) − m ( u δ , η ) − m ( w δ , η ) = 0 , (CH δ o )and the initial condition u δ (0) = u almost everywhere in Γ . Remark 5.19.
Note that in this case we automatically obtain that the energy of the initial condition u ∈ H (Γ ) is bounded.This problem fits the framework of the general smooth potential we studied in the first section. Infact, it is easy to check that, for sufficiently small δ > , F obs δ satisfies the assumptions (A1)-(A2) inSection 4, and thus Theorem 4.12 implies that the problem (CH δ o ) is well-posed. We again observethat Remark 4.6 holds for the Galerkin approximation of the problem above. As before, the a prioribounds are not independent of δ , and thus we must obtain obtain new estimates before passing to thelimit.The Galerkin approximation for (CH δ o ) is the following problem. Problem 5.20.
Given δ > and u ∈ I , find ( u Mδ , w Mδ ) ∈ L V M with ∂ • u Mδ ∈ L V M satisfying, for all η ∈ L V M and t ∈ [0 , T ] , m ∗ ( ∂ • u Mδ , η ) + g ( u Mδ , η ) + a N ( u Mδ , δ ) + a S ( w Mδ , η ) = 0 ,a S ( u Mδ , η ) + m ( ϕ δ ( u Mδ ) , η ) − m ( u Mδ , η ) − m ( w Mδ , η ) = 0 , (CH M,δ o )and u Mδ (0) = P M u almost everywhere in Γ .The a priori estimates for the double obstacle potential are analogous to those in the previous section.As in Lemma 5.6, we start by obtaining a uniform bound for the initial energy. We reuse the notation E CH δ [ u ] = Z Γ( t ) |∇ Γ u | + F δ ( u ) . AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 29
Lemma 5.21 (A priori estimates for (CH
M,δ log )) . Let ( u Mδ , w Mδ ) be the unique solution of (CH M,δ o ) .There exists a constant C > , independent of both M and δ , such that E CH δ [0; u Mδ (0)] ≤ C .Proof. The proof follows as in Lemma 5.6. Observe that in this case F obs δ ( u ) = F obs ( u ) = 0 . (cid:3) We now obtain the a priori estimates for (CH δ o ). Proposition 5.22 (A priori estimates for (CH δ o )) . Let ( u δ , w δ ) be the unique solution of (CH δ o ) . (a) There exists a constant
C > , independent of δ , such that sup t ∈ [0 ,T ] E CH δ [ t ; u δ ( t )] + Z T k∇ Γ( t ) w δ ( t ) k L (Γ( t )) ≤ C ; (b) u δ and w δ are uniformly bounded in L ∞ H and L H , respectively; (c) ∂ • u δ is uniformly bounded in L H − .Proof. The proof of this result is exactly the same as that of Proposition 5.7. Observe that, also forthe approximation of the double obstacle potential, we have for any r ∈ R F obs δ ( r ) ≥ , rϕ δ ( r ) ≥ , | ϕ ′ δ ( r ) | ≤ rϕ ′ δ ( r ) + 1 , and hence the whole proof carries over unchanged to this case. (cid:3) Passage to the limit.
As before, we have u ∈ L H with ∂ • u ∈ L H − and w ∈ L H , such that, as δ → (up to a subsequence) u δ ⇀ u weakly in L H , ∂ • u δ ⇀ ∂ • u weakly in L H − , w δ ⇀ w weakly in L H . (5.2.2)We can use the Aubin-Lions Lemma in Proposition 2.4(c) to further obtain u δ → u strongly in L L ,and so also up to a subsequence (non-relabeled) we have, for almost every t ∈ [0 , T ] , u δ ( t ) → u ( t ) almost everywhere in Γ( t ) .Before passing to the limit in the problem we need to ensure u ∈ K . First, let us introduce somenotation: for δ > , define β δ ( r ) := δϕ δ ( r ) , r ∈ R , and β ( r ) := lim δ → β δ ( r ) = r − , for r ≥ , for | r | ≤ r + 1 , for r ≤ − . (5.2.3)Observe that β is Lipschitz continuous, and that we have, for any r ∈ R , | β ( r ) − β δ ( r ) | ≤ δ , ≤ β ′ δ ( r ) ≤ . (5.2.4) Lemma 5.23.
With the notation above, u ∈ K .Proof. Test the second equation of (CH δ o ) with η ( t ) = β δ ( u δ ( t )) ∈ H (Γ( t )) to obtain, for a.e. t ∈ [0 , T ] , a S ( u δ , β δ ( u δ )) + 1 δ k β δ ( u δ ) k L ≤ C (cid:0) k u δ k L + k w δ k L (cid:1) + 12 δ k β δ ( u δ ) k L , and hence from the uniform bounds we obtain C , C > such that a S ( u δ , β δ ( u δ )) + 12 δ k β δ ( u δ ) k L ≤ C + C k w δ k L . But note that, since β ′ δ ≥ , a S ( u δ , β δ ( u δ )) ≥ , and therefore we obtain δ k β δ ( u δ ) k L ≤ C + C k w δ k L , from where Z T k β δ ( u δ ) k L ≤ C δ + C δ Z T k w δ k L dt ≤ C δ, for some constant C > . Thus β δ ( u δ ) → in L L as δ → .Now let η ∈ L L . We have Z T m ( β ( u ) , η ) ≤ Z T ( k β ( u ) − β ( u δ ) k L + k β ( u δ ) − β δ ( u δ ) k L + k β δ ( u δ ) k L ) k η k L and using (5.2.4) it follows that the right hand side converges to as δ → . Thus β ( u ( t )) = 0 almosteverywhere in Γ( t ) for almost all t ∈ [0 , T ] , which means that u ∈ K , as desired. (cid:3) Remark 5.24.
In contrast with the logarithmic potential problem, in which the solutions do not touchthe pure phases ± (see Lemma 5.8), for the double obstacle case solutions are expected to attain thevalues ± on sets of positive measure.We finally turn to the well-posedness of (CH o ). Lemma 5.25.
The pair ( u, w ) satisfies, for all η ∈ L H and a.e. t ∈ [0 , T ] , m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 , and u (0) = u almost everywhere in Γ .Proof. The proof of this result is identical to that of the first part of Proposition 4.9, by replacing M → ∞ with δ → , so we omit it. The proof for the initial condition uses only the first equation ofthe system, and hence it follows also as in Proposition 4.9. (cid:3) Lemma 5.26.
The pair ( u, w ) satisfies the variational inequality.Proof. Let η ∈ K and ξ ∈ C ([0 , T ]) satisfy ξ ≥ . Testing the second equation with η − u δ , integratingover [0 , T ] and noting that β δ ( η ) = 0 we have, for almost every t , Z T ξ a S ( u δ , η − u δ ) − Z T ξ m ( w δ + u δ , η − u δ ) = − δ Z T ξ m ( β δ ( u δ ) , η − u δ )= 1 δ Z T ξ m ( β δ ( η ) − β δ ( u δ ) , η − u δ ) ≥ , by monotonicity of β δ . To let δ → , we simply observe that, by the convergences in (5.2.2), Z T ξ m ( w δ + u δ , η − u δ ) → Z T ξ m ( w + u, η − u ) . Also, the weak convergence ∇ Γ u δ ( t ) ⇀ ∇ Γ u ( t ) gives k∇ Γ u ( t ) k L (Γ( t )) ≤ lim inf δ → k∇ Γ u δ ( t ) k L (Γ( t )) , for almost every t ∈ [0 , T ] , and hence lim sup δ → Z T ξ a S ( u δ , η − u δ ) = Z T ξa S ( u, η ) − lim inf δ → Z T ξ k∇ Γ u δ k L ≤ Z T a S ( u, η − u ) . Therefore, Z T ξa S ( u, η − u ) − Z T ξm ( w + u, η − u ) ≥ lim sup δ → Z T ξ ( a S ( u δ , η − u δ ) − m ( w δ + u δ , η − u δ )) ≥ , concluding the proof. Since ξ ≥ is arbitrary, it follows that a S ( u, η − u ) − m ( u, η − u ) ≥ m ( w, η − u ) , as desired. (cid:3) Before proving uniqueness, we require the following auxiliary result. For the meaning of inequalitiesin the H sense, see for instance [KS80, Section II.5]. Lemma 5.27.
Let ( u, w ) be a solution pair for (CH o ) and consider the open set U ( t ) = { x ∈ Γ( t ) : | u ( t, x ) | < in the sense of H } . Then U ( t ) is non-empty for almost all t ∈ [0 , T ] .Proof. Since u ∈ L ∞ H , we have u ( t ) ∈ H (Γ( t )) for almost all t ∈ [0 , T ] , and so the set U ( t ) iswell defined for almost all t ∈ [0 , T ] . Fix one such t , and suppose for the sake of contradiction that U ( t ) = ∅ . Then | u ( t, x ) | ≥ in the sense of H for all x ∈ Γ( t ) , which implies that | u ( t, x ) | ≥ almosteverywhere in Γ( t ) . Since u ∈ K , also | u ( t, x ) | ≤ almost everywhere in Γ( t ) , and hence | u ( t ) | = 1 almost everywhere. As a consequence, ∇ Γ u = 0 almost everywhere (see [GT98, Section 7.4]). AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 31
We now have three cases to consider. In the cases where u = 1 a.e. or u = − a.e., then we have,respectively, m u ( t ) = 1 | Γ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)Z Γ u (cid:12)(cid:12)(cid:12)(cid:12) = 1 | Γ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ( t ) u ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 which contradicts u ∈ I . In the case when the sets U ( t ) = { x ∈ Γ( t ) : u ( t, x ) = 1 } , U ( t ) = { x ∈ Γ( t ) : u ( t, x ) = − } both have positive measure, then we have, using the generalized Poincaré inequality with both sets U and U (see e.g. [Leo09, Theorem 12.23]) and the fact that ∇ Γ u = 0 a.e., Z Γ( t ) | u − ( u ) U | ≤ and Z Γ( t ) | u − ( u ) U | ≤ , from where u ( t ) ≡ ( u ) U > and u ( t ) ≡ ( u ) U < , which is also a contradiction. Hence U ( t ) cannotbe empty, proving the result. (cid:3) Lemma 5.28.
The solution pair ( u, w ) is unique.Proof. Suppose ( u , w ) and ( u , w ) are two solution pairs and let us denote ξ u = u − u , ξ w = w − w .Exactly as in the proof of Proposition 4.11, we obtain m ∗ ( ∂ • ξ u , η ) + g ( ξ u , η ) + a N ( ξ u , η ) + a S ( ξ w , η ) = 0 , ∀ η ∈ L H , (5.2.5)from where we deduce ddt k ξ u k − + a N ( ξ u , G ξ u ) + m ( ξ w , ξ u ) = m ( ξ u , ∂ • G ξ u ) . (5.2.6)Now, using the variational inequalities for both pairs we directly get k∇ Γ ξ u k L ≤ m ( ξ w , ξ u ) + k ξ u k L , (5.2.7)and combining (5.2.6) and (5.2.7) yields, as in Proposition 4.11, ddt k ξ u k − + 1 C k∇ Γ ξ u k L ≤ C k ξ u k − . An application of Gronwall’s inequality gives uniqueness for u .We now prove uniqueness for w . Note that from (5.2.5) we conclude that that w is unique up toa constant. Since u ∈ L ∞ H , for almost every t ∈ [0 , T ] we have u ( t ) ∈ H (Γ( t )) . Fix one such t anddefine U ( t ) = { x ∈ Γ( t ) : | u ( t, x ) | < in the sense of H } . We proved in Lemma 5.27 that U ( t ) is a non-empty open set. Choose ϕ ∈ C ∞ ( U ( t )) and τ > sufficiently small so that η ± := u ± τ ϕ ∈ K . Testing the second equation with both η + and η − gives,for almost all t , the equalities a S ( u, ϕ ) = m ( u + w , ϕ ) and a S ( u, ϕ ) = m ( u + w , ϕ ) , and subtracting these equations gives m ( ξ w , ϕ ) = 0 . But ξ w is constant and ϕ is arbitrary, so it mustbe ξ w = 0 , i.e. w = w . (cid:3) We collect the results of this section in the next theorem.
Theorem 5.29.
Let u ∈ I . There exists a unique pair ( u, w ) such that u ∈ H H − ∩ L ∞ H ∩ K and w ∈ L H satisfying, for almost every t ∈ [0 , T ] , m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 , ∀ η ∈ L H ,a S ( u, η − u ) − m ( u, η − u ) ≥ m ( w, η − u ) , ∀ η ∈ K, and u (0) = u almost everywhere in Γ . The solution u satisfies the additional regularity u ∈ C L ∩ L ∞ L p , for all p ∈ [1 , + ∞ ) . Furthermore, if u , v ∈ I satisfy ( u ) Γ = ( v ) Γ , and u, v are the solutions of the system with u (0) = u , v (0) = v , then there exist constants C, K > independent of t such that, for almost all t ∈ [0 , T ] , k u ( t ) − v ( t ) k − ≤ Ce Kt k u − v k − . Extra regularity.
As in the previous sections, we conclude by establishing extra regularity forthe solution u . Theorem 5.30.
Let ( u, w ) be the solution pair given by Theorem 5.29. Then we have the regularity u ∈ L H .Proof. Let us consider the solution ( u Mδ , w Mδ ) of the Galerkin approximation (CH M,δ o ). The secondequation can be rewritten as a S ( u Mδ , η ) + 1 δ m ( β δ ( u Mδ ) , η ) − m ( u Mδ , η ) − m ( w Mδ , η ) = 0 , ∀ η ∈ L V M , where β δ is as in (5.2.3). Testing the above with η = − ∆ Γ u Mδ leads to k ∆ Γ u Mδ k L + 1 δ m (cid:0) β ′ δ ( u Mδ ) ∇ Γ u Mδ , ∇ Γ u Mδ ) (cid:1) = a S ( w Mδ , u Mδ ) + k∇ Γ u Mδ k L ≤ C k w Mδ k H + C k u Mδ k H and since β ′ δ ≥ this implies that k ∆ Γ u Mδ k L ≤ C k w Mδ k H + C k u Mδ k H . The uniform bounds for u Mδ in L ∞ H and for w Mδ in L H yield k ∆ Γ u Mδ k L L ≤ C, for some C > , and elliptic regularity theory then imply that u Mδ is uniformly bounded in L H . It follows that u ∈ L H ,as desired. (cid:3) Lemma 2.2d) also implies u ( t ) ∈ C α (Γ( t )) , for a.a. t ∈ [0 , T ] and all α ∈ (0 , .To deduce higher regularity properties of u from the second equation, we need regularity results forthe solution of the obstacle problem − ∆ Γ u − u ≥ w, − ≤ u ≤ . Even in the classical obstacle problem with a smooth right hand side, one obtains at most H -regularityfor the solution (which corresponds to our result in Theorem 5.30), and extra smoothness on the setwhere the solution does not coincide with the obstacle. As for the logarithmic potential, we leave suchanalysis for future work.5.3. Relating the two singular models.
Recalling the definitions for the logarithmic free energy F θ ( r ) = θ θ c ((1 + r ) log(1 + r ) + (1 − r ) log(1 − r )) + 1 − r , (5.3.1)and the double obstacle potential F o ( r ) = ( (1 − r ) , if | r | ≤ ∞ , if | r | > , (5.3.2)we can formally see (5.3.1) as the limit of (5.3.2) as the temperature θ → , and we refer to thedouble obstacle problem (CH o ) as the deep quench limit of the logarithmic problem (CH log ). It isthen natural to ask whether solutions ( u θ , w θ ) , where we now explicitly write the dependence on θ , to(CH log ) converge, as θ → , to the unique solution of (CH o ). Our aim in this short section is to provethat this is indeed the case.We start by noticing that, from the estimates in Section 5.1, it follows that u θ , w θ and ∂ • u θ areuniformly bounded in L ∞ H , L H and L H − , respectively, from where we obtain functions u, w ∈ L H with ∂ • u ∈ L H − and we have convergences, as θ → , u θ ∗ ⇀ u in L ∞ H , w θ ⇀ w in L H , ∂ • u θ ⇀ ∂ • u in L H − . By the Aubin-Lions Lemma in Proposition 2.4(c) we also obtain the strong convergence u θ → u in L L . Theorem 5.31.
The limit pair ( u, w ) is the unique solution of the double obstacle problem (CH o ) . AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 33
Proof.
It follows immediately that ( u, w ) satisfies, for all η ∈ L H , m ∗ ( ∂ • u, η ) + g ( u, η ) + a N ( u, η ) + a S ( w, η ) = 0 , and combining u θ → u pointwise a.e. with | u θ | < we also obtain u ∈ K .To show that ( u, w ) also satisfies the variational inequality, we consider η ∈ K , a small parameter α ∈ (0 , and define η α = (1 − α ) η , so that | η | ≤ − α < almost everywhere. Fix also ξ ∈ C ([0 , T ]) such that ξ ≥ . Testing the second equation with η = ξ (cid:0) η α − u θ (cid:1) and integrating over [0 , T ] leads to Z T ξ Z Γ( t ) ∇ Γ u θ · ∇ Γ ( η α − u θ ) − Z T ξ Z Γ( t ) u θ ( η α − u θ ) − Z T ξ Z Γ( t ) w θ ( η α − u θ )= Z T ξ Z Γ( t ) ϕ θ ( u θ )( u θ − η α )= Z T ξ Z Γ( t ) (cid:16) ϕ θ ( u θ ) − ϕ θ ( η α ) (cid:17) (cid:16) u θ − η α (cid:17) + Z T ξ Z Γ( t ) ϕ θ ( η α ) (cid:16) u θ − η α (cid:17) ≥ Z T ξ Z Γ( t ) ϕ θ ( η α )( u θ − η α ) , due to monotonicity of ϕ θ .We now pass to the limit θ → . For the integral on the right hand side, we use | ϕ θ ( η α ) | ≤ C α θ , sothat (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T ξ Z Γ( t ) ϕ θ ( η α )( u θ − η α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C α,ξ θ Z T Z Γ( t ) (cid:12)(cid:12)(cid:12) u θ − η α (cid:12)(cid:12)(cid:12) → as θ → , since u θ → u strongly in L L . As for the terms on the left hand side, we have directly from theconvergence results for u θ and w θ that Z T ξ Z Γ( t ) u θ ( η α − u θ ) → Z T ξ Z Γ( t ) u ( η α − u ) and Z T ξ Z Γ( t ) w θ ( η α − u θ ) → Z T ξ Z Γ( t ) w ( η α − u ) , and from weak lower semicontinuity of the norm we also have Z T ξ Z Γ( t ) ∇ Γ u · ∇ Γ ( η α − u ) ≥ lim inf θ → Z T ξ Z Γ( t ) ∇ Γ u θ · ∇ Γ ( η α − u θ ) . Combining the three displayed identities above leads to Z T ξ Z Γ( t ) ∇ Γ u · ∇ Γ ( η α − u ) − Z T ξ Z Γ( t ) u ( η α − u ) ≥ Z T ξ Z Γ( t ) w ( η α − u ) , and α → yields the variational inequality Z T ξ Z Γ( t ) ∇ Γ u · ∇ Γ ( η − u ) − Z T ξ Z Γ( t ) u ( η − u ) ≥ Z T ξ Z Γ( t ) w ( η − u ) , Since ξ ≥ is arbitrary we obtain, for almost all t ∈ [0 , T ] , the desired variational inequality Z Γ( t ) ∇ Γ u · ∇ Γ ( η − u ) − Z Γ( t ) u ( η − u ) ≥ Z Γ( t ) w ( η − u ) , finishing the proof. (cid:3) Global well-posedness for a related model
The aim of this section is to analyse an alternative derivation of the Cahn-Hilliard equation on anevolving surface presented in the recent article [ZTL +
19] (we refer the reader to Sections 4.1, 4.2, 5.2,and Appendix A in particular). For this alternative problem, we establish, using the same techniques asabove, global well-posedness results which we can now prove, even for the singular potentials, withoutany additional assumptions on the evolution of the surfaces or the initial data. For simplicity, andsince this does not change any of the results, we assume that there are no tangential velocities in themodel (i.e. we take V τ = V a = 0 ). Well-posedness.
As explained in the Introduction, we are interested in finding a pair ( c, w ) satisfying, for all η ∈ L H , h ρ∂ • c, η i H − , H + Z Γ( t ) ρ ∇ Γ w · ∇ Γ η = 0 , Z Γ( t ) ∇ Γ c · ∇ Γ η + Z Γ( t ) ρF ′ ( c ) η = Z Γ( t ) ρwη, (CH ρ )where the weight function ρ is determined by the differential equation ∂ • ρ + ρ ∇ Γ · V = 0 . (6.1.1)A detailed derivation of the model above can be found in [ZTL + ρ represents the total density of the system, and (6.1.1) is simplyconservation of total mass. The system of equations above is obtained by considering a conservationlaw now for the quantity Z Γ( t ) ρc, which is, as a consequence, preserved along the system (simply test the first equation with η = 1 ). Thechemical potential can be seen as the functional derivative of the following altered Cahn-Hilliard freeenergy functional E ρ CH ( c ) := Z Γ( t ) |∇ Γ c | ρF ( c ) . (6.1.2)Before we establish the a priori bounds for (CH ρ ), we note that (6.1.1) gives an explicit formula forthe weight function ρ . Indeed, by evaluating the equation along the flow, we obtain, for all p ∈ Γ and t ∈ [0 , T ] , ddt ρ ( t, Φ( t, p )) + ρ ( t, Φ( t, p )) ∇ Γ · V ( t, Φ( t, p )) = 0 , from where ρ ( t, Φ( t, p )) = exp (cid:26) − Z t ∇ Γ · V ( s, Φ( s, p )) (cid:27) . (6.1.3)In terms of elements x ∈ Γ( t ) at time t ∈ [0 , T ] , the above reads as ρ ( t, x ) = exp (cid:26) − Z t ∇ Γ · V ( s, x ) (cid:27) . In particular, there exists C ρ > such that < C ρ ≤ ρ ≤ C ρ and |∇ Γ ρ | ≤ C ρ . It is also worthwhile comparing (6.1.3) with (2.1.4), which shows that ρ ( t, Φ( t, p )) = ( J t ( p )) − , ∀ p ∈ Γ (6.1.4)and therefore the change of variables formulae in (2.1.3) imply that Z Γ( t ) c ( x ) = Z Γ ˜ c ( p ) ρ ( t, Φ( t, p )) , Z Γ ˜ c ( p ) = Z Γ( t ) c ( x ) ρ ( t, x ) . This makes the model particularly interesting for the cases of a logarithmic and a double obstaclepotential. Recall that we started both Sections 5.1, 5.2 by showing that a condition relating the initialdata and the areas of the surfaces was necessary for well-posedness of the systems, and this was aconsequence of the fact that the integral of the solution was preserved along the evolution. This isintuitively clear: preservation of the integral together with a decrease in the areas of the surfaceswould force an increase on the magnitude of the solutions, which is precluded by the condition | u | ≤ in either case. This simple argument no longer holds true for the problem (CH ρ ). Due to (6.1.4),preservation of the integral of ρc can be compatible with restrictions on | c | , since the term ρ can nowaccount for local stretching or compressing of the surfaces.We now show that indeed the presence of the weight term allows for global well-posedness results forthe system (CH ρ ) for the three different types of nonlinearities we considered in the previous sections. AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 35
A priori estimates.
In this section we obtain some a priori estimates for solutions of the problemabove. The calculations below are just formal, but can be made rigorous by arguing as in the previoussections (e.g rerasoning by approximation). So let us denote by ( c, w ) a solution pair to (CH ρ ). Bydifferentiating the energy (6.1.2) along this solution pair, we have ddt E ρ CH ( c ) = Z Γ( t ) ∇ Γ c · ∇ Γ ∂ • c + ρF ′ ( c ) ∂ • c d Γ + Z Γ( t ) B ( V ) |∇ Γ c | = Z Γ( t ) ρw∂ • c + Z Γ( t ) B ( V ) |∇ Γ c | = − Z Γ( t ) ρ |∇ Γ w | + Z Γ( t ) B ( V ) |∇ Γ c | , and thus ddt E ρ CH ( c ) + Z Γ( t ) ρ |∇ Γ w | = Z Γ( t ) B ( V ) |∇ Γ c | ≤ C V Z Γ( t ) |∇ Γ c | ≤ C E ρ CH ( c ) . Recalling that ρ is strictly positive, an application of Gronwall’s inequality immediately gives theenergy estimate ess sup t ∈ [0 ,T ] E ρ CH ( c ) + Z T Z Γ( t ) |∇ Γ w | ≤ E ρ CH ( c ) + C, (6.2.1)Asssuming E ρ CH ( c ) to be finite, this implies in particular ∇ Γ c ∈ L ∞ L and ∇ Γ w ∈ L L We make use of (6.2.1) to estimate the remaining quantities. Observe that, using the first equationwith η = ρ − , ddt Z Γ( t ) c = Z Γ( t ) ∂ • c + Z Γ( t ) c ∇ Γ · V = − Z Γ( t ) ρ ∇ Γ w · ∇ Γ (cid:18) ρ (cid:19) − Z Γ( t ) ∇ Γ c · V , so that Z Γ( t ) c = Z Γ c − Z t Z Γ( s ) ρ ∇ Γ w · ∇ Γ (cid:18) ρ (cid:19) − Z t Z Γ( t ) ∇ Γ c · V and thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ( t ) c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z Γ c (cid:12)(cid:12)(cid:12)(cid:12) + CT ≤ C ′ . Combining the above, (6.2.1), and Poincaré’s inequality then implies c ∈ L ∞ H . (6.2.2)Using the Sobolev embedding H (Γ( t )) ֒ → L p (Γ( t )) , for all t ∈ [0 , T ] and p ∈ [1 , + ∞ ) , this in particularimplies c ∈ L ∞ L p , for all p ∈ [1 , + ∞ ) . We can also easily estimate the material derivative of c by noticing that, for any η ∈ L H , h ∂ • c, η i L H − , L H = Z T Z Γ( t ) ∂ • c ( t ) η ( t )= − Z T Z Γ( t ) ρ ∇ Γ( t ) w ( t ) · ∇ Γ( t ) (cid:18) η ( t ) ρ ( t ) (cid:19) = − Z T Z Γ( t ) ∇ Γ w · ∇ Γ η + Z T Z Γ( t ) ηρ ∇ Γ w · ∇ Γ ρ ≤ C k∇ Γ η k L L + C k η k L L ≤ C k η k L H , which gives ∂ • c ∈ L H − . (6.2.3)Combining (6.2.2) and (6.2.3) also gives the extra regularity c ∈ C L . The estimate for w in L L is slightly more involved.In the case that the nonlinearity F satisfies Assumptions (A1)-(A2) as in Section 4, we obtain fromthe second equation with η = 1 /ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ( t ) w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ( t ) ∇ Γ c · ∇ Γ (cid:18) ρ (cid:19) + Z Γ( t ) F ′ ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C k c k qH ≤ C , which combined with Poincaré’s inequality and the energy estimate implies a bound for w in L H .Let us now focus in the case of a singular potential, where we assume ( c ) Γ ∈ ( − , . We assumethe nonlinearity satisfies, for any r , | F ′ ( r ) | ≤ rF ′ ( r ) + 1 . This is satisfied by the approximating potentials in Sections 5.1, 5.2 (and also the logarithmic potential(5.1.1)). We now note that if we can show that, for some
C > , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρcw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C, (6.2.4)then using the second equation tested with η = c leads to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρF ′ ( c ) c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρcw − Z T Z Γ( t ) |∇ Γ c | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρcw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) |∇ Γ c | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C, and from (6.2.4) it follows that Z T Z Γ( t ) | F ′ ( c ) | ≤ Z T Z Γ( t ) cF ′ ( c ) + 1 ≤ C ρ Z T Z Γ( t ) ρ ( cF ′ ( c ) + 1) ≤ C ρ Z T Z Γ( t ) ρcF ′ ( c ) + C ρ Z T Z Γ( t ) ρ ≤ C. In particular, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ∇ Γ c · ∇ Γ (cid:18) ρ (cid:19) + Z T Z Γ( t ) F ′ ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C. Together with Poincaré’s inequality and the energy estimate this implies the desired bound for w .So it remains to establish (6.2.4). Our strategy will be to pullback the integrals to the referencedomain Γ and estimate the involved quantities there. This is helpful since we now have no control onthe size of ( c ( t )) Γ( t ) = 1 | Γ( t ) | Z Γ( t ) c ( t ) AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 37 but rather on its pullback (˜ c ( t )) Γ = 1 | Γ | Z Γ ˜ c ( t ) = 1 | Γ | Z Γ( t ) c ( t ) ρ ( t ) = 1 | Γ | Z Γ c = ( c ) Γ ∈ ( − , . We then write Z T Z Γ( t ) ρcw = Z T Z Γ ˜ c ˜ w = Z T Z Γ ˜ w (cid:0) ˜ c − (˜ c ) Γ (cid:1) + Z T (˜ c ) Γ Z Γ ˜ w = Z T Z Γ (cid:0) ˜ w − ( ˜ w ) Γ (cid:1)(cid:0) ˜ c − (˜ c ) Γ (cid:1) + ( c ) Γ Z T Z Γ( t ) ρw so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρcw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ Γ ˜ w k L (0 ,T ; L (Γ )) k∇ Γ ˜ c k L (0 ,T ; L (Γ )) + | ( c ) Γ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ′ + | ( c ) Γ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . But we also have, testing the second equation with η = 1 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρF ′ ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρF ′ ( c ) c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρcw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and thus combining the identities above leads to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρcw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C + | ( c ) Γ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , from where (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ρcw (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C − | ( c ) Γ | , which is (6.2.4). Therefore the integral of w is uniformly bounded, and again Poincaré’s inequalitycombined with (6.2.1) implies w ∈ L H . The a priori bounds being established, we can then argue as in Sections 4, 5.1, 5.2 to obtain well-posedness results for (CH ρ ). We denote ( · ) ρ, Γ := ( ρ · ) Γ and k · k ρ, − := k ρ · k − . We then have the analogue of Theorem 4.12.
Theorem 6.1.
Let c ∈ H (Γ ) , F : R → R be a smooth potential satisfying (A1)-(A2), and let thedensity function ρ satisfy ∂ • ρ + ρ ∇ Γ · V = 0 . Then, there exists a unique pair ( c, w ) with c ∈ H H − ∩ L ∞ H and w ∈ L H satisfying the system (CH ρ ) , and the initial condition c (0) = c almost everywhere in Γ . The solution c satisfies the additional regularity c ∈ C L ∩ L H ∩ L ∞ L p , for all p ∈ [1 , + ∞ ) . Furthermore, if c , , c , ∈ H (Γ ) satisfy ( c , ) ρ, Γ = ( c , ) ρ, Γ , and c , c are the solutions of thesystem with c (0) = c , , c (0) = c , , then there exists a constant C > independent of t such that,for almost all t ∈ [0 , T ] , k c ( t ) − c ( t ) k ρ, − ≤ e Ct k c , − c , k ρ, − . The proof is essentially the same as that of Theorem 4.12 apart from the uniqueness result, whereone should now work with the weighted inverse Laplacian defined, for any f ∈ H − (Γ( t )) such that m ρ ∗ ( f,
1) = 0 , as the unique solution c = (∆ Γ ,ρ ) − f ∈ H (Γ( t )) to the problem a S ( c, η ) = m ρ ∗ ( f, η ) m ( c,
1) = 0 . Under further assumptions on both F and the evolution of the surfaces, we can also obtain extraregularity for c as in Theorem 4.13.Similarly, we have a well-posedness result for the logarithmic potential model as in Theorem 5.14. Theorem 6.2.
Let c ∈ H (Γ ) satisfy | c | ≤ , ( c ) Γ ∈ ( − , , and E ρ CH ( c ) < ∞ , F : [ − , → R be the logarithmic potential (5.1.1) , and let the density function ρ satisfy ∂ • ρ + ρ ∇ Γ · V = 0 . Then, there exists a unique pair ( c, w ) with u ∈ H H − ∩ L ∞ H and w ∈ L H satisfying, for almost all t ∈ [0 , T ] , | c ( t ) | < a.e. in Γ( t ) , the system (CH ρ ) , and the initial condition c (0) = c almost everywhere in Γ . The solution c satisfies the additional regularity c ∈ C L ∩ L H ∩ L ∞ L p , for all p ∈ [1 , + ∞ ) . Furthermore, if c , , c , ∈ H (Γ ) satisfy ( c , ) ρ, Γ = ( c , ) ρ, Γ , and c , c are the solutions of thesystem with c (0) = c , , c (0) = c , , then there exists a constant C > independent of t such that,for almost all t ∈ [0 , T ] , k c ( t ) − c ( t ) k ρ, − ≤ e Ct k c , − c , k ρ, − . Also in this case the proof carries over unchanged from that of Theorem 5.14, with the exception ofthe proof of uniqueness which should use, as explained before, the weighted inverse Laplacian (∆ Γ ,ρ ) − .Finally, we state the result for the double obstacle potential which is analogous to Theorem 5.29.Reusing the notation K := { η ∈ L H : | η ( t ) | ≤ a.e. on Γ( t ) for a.e. t ∈ [0 , T ] } , the second equation becomes a variational inequality due to non-differentiability of the potential, andwe consider the problem h ρ∂ • c, η i H − , H + Z Γ( t ) ρ ∇ Γ w · ∇ Γ η = 0 , ∀ η ∈ L H , Z Γ( t ) ∇ Γ c · ∇ Γ ( η − c ) − Z Γ( t ) c ( η − c ) ≥ Z Γ( t ) ρw ( η − c ) , ∀ η ∈ K. (6.2.5) Theorem 6.3.
Let c ∈ H (Γ ) satisfy | c | ≤ and ( c ) Γ ∈ ( − , , and let the density function ρ satisfy ∂ • ρ + ρ ∇ Γ · V = 0 . There exists a unique pair ( c, w ) with c ∈ H H − ∩ K ∩ L ∞ H and w ∈ L H satisfying (6.2.5) and c (0) = c almost everywhere in Γ . The solution c satisfies the additional regu-larity c ∈ C L ∩ L H ∩ L ∞ L p , for all p ∈ [1 , + ∞ ) . Furthermore, if c , , c , ∈ H (Γ ) satisfy ( c , ) ρ, Γ = ( c , ) ρ, Γ , and c , c are the solutions of thesystem with c (0) = c , , c (0) = c , , then there exists a constant C > independent of t such that,for almost all t ∈ [0 , T ] , k c ( t ) − c ( t ) k ρ, − ≤ e Ct k c , − c , k ρ, − . Again, most proofs carry over with minor adaptations.7.
Examples
In this section, we collect five simple examples to illustrate our main results.
AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 39
Example 7.1.
The simplest example to consider is the case V ≡ , which implies that Γ( t ) ≡ Γ ,for all t ∈ [0 , T ] , and our work provides a proof for well-posedness of the Cahn-Hilliard system onthe stationary surface Γ for initial data u ∈ H (Γ ) and satisfying, for the singular potentials, thecondition | ( u ) Γ | = m u (0) < . Example 7.2.
Consider a velocity field V satisfying ∇ Γ · V = 0 , implying | Γ( t ) | = | Γ | for all t ∈ [0 , T ] . This assumption arises in models for inextensible membranes. In this case, our workproves well-posedness for the Cahn-Hilliard system an evolving surface { Γ( t ) } t ∈ [0 ,T ] for initial data u ∈ H (Γ ) and satisfying, for the singular potentials, the condition | ( u ) Γ | = m u (0) < . Asfor the model in Section 6, in this case we obtain ∂ • ρ = 0 , and therefore (up to multiplication by aconstant) the two models we presented are the same. This is also the setting considered in [YQO19],and our work completes their results by providing a proof of well-posedness for the continuous model. Example 7.3.
Let the nonlinearity F be the usual quartic potential F ( r ) = ( r − r − r F ( r ) + F ( r ) , r ∈ R . (7.1)Then it is easy to check that assumptions (A1)-(A2) are satisfied with (e.g.) the parameters α = α = 1 / , α = 3 , α = 1 ,β = β = β = β = 0 , β = 1 / ,q = 4 , and therefore Theorem 4.12 implies that, given any initial data u ∈ H (Γ ) , there exists a unique weaksolution pair to the Cahn-Hilliard system with nonlinearity given by (7.1). This recovers in particularthe result in [ER15] regarding well-posedness of the continuous problem. Example 7.4.
Let
T > and suppose that the evolving surface { Γ( t ) } t ∈ [0 ,T ] is such that Γ = S (1) and, at some t ∈ [0 , T ] , Γ( t ) = S (1 / , where we denote by S ( r ) the sphere of radius r > in R . Ifwe take u ≡ / ∈ H (Γ ) , then m u ( t ) = 1 | Γ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)Z Γ u (cid:12)(cid:12)(cid:12)(cid:12) = 1 π π > , and so Proposition 5.1 (or Corollary 5.2) implies that there is no (global in time) solution to theCahn-Hilliard system with a logarithmic potential. Example 7.5.
For a given
T > , suppose { Γ( t ) } t ∈ [0 ,T ] evolves in such a way that | Γ( t ) | ≥ | Γ | for all t ∈ [0 , T ] . Then for any u ∈ H (Γ ) with finite initial energy and | ( u ) Γ | < , we have m u ( t ) = 1 | Γ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)Z Γ u (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Γ | (cid:12)(cid:12)(cid:12)(cid:12)Z Γ u (cid:12)(cid:12)(cid:12)(cid:12) < , so that the Cahn-Hilliard systems with both the logarithmic and the double obstacle potentials withinitial condition u are well-posed.For some more concrete examples and illustrations we refer to [YQO19, Section 4], where the authorspresent some numerical simulations and plots of the evolution of solutions to the Cahn-Hilliard system.8. Concluding remarks
Our aim in this work has been to present a rigorous derivation of Cahn-Hilliard equations on anevolving surface, to establish well-posedness for the typical smooth, logarithmic and double obstaclepotentials and to analyse the effect of the evolving nature of the domains in the solutions. Thisevolution for the surfaces is assumed to be given a priori .We found that the system with regular potentials is well-posed for any choice of initial data u ∈ H (Γ ) , and this has been proved by an evolving space Galerkin method. The class of nonlinearitiesconsidered includes the usual quartic potentials considered in the literature. As we have mentionedalready, even though these do not produce physically meaningful solutions (as u might leave theinterval [ − , even if | u | ≤ ), the quartic potentials are considered good approximations for themodel describing a shallow quench of the system.The case of the singular potentials is more interesting, as it turns out that well-posedness of thesystems relies on an interplay between the evolution of the surfaces, the initial data and the Cahn-Hilliard dynamics which force preservation of the integral of the solution. Here we identified a set of admissible initial conditions for which the system is well-posed, which essentially forces the mean valueof the solution to remain in the interval ( − , . The proof for the logarithmic potential is achievedby regularisation of the nonlinearity close to the singularities, and for the double obstacle potential byuse of a penalty method. We also showed that the double obstacle system can be obtained as the deepquench limit of the logarithmic problem.In Section 6, we considered an alternative derivation for the model similar to what is proposed inrecent work [OXY20, YQO19, ZTL +
19] which allows for general well-posedness statements in all caseswithout restricting the set of initial conditions. The reason for this is that while our first model preservesthe integral of the order parameter u , which in the singular cases lies in ( − , , this alternative systemconserves the integral of ρc , where now c ∈ ( − , and the weight function ρ counterbalances thechanges in the domains.We finish with some comments on our results and some questions that remain to be addressed.All of the work in this article was done for surfaces, i.e. in dimension , as this is the most relevantdimension for applications. No changes are needed for curves in R and in higher dimensions onlyminor adaptations need to be made. For the regular potentials, some conditions are needed on thepolynomial growth order; q needs to be such that | u | q is integrable. For the singular potentials, sincethe regularisations are quadratic, no extra conditions are needed. Of course, the regularity results forall cases would also change, and this would also restrict the exponents p ∈ [1 , + ∞ ) for which u ∈ L ∞ L p .Throughout Sections 5.1 and 5.2 we have always assumed that the initial condition u satisfies ( u ) Γ ∈ ( − , , which excludes the cases u ≡ ± . In a stationary domain, since preservation ofintegral implies u ≡ for all times, there is no solution for the logarithmic problem, and the solutionremains constant for the double obstacle problem. This is consistent with the physical interpretationof the model, and it would be interesting to establish the same result in our framework. This isanother manifestation of the fact that the surfaces evolve with time – preservation of the integral isnot equivalent to preservation of the mean value.Our result does not cover either the case in which m u never hits the value on a set of zeromeasure. As a concrete example, suppose there exists t ∗ ∈ (0 , T ) such that m u ( t ∗ ) = 1 but m u < on [0 , t ∗ ) ∪ ( t ∗ , T ] . We have well-posedness on [0 , t ∗ ] , and it is natural to ask whether the solution canbe extended to [0 , T ] , effectively resolving a ’singularity’ at t ∗ . This is an interesting question, whichwe aim to address in the future.We conclude by mentioning some possible future research directions.It would be interesting to consider more general initial data, say u ∈ L (Γ ) ; for this case we expectwell-posedness under the same type of assumptions and to observe instantaneous smoothing of thesolutions. We have also to address higher regularity results for the logarithmic potential, which as weexplained requires proving separation of the solutions from the pure phases, and for the double obstaclecase which leads to the study of regularity for variational inequalities. Another natural question is thatof long-term dynamics of the system. For instance, if Γ( t ) are defined for all t ∈ [0 , + ∞ ) and convergeto some surface Γ ∞ as t → ∞ , it is natural to try to identify the possible limits of the solution u ( t ) .In this article we have also only considered a constant mobility for the system, and it is an interestingand challenging problem to formulate and analyse the system with the phase-dependent mobility M ( u ) = 1 − u , which leads to the degenerate Cahn-Hilliard equation. Even in the classical setting,there are still many open questions on this problem, as well as for general degenerate fourth orderparabolic PDEs. Another possible avenue is to drop the assumption that the evolution of the domainsis given a priori , and to couple the surface motion with the Cahn-Hilliard system on the surfaces, forinstance by considering ( L , H − ) -gradient flow for the Cahn-Hilliard energy E CH = E CH (Γ , u ) . Weleave all of these for future work. Acknowledgements.
The work of CME was partially supported by the Royal Society via a WolfsonResearch Merit Award. The research of DC was funded by the Engineering and Physical SciencesResearch Council grant EP/H023364/1 under the MASDOC Centre for Doctoral Training at the Uni-versity of Warwick.
Appendix A. A generalised Gronwall Lemma
In this appendix, we present a generalised Gronwall lemma.
Lemma A.1 (Generalised Gronwall Lemma) . Fix T ∈ (0 , ∞ ) and t ∗ ∈ (0 , T ) . For M ∈ N , let α M : [0 , t ∗ ] → [0 , + ∞ ) be a sequence of nonnegative differentiable functions. Suppose that there exists AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 41 ˜ M ∈ N such that, for any t ∈ [0 , t ∗ ] and M ≥ ˜ M , α ′ M ( t ) ≤ C (cid:16) C + α M ( t ) + ε α q +1 M ( t ) (cid:17) ,α M (0) = α ≥ , where C > , C ≥ , ε > , and q ∈ N are independent of M and t ∗ . If ε is small enough so that ε e T Cq ( α + C ) q < , (A.1) it holds that, for M ≥ ˜ M , α M ( t ) ≤ ( α + C ) e T C q p − ε e T Cq ( α + C ) q − C . Proof.
Suppose first that C = 0 , then observe that q ddt log (cid:18) α M ( t ) q ε α M ( t ) q (cid:19) = α ′ M ( t ) α M ( t ) + ε α M ( t ) q +1 ≤ C, and we can integrate the above over [0 , t ] , for any t ∈ [0 , t ∗ ] , to obtain log (cid:18) α M ( t ) q ε α M ( t ) q (cid:19) ≤ log (cid:18) α q ε α q (cid:19) + t ∗ Cq ≤ log (cid:18) α q ε α q (cid:19) + T Cq
Now this implies that α M ( t ) q ε α M ( t ) q ≤ e T Cq α q ε α q ≤ e T Cq α q , which we can solve for α M ( t ) to obtain (cid:0) − ε e T Cq α q (cid:1) α M ( t ) q ≤ α q e T Cq . Using (A.1), it follows that α M ( t ) ≤ α e T C q p − ε e T Cq α q ≤ α e T C q p − ε e T Cq α q , for M ≥ f M , as desired.If C = 0 , then define e α M ( t ) := α M ( t ) + C , so that e α ′ M ( t ) = α ′ M ( t ) ≤ C (cid:16)e α M ( t ) + ε α q +1 M (cid:17) ≤ C (cid:16)e α M ( t ) + ε e α q +1 M (cid:17) , and the previous case gives, for some f M ∈ N , e α M ( t ) ≤ e α e T C q p − ε e T Cq e α q = ( α + C ) e T C q p − ε e T Cq ( α + C ) q , ∀ M ≥ ˜ M, which translates to α M ( t ) ≤ ( α + C ) e T C q p − ε e T Cq ( α + C ) q − C , (cid:3) Appendix B. Dominated convergence theorems on evolving surfaces
In this Appendix we return to the setting of an evolving surface { Γ( t ) } t ∈ [0 ,T ] under the same as-sumptions as in Section 2. We start by establishing an evolving space version of Lebesgue’s classicaldominated convergence theorem. Theorem B.1 (Dominated Convergence Theorem) . Suppose ( f M ) M ⊂ L L be a sequence satisfying (i) for almost all t ∈ [0 , T ] , we have f M ( t, x ) → f ( t, x ) as M → ∞ for a.e. x ∈ Γ( t ) ; (ii) there exists g ∈ L L such that, for a.a. t ∈ [0 , T ] and all M ∈ N , | f M ( t, x ) | ≤ g ( t, x ) for a.e. x ∈ Γ( t ) .Then f ∈ L L and k f M k L L → k f k L L as M → ∞ .Proof. The proof follows from consecutive applications of the classical dominated convergence theorem(first in space, then in time).
For the first step, by fixing t in the set of positive measure for which f M ( t ) → f ( t ) a.e. in Γ( t ) , | f M ( t ) | ≤ | g ( t ) | and g ( t ) ∈ L (Γ( t )) we can apply the DCT to obtain f ( t ) ∈ L (Γ( t )) and k f M ( t ) k L (Γ( t )) → k f ( t ) k L (Γ( t )) , for almost all t ∈ [0 , T ] .For the second step, define ˜ f M ( t ) = k f M ( t ) k L (Γ( t )) and ˜ f ( t ) = k f ( t ) k L (Γ( t )) . Then the first stepshows that ˜ f M ( t ) → ˜ f ( t ) for a.a. t ∈ [0 , T ] , and by assumption | ˜ f M ( t ) | ≤ k g ( t ) k L (Γ( t )) ∈ L (0 , T ) , sothat again an application of the DCT implies that ˜ f ∈ L (0 , T ) and k ˜ f M k L (0 ,T ) → k ˜ f k L (0 ,T ) . Butthis is exactly the statement that f ∈ L L with f M → f in L L , concluding the proof. (cid:3) We now prove a version of Lebesgue’s dominated convergence theorem for weak convergence. Thestatement and proof are a direct adaptation of [Rob01, Lemma 8.3].
Theorem B.2 (Generalised Dominated Convergence Theorem) . Suppose that ( g M ) M ⊂ L L is suchthat there exists C > such that, for all M ∈ N , k g M k L L ≤ C. (B.1) If g ∈ L L and g M → g almost everywhere, then g M ⇀ g in L L .Proof. Define, for each M ∈ M and t ∈ [0 , T ] , the family of sets Γ M ( t ) := { x ∈ Γ( t ) : | g j ( t, x ) − g ( t, x ) | ≤ , for all j ≥ M } . It is clear that Γ M ( t ) ⊂ Γ M +1 ( t ) and since g M ( t ) → g ( t ) almost everywhere we have | Γ M ( t ) | ր | Γ( t ) | as M → ∞ . Now consider the sets U M ( t ) = (cid:8) ϕ ∈ L L : supp ϕ ( t ) ⊂ Γ M ( t ) (cid:9) and U ( t ) = ∞ [ M =1 U M ( t ) , and define also L U M := { ϕ ∈ L L : ϕ ( t ) ∈ U M ( t ) } and L U = [ M ∈ N L U M . We start by proving that L U is dense in L L . Let ϕ ∈ L L and define a sequence ϕ M ( t ) = χ Γ M ( t ) ϕ ( t ) ,where χ Γ M ( t ) denotes the characteristic function of V Γ M ( t ) , so that ϕ M ∈ L U M . For a.a. t ∈ [0 , T ] , thedefinition of Γ M ( t ) implies that ϕ M ( t ) → ϕ ( t ) a.e. in Γ( t ) , and we also have | ϕ M ( t ) | ≤ | ϕ ( t ) | and ϕ ∈ L L , and therefore by the Dominated Convergence Theorem B.1 it follows that ϕ M → ϕ in L (Γ) .Now let ϕ ∈ L U . We claim that as M → ∞ we have Z T Z Γ( t ) ϕ ( g M − g ) → . (B.2)This is again an application of the DCT; there exists M ∈ N such that ϕ ∈ L U M , and thus | ϕ ( g M − g ) | ≤ | ϕ | ∈ L L , for M ≥ M , and of course ϕ ( g M − g ) → a.e., so (B.2) follows again from Theorem B.1.To establish the weak convergence g M ⇀ g we combine (B.2) with the density of L U in L L . Let ε > and η ∈ L L be arbitrary, and pick ϕ ∈ L U such that k η − ϕ k L L ≤ ε C , where C is the constant in (B.1). By (B.2) we can also take M ∈ N large enough so that Z T Z Γ( t ) ϕ ( g M − g ) ≤ ε , for M ≥ M . Finally, we also apply Fatou’s lemma combined with (B.1) to obtain k g k L L ≤ lim inf M →∞ k g M k L L ≤ C, and therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) η ( g M − g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ( η − ϕ )( g M − g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ( t ) ϕ ( g M − g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 43 ≤ k η − ϕ k L L (cid:16) k g M k L L + k g k L L (cid:17) + ε ≤ ε C C + ε ε, as desired. (cid:3) Remark B.3.
The statements above can be extended to the more general Banach spaces L pL q , with p, q ∈ [1 , + ∞ ] , with similar proofs, but we only need the case p = q = 2 . Although we have notintroduced these spaces in Section 2, they are defined in exactly the same way with the naturalchanges (see [AES15a]). The assumptions that the bounds in both results hold for all M ∈ N can ofcourse be relaxed; it suffices that they hold only for large enough M . Appendix C. The inverse Laplacian
Fix t ∈ [0 , T ] . For each z ∈ H − (Γ( t )) with h z, i H − , H = 0 we define the inverse Laplacian of z , which we denote by G ( t ) z , the unique solution of the problem Z Γ( t ) ∇ Γ G ( t ) z · ∇ Γ η = h z, η i H − , H ∀ η ∈ H (Γ( t )) , Z Γ( t ) G ( t ) z = 0 . This defines a map G ( t ) : H − (Γ( t )) → H (Γ( t )) , which we use to define a norm in H − (Γ( t )) by k z k − ,t := k∇ Γ( t ) G ( t ) z k L (Γ( t )) = m ∗ ( t ; z, G ( t ) z ) . More generally, setting L H , := ( u ∈ L H : Z T Z Γ( t ) u ( t ) = 0 ) , L H − , := n z ∈ L H − : h z, i L H − , L H = 0 o the above allows us to define a map G : L H − , → L H , , ( G z )( t ) := G ( t ) z ( t ) . The following result will be useful to prove uniqueness of solutions.
Lemma C.1. If z ∈ H H − , then G z ∈ H H , i.e. G z ∈ H H − with ∂ • G z ∈ L H . For a proof, see [ER15, Lemma 4.3].
References [AES15a] A. Alphonse, C. M. Elliott, and B. Stinner. An abstract framework for parabolic PDEs on evolving spaces.
Portugaliae Mathematica , 71(1):1–46, 2015.[AES15b] A. Alphonse, C. M. Elliott, and B. Stinner. On some linear parabolic PDEs on moving hypersurfaces.
Inter-faces and Free Boundaries , 17:157–187, 2015.[AET17] A. Alphonse, C. M. Elliott, and J. Terra. A coupled ligand-receptor bulk-surface system on a moving domain:Well posedness, regularity, and convergence to equilibrium.
SIAM Journal on Mathematical Analysis , 50:1544–1592, 2017.[Bai65] C. Baiocchi. Regolarità e unicità della soluzione di una equazione differenziale astratta.
Rendiconti del Sem-inario Matematico delle Università di Padova , 35(2):380–417, 1965.[BE91] J. F. Blowey and C. M. Elliott. The Cahn–Hilliard gradient theory for phase separation with non–smoothfree energy Part I: Mathematical Analysis.
European J. Applied Mathematics , 2:233–280, 1991.[BE92] J. F. Blowey and C. M. Elliott. The Cahn–Hilliard gradient theory for phase separation with non–smoothfree energy Part II: Numerical Analysis.
European J. Applied Mathematics , 3:147–179, 1992.[BEM11] R. Barreira, C. M. Elliott, and A. Madzvamuse. The surface finite element method for pattern formation onevolving biological surfaces.
Journal of Mathematical Biology , 63:1095–1119, 2011.[Cah61] J. W. Cahn. On spinodal decomposition.
Acta Metall. Mater. , 9:795–801, 1961.[CENC96] J. W. Cahn, C. M. Elliott, and A. Novick-Cohen. The Cahn–Hilliard equation with a concentration dependentmobility: motion by minus the laplacian of the mean curvature.
European J. Applied Mathematics , 7:287–301,1996.[CH58] J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system I. Interfacial free energy.
J. Chem. Phys. ,28:258–267, 1958. [CMZ11] L. Cherfile, A. Miranville, and S. Zelik. The Cahn-Hilliard Equation with Logarithmic Potentials.
MilanJournal of Mathematics , 79(2):561–596, Dec 2011.[DD95] A. Debussche and L. Dettori. On the Cahn-Hilliard equation with a logarithmic free energy.
Nonlinear Anal.- Theor. , 24(10):1491–1514, 1995.[DD14] S. Dai and Q. Du. Coarsening mechanism for systems governed by the Cahn-Hilliard equation with degeneratediffusion mobility.
Multiscal Model. Simul. , 12(4):1870–1889, 2014.[DD16] S. Dai and Q. Du. Weak solutions for the Cahn-Hilliard equation with degenerate mobility.
Arch. RationalMech. Anal , 219:1161–1184, 2016.[DDE05] K. Deckelnick, G. Dziuk, and C. M. Elliott. Computation of geometric partial differential equations and meancurvature flow.
Acta Numerica , 14:139–232, 2005.[DE07] G. Dziuk and C. M. Elliott. Finite elements on evolving surfaces.
IMA Journal Numerical Analysis , 25:385–407, 2007.[DE13] G. Dziuk and C. M. Elliott. Finite element methods for surface partial differential equations.
Acta Numerica ,22:289–396, 2013.[EAK +
01] J. Erlebacher, M.J. Aziz, A. Karma, N. Dimitrov, and K. Sieradzki. Evolution of nanoporosity in dealloying.
Nature , 410:450–453, 2001.[EE08] C. Eilks and C. M. Elliott. Numerical simulation of dealloying by surface dissolution via the evolving surfacefinite element method.
Journal of Computational Physics , 227(23):9727–9741, 12 2008.[EF89] C. M. Elliott and D. A. French. A nonconforming finite element method for the two-dimensional Cahn-Hilliardequation.
SIAM J. Numer. Anal. , 24(4):884–903, 1989.[EFM89] C. M. Elliott, D. A. French, and F. A. Milner. A second order splitting method for the Cahn-Hilliard equation.
Numerische Mathematik , 54(5):575–590, 1989.[EG96] C. M. Elliott and H. Garcke. On the Cahn-Hilliard equation with degenerate mobility.
SIAM Journal onMathematical Analysis , 27(2):404–423, 1996.[EL91] C. M. Elliott and S. Luckhaus. A generalised diffusion equation for phase separation of a multi-componentmixture with interfacial free energy.
IMA Preprint Series , (887), 1991.[EL92] C. M. Elliott and S. Larsson. Error estimates with smooth and nonsmooth data for a finite element methodfor the Cahn-Hilliard equation.
Math. Comput. , 58(198):603–630, 1992.[Ell89] C. M. Elliott.
The Cahn-Hilliard Model for the Kinetics of Phase Separation , pages 35–73. Number 88 inInternational Series of Numerical Mathematics. Birkhäuser Verlag, Basel, Germany, 1989.[ER15] C. M. Elliott and T. Ranner. Evolving surface finite element method for the Cahn–Hilliard equation.
Nu-merische Mathematik , 129(3):483–534, Mar 2015.[ER17] C. M. Elliott and T. Ranner. A unified theory for continuous in time evolving finite element space approxi-mations to partial differential equations in evolving domains. arXiv e-prints , (1703.04679), 2017.[ESV12] C. M. Elliott, B. Stinner, and C. Venkataraman. Modelling cell motility and chemotaxis with evolving surfacefinite elements.
Journal of the Royal Society Interface , 9(76):3027–3044, 2012.[EZ86] C. M. Elliott and S. Zheng. On the Cahn-Hilliard equation.
Archive for Rational Mechanics and Analysis ,96(4):339–357, 1986.[Gar13] H. Garcke. Curvature driven interface evolution.
Jahresbericht der Deutschen Mathematiker-Vereinigung ,115(2):63–100, Sep 2013.[GK18] H. Garcke and P. Knopf. Weak solutions of the Cahn–Hilliard system with dynamic boundary conditions: Agradient flow approach. arXiv e-prints , page arXiv:1810.09817, Oct 2018.[GLS13] H. Garcke, K. F. Lam, and B. Stinner. Diffuse interface modelling of soluble surfactants in two-phase flow. arXiv e-prints , page arXiv:1303.2559, Mar 2013.[GT98] D. Gilbarg and N. S. Trudinger.
Elliptic partial differential equations of second order . Grundlehren der math-ematischen Wissenschaften. Springer Verlag, 1998.[Hei15] M. Heida. Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation.
Applica-tions of Mathematics , 60(1):51–90, 2015.[KS80] D. Kinderlehrer and G. Stampacchia.
An introduction to variational inequalities and their applications . Aca-demic Press, 1980.[Leo09] G. Leoni.
A First Course in Sobolev Spaces . American Mathematical Society, 2009.[Lio57] Jacques-Louis Lions. Sur les problèmes mixtes pour certains systèmes paraboliques dans les ouverts noncylindriques.
Annales de l’institut Fourier , 7:143–182, 1957.[LSTT20] D. Lan, D. T. Son, B. Q. Tang, and L. T. Thuy. Quasilinear parabolic equations with first order terms and L -data in moving domains. arXiv e-prints , arXiv:2003.00064v1, 2020.[Mir19] A. Miranville. The Cahn-Hilliard equation: Recent Advances and Applications . Society for Industrial andApplied Mathematics, 2019.[Nae15] P. Naegele.
Monotone operator theory for unsteady problems on non-cylindrical domains . PhD thesis, Uni.Freiburg, 2015.[NCS84] A. Novick-Cohen and L. A. Segel. Nonlinear aspects of the Cahn-Hilliard equation.
Physica D: NonlinearPhenomena , 10(3):277 – 298, 1984.[OS16] D. O’Connor and B. Stinner. The Cahn-Hilliard equation on an evolving surface. arXiv e-prints , pagearXiv:1607.05627, Jul 2016.[OXY20] M. Olshanskii, X. Xu, and V. Yushutin. A finite element method for Allen-Cahn equation on deformingsurface. arXiv e-prints , (arXiv:2007.09531v1), 2020.
AHN-HILLIARD EQUATIONS ON AN EVOLVING SURFACE 45 [PD96] G. Da Prato and A. Debussche. Stochastic Cahn-Hilliard equation.
Nonlinear Analysis: Theory, Methods &Applications , 26(2):241 – 263, 1996.[Peg86] R. L. Pego. Front migration in the nonlinear Cahn-Hilliard equation.
Proceedings of the Royal Society ofLondon. Series A, Mathematical and Physical Sciences , 422, 1986.[Rob01] J. C. Robinson.
Infinite dimensional dynamical systems . Cambridge Texts in Apllied Mathematics. Cambridge,2001.[Vie11] M. Vierling. On control-constrained parabolic optimal control problems.
ArXiv , abs/1106.0622, 2011.[VSG +
11] C. Venkataraman, T. Sekimura, E. A. Gaffney, P. K. Maini, and A. Madzvamuse. Modeling parr-mark patternformation during the early development of amago trout.
Phys. Rev. E , 84:041923, Oct 2011.[YQO19] V. Yushutin, A. Quaini, and M. Olshanskii. Numerical modelling of phase separation on dynamic surfaces. arXiv e-prints , page arXiv:1907.11314, Jul 2019.[ZTL +
19] C. Zimmermann, D. Toshniwal, C. M. Landis, T. J. R. Hughes, K. K. Mandadapu, and R. A. Sauer. Anisogeometric finite element formulation for phase transitions on deforming surfaces.