GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION: GLOBAL EXISTENCE AND LATENT SINGULARITY

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1 1 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION: GLOBAL EXISTENCE AND LATENT SINGULARITY 3 YUAN GAO, JIAN-GUO LIU, AND XIN YANG LU Abstract. In this work, we study a fourth order exponential equation, u t = e u, derived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow method in metric space to characterize the latent singularity in global strong solution, which is intrinsic due to high degeneration. We define a suitable functional, which reveals where the singularity happens, and then prove the variational inequality solution under very weak assumptions for initial data. Moreover, the existence of global strong solution is established with regular initial data Introduction 1.1. Background. Thin film growth on crystal surface includes kinetic processes by which adatoms detach from above, diffuse on the substrate and then are absorbed at a new position. These processes drive the morphological changes of crystal surface, which is related to various nanoscale phenomena [11, ]. Below the roughing temperature, crystal surfaces consist of facets and steps, which are interacting line defects. At the macroscopic scale, the evoluion of those interacting line defects is generally formulated as nonlinear PDEs using macroscopic variables; see [5, 7, 1, 18, 1, 4, 5]. Especially from rigorously mathematical level, [6, 10, 8, 9, 1, 17] focus on the existence, long time behavior, singularity and self-similarity of solutions to various dynamic models under different regimes. Let us first review the continuum model with respect to the surface height profile u(t, x). Consider the general surface energy, G(u) := (β 1 u + β p u p ) dx, (1) where is the step locations area we are concerned with. Then the chemical potential µ, defined as the change per atom in the surface energy, can be expressed as µ := δg ( δu = u ) β 1 u + β u p u. Date: October 19, Mathematics Subject Classification. 35K65, 35R06, 49J40. Key words and phrases. Fourth-order exponential parabolic equation, Radon measure, global strong solution, latent singularity, curve of maximal slope. 1

2 YUAN GAO, JIAN-GUO LIU, AND XIN YANG LU Now by conservation of mass, we write down the evolution equation for surface height of a solid film u(t, x): where u t + J = 0, J = M( u) ρ s is the adatom flux by Fick s law [18], the mobility function M( u) is a functional of the gradients in u and ρ s is the local equilibrium density of adatoms. By the Gibbs-Thomson relation [14, 0, 18], which is connected to the theory of molecular capillarity, the corresponding local equilibrium density of adatoms is given by ρ s = ρ 0 e µ kt, where ρ 0 is a constant reference density, T is the temperature and k is the Bolzmann constant. Notice those parameters can be absorbed in the scaling of the time or spatial variables. The evolution equation for u can be rewritten as ) u t = (M( u) e δg δu. () It should be pointed out that in past, the exponential of µ/kt is typically linearized under the hypothesis that µ kt ; see for instant [13, 15, 3] and most rigorous results in [6, 10, 8, 9, 1, 17] are established for linearized Gibbs-Thomson relation. This simplification, e µ 1 + µ, yields the linear Fick s law for the flux J in terms of the chemical potential J = M( u) µ The resulting evolution equation is ( ( )) u δg t = M( u), (3) δu which is widely studied when the mobility function M( u) takes distinctive forms in different limiting regimes. For example, in the diffusion-limited (DL) regime, where the dynamics is dominated by the diffusion across the terraces and M is a constant M 1, Giga and Kohn [10] rigorously showed that with periodic boundary conditions on u, finite-time flattening occurs for β 1 0. A heuristic argument provided by Kohn [1] indicates that the flattening dynamics is linear in time. While in the attachment-detachment-limited (ADL) case, i.e. the dominant processes are the attachment and detachment of atoms at step edges and the mobility function [1] takes the form M( u) = u 1, we refer readers to [1, 1, 8, 9] for analytical results. Note that the simplifed version of PDE (3), which linearizes the Gibbs-Thomson relation, does not distinguish between convex and concave parts of surface profiles. However the convex and concave parts of surface profiles actually have very different dynamic processes due to the exponential effect, which is explained in Section 1. below; see also numerical simulations in [16].

3 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION 3 3 Now we consider the original exponential model () in DL regime ) u t = ( e δg δu ( ) u p u = e, (4) with surface energy G := 1 p u p dx, p 1. The physical explanation of the p-laplacian surface energy can be found in [19]. From the atomistic scale of solid-on-solid (SOS) model, the transitions between atomistic configurations are determined by the number of bonds that each atom would be required to break in order to move. It worth noting for p = 1 [16] developed an explicit solution to characterize the dynamics of facet position in one dimensional, which is also verified by numerical simulation. In this work, we focus on the case p = for high dimensional and use the gradient flow approach to study the strong solution with latent singularity to (4). We will see clearly the different performs between convex and concave parts of the surface. Explicitly, given T > 0 and an open, bounded, connected, spatial domain R d with smooth boundary, we consider the evolution problem u t = e u in [0, T ], u ν = e u ν = 0 on [0, T ], (5) u(x, 0) = u 0 (x) on, where ν denotes the outer unit normal vector to. The main results of this work is to prove the existence of variational inequality solution to (5) under weak assumptions for initial data and also the existence of strong solution to (5) under strong assumptions for initial data; see Theorem 10 and Theorem 13 separately. 1.. Formal observations. We first show some a-priori estimates to see the mathematical structures of (5). On one hand, formally define the energy F (u) := e u dx, so we can rewrite the original equation as a gradient flow u t = δf δu = e u (6) and T F (T ) + δf dx dt = F (0) 0 δu for any T > 0. Notice boundary condition u ν = 0. We have u dx = 0, which gives ( u) + L 1 () = ( u) L 1 () = u L 1 (), (7)

4 4 YUAN GAO, JIAN-GUO LIU, AND XIN YANG LU where ( u) + := max{0, u} is the positive part of u and ( u) := min{0, u} is the negative part of u. Since ( u) L 1 = ( u) dx e ( u) dx e ( u)+ +( u) dx = F (u) F (u 0 ) < +, we know u L 1 () F (u 0 ) < +. However, since L 1 is non-reflexive Banach space, the uniform bound of L 1 norm for u dose not prevent it being a Radon measure. In fact, from F (u) = e u dx and (6), we can see a positive singularity in u should be allowed for the dynamic model; also see an example in [17, p.6] for a stationary solution with singularity. We will introduce the latent singularity in ( u) + officially in Section.1. On the other hand, since d u t dx = u t ( e u ) t dx = u t (e u ) t dx = ( u t ) e u dx 0, dt we have high order a-priori estimate u t dx = ( e u ) dx C(u 0 ), where C(u 0 ) is a constant depending only on u 0 ; see also [17]. Notice that e u dx e u dx 1 e u dx + c e u dx. (8) Then by [17, Lemma 1], we have D e u dx This, together with (8), implies ( e u ) dx C(u 0 ). e u H () C 1 (u 0 ). Although these are formal observations for now, we will prove them rigorously later Overview of our method and related method. Although from formal observations in Section 1. the original problem can be recast as a standard gradient flow, the main difficulty is how to characterize the latent singularity in ( u) + and choose a natural working space. As we explained before, the possible existence of singular part for u is intrinsic, so the best regularity we can expect for u is Radon measure space. To get the uniform bound of u M(), we need to first construct an invariant ball, which is the indicator functional ψ defined in (1), then get rid of ψ after we obtain the variational inequality solution; see Theorem 10 and Corollary 11. After we choose the working space M() for u, we can define the energy functional φ rigorously in (11) using Lebesgue decomposition. To avoid the extra technical difficulties brought by non-reflexive Banach space, we first ignore the Banach space structure and use the gradient flow approach in metric space introduced by []. We consider a curve of maximal slope of the energy functional φ + ψ and try to gain the variational inequality solution defined in Definition under weak assumptions for the initial data following [, Theorem 4.0.4]. However, since the

5 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION functional φ is defined only on the absolutely continuous part of u, it is not easy to verify the lower semi-continuity and convexity of φ, which is developed in Section.3 and Section.4. Finally, when the initial data have enough regularities, we prove the variational inequality solution has higher regularities and is also strong solution to (5) defined in Definition 1. Recently, [17] also studies the same problem (5) using the method of approximating solutions. Their method based on carefully chosen regularization, which is delicate but the construction is subtle to reveal the mathematical structure of our problem. Instead, our method using gradient flow structure is natural and more general, which is flexible to wide classes of dynamic systems with latent singularity. When proving the variational inequality solution to (5), we also provide an additional understanding for the evolution of thin film growth, i.e., the solution u is a curve of maximal slope of the well-defined energy functional φ + ψ; see Definition 9. The rest of this work is devoted to first introduce the abstract setup of our problem in Section.1 and Section.. Then in Section.3,.4 and Section.5, we prove the variational inequality solution following [, Theorem 4.0.4]. In Section 3, under more assumptions on initial data, we finally obtain the strong solution to (5). 89. Gradient flow approach and variational inequality solution Preliminaries. We first introduce the spaces we will work in. Since we are not expecting 91 classical solution to (5), the boundary condition in (5) can not be recovered exactly. Instead, we equip the boundary condition in the space H, Ṽ defined blow. Let { } H := u L () : u dx = 0, 93 endowed with the standard scalar product u, v H := uv dx. 94 Since L 1 is not reflexive Banach space and has no weak compactness, those a-priori estimates 95 in Section 1. can not guarantee the W,1 ()-regularity of solutions to (5). Hence we define the 96 space Ṽ as Ṽ := {u H; u L (), u M(), ϕ d( u) = u ϕ dx for any ϕ C b ()}, (9) where M is the space of finite signed Radon measures and C b () is all the bounded continuous functions on. Endow Ṽ with the norm v Ṽ := v L () + v M() Here M() denotes the total mass of the measure. Next, since u can be a Radon measure, we need to make those formal observations in Section 1. rigorous. For any µ M, from [4, p.4], we have the decomposition µ = µ + µ (10)

6 6 YUAN GAO, JIAN-GUO LIU, AND XIN YANG LU with respect to the Lebesgue measure, where µ L 1 () is the absolutely continuous part of µ and µ is the singular part, i.e., the support of µ has Lebesgue measure zero. Define the function φ : H [0, + ], φ(u) := e ( u)+ +( u) dx, if u Ṽ, (11) + otherwise, where ( u) denotes the absolutely continuous part of u, ( u) := min{0, u} is the negative part of u. We call the singular part ( u) + latent singularity in solution u. Remark 1. Although the singularity vanishes in the energy functional φ, it is not a removable singularity in the dynamics. Indeed, noticing the boundary condition, we can not recover a new solution v by removing the singularity such that v = ( u) + ( u) and v t = e v. So the singularity in solution ( u) + actually have effect on u t and we refer it as latent singularity. In view of the a priori estimate on the mass of the measure u, we introduce the indicator function 0 if u ψ : H {0, + }, ψ(u) := Ṽ, u M() C, (1) + otherwise. Here C is a fixed constant, which is determined by the initial datum later. From Section 1. we know the bound for u M() (1) is not artificial... Euler schemes. Even if (5) has a nice variational structure, and V has Banach space structure, the non-reflexivity of V imposes extra technical difficulties. Instead of arguing with maximal monotone operator like in [9], we try to use the result [, Theorem 4.0.4] by Ambrosio, Gigli and Savaré. After defining the energy functional rigorously, we take the counterintuitive approach of ignoring the variational structure of (5) and the Banach space structure of W,1 (). In other words, we consider the gradient flow evolution in the metric space (H, dist), with distance dist(u, v) := u v H. Let u 0 H be a given initial datum and 0 < τ 1 be a given parameter. We consider a sequence {x τ n} which satisfies the following unconditional-stable backward Euler scheme { x (τ) n argmin x H (φ + ψ)(x ) + 1 } τ x x (τ) n 1 H n 1, (13) x (τ) 0 := u 0 H. The existence and uniqueness of the sequence {x τ n} will proved later in Proposition 8. Thus we are considering the gradient descent with respect to φ + ψ in the space (H, dist). Now for any 0 < τ 1 we define the operator { J τ [u] := argmin v H (φ + ψ)(v) + 1 } τ v u H,

7 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION then the variational approximation obtained by Euler scheme (13) is u n := (J t/n ) n [u 0 ]. (14) The results for gradient flow in metric space [, Theorem 4.0.4] establish the convergence of the variational approximation u n to variational inequality solution to (5), which is defined below. Definition. Given initial data u 0 H, we call u : [0, + ) H a variational inequality solution to (5) if u(t) is a locally absolutely continuous curve such that lim t 0 u(t) = u 0 in H and 1 d dt u(t) v (φ + ψ)(v) (φ + ψ)(u(t)), for a.e. t > 0, v D(φ + ψ). (15) Before proving the existence of variational inequality solution to (5), we first study some properties of the functional φ + ψ in Section.3 and Section Weak-* lower semi-continuity for functional φ in Ṽ. For any µ M(), we denote 133 µ L d if µ is absolutely continuous with respect to Lebesgue measure and denote µ := dµ as dl d 134 the density of µ. For notational simplification, denote µ (resp. µ ) as the absolutely continuous 135 part (resp. singular part) of µ with respect to Lebesgue measure. 136 Let us first give the following proposition claiming weak-* lower semi-continuity for functional 137 φ in Ṽ, which will be used in Lemma 6. Proposition 3. Let un, u Ṽ. If un u in M(), we have lim inf φ(u n) φ(u). (16) n + Before proving Proposition 3, we first state some lemmas. From now on, we identify µ n L d with its density µ n := dµn and do not distinguish them for dl d brevity. Given N > 0 and a sequence of measures µ n such that µ n L d, observe that µ n = min{µ n, N} + max{µ n, N} N. (17) First we state a lemma about the limit of the truncated measure min{µ n, N}. 143 Lemma 4. Given a sequence of measures µ n such that µ n L d, we assume moreover that µ n µ 144 and sup n + φ(µ n ) < +. Then there exists a measure µ down L d such that N µ down 145 and min{µ n, N} µ down. 146 Proof. Since µ n µ, we know there exists µdown M() such that min{µ n, N} µ down. From N min{µ n, N} 0 we have N µ down 0. Moreover we claim µ down L d. Indeed, if µ down contained a negative singular measure, then sup n µ n L () = +. However, since µ n N, this forces + > e µn dx µ n dx +. Hence we have µ down L d. {µ n 1} {µ n 1}

8 8 YUAN GAO, JIAN-GUO LIU, AND XIN YANG LU We also need the following useful Lemma to clarify the relation between µ down and the weak- limit of µ n. Lemma 5. Given a sequence of measures µ n such that µ n L d 149, we assume moreover that µ n µ 150 and sup n + φ(µ n ) < +. Then for any N > 0, there exist µ down, µup M(), such that min{µ n, N} µ down, µ down L d, µ down µ, (18) max{µ n, N} µup, (µ up ) N, (19) where µ (resp. µ ) is the absolutely continuous part (resp. singular part) of µ. Moreover, e µ dx e µ down dx. (0) Proof. From Lemma 4 we know, upon subsequence, min{µ n, N} µ down for some measure µ down satisfying µ down L d and N µ down. By Lebesgue decomposition theorem, there exist unique measures µ L d and µ L d such that µ = µ + µ. The decomposition (17) then gives 0 µ n min{µ n, N} = max{µ n, N} N µ µ down Taking µup := µ µ down +N, as the sequence max{µ n, N} N 0, we obtain max{µ n, N} µup and (µ µ down ) = µup N 0. Besides, since e µ is decreasing with respect to µ and µ µ down, we obtain (0). Now we can start to prove Proposition 3. Proof of Proposition 3. Assume u n u in M. Denote fn := u n and f := u. Without loss of generality we assume sup n + φ(u n ) < + so fn, f L d. This implies immediately that we only need to consider two cases: (i) there are some f n are positive measures, i.e. f n 0, and f n g1 L d, f n g 0 and g 1 + g = f ; or (ii) all f n are absolutely continuous and f n = f n may weakly-* converge to a singular measure. For case (i), if we have f n g1 L d, f n g 0 and g 1 + g = f, then since e f is decreasing with respect to f, we have e g 1 dx e f dx. On the other hand, we know ϕ(v) := e v dx is lower-semicontinuous on L 1 () with respect to the strong topology. Hence by the convexity of ϕ(v) := e v dx on L 1 () and [3, Corollary 3.9], we know ϕ(v) is l.s.c on L 1 () with respect to the weak topology. So f n g1 L d gives f n g 1 in L 1 () and lim inf φ(u n ) = lim inf e f n dx e g 1 dx e f dx = φ(u) (1) n n which ensure (16) holds. Now we only concern the case (ii): f n = 0 and f n = f n may weakly-* converge to a singular measure. For any N > 0 large enough, denote φ N (u n ) := e min{fn,n} dx. Then the truncated

9 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION measures min{f n, N} satisfy φ N (u n ) = = e min{fn,n} dx {f n N} {f n N} The second equality also shows φ N (u n ) e N L d ({f n > N}) = Hence we obtain e min{fn,n} dx + e N L d ({f n > N}) e fn dx + e fn dx = φ(u n ). {f n>n} {f n N} e min{fn,n} dx e fn dx = φ(u n ). φ(u n ) φ N (u n ) e N L d ({f n > N}) e N. () From Lemma 5, we know the truncated sequence min{f n, N} satisfies min{f n, N} f down, f down L d, e f down dx e f dx. (3) Since min{f n, N} f down in L 1 (), using the same argument with (1), we obtain lim inf e min{fn,n} dx e f down dx e f dx = φ(u). (4) n + Combining this with (), we obtain lim inf φ(u n) lim inf φ N(u n ) e N n + n + = lim inf e min{fn,n} dx e N n + φ(u) e N, and thus we complete the proof of Proposition 3 by the arbitrariness of N..4. Convexity and lower semi continuity of functional φ + ψ in H. Lemma 6. The sum φ + ψ : H [0, + ] is proper, convex, lower semicontinuous in H and satisfies coercivity defined in [, (.4.10)]. Proof. Clearly since u 0 D(φ + ψ), D(φ + ψ) = {φ + ψ < + } is non empty, hence φ + ψ is proper. Due to the positivity of φ, ψ, coercivity [, (.4.10)] is trivial. (5) Convexity. Note that since both φ, ψ 0, we have D(φ + ψ) = D(φ) D(ψ). Given u, v H, t (0, 1), without loss of generality assume u, v D(φ + ψ), otherwise convexity inequality is

10 10 YUAN GAO, JIAN-GUO LIU, AND XIN YANG LU trivial. Thus (1 t)u + tv D(ψ), and the measure [(1 t)u + tv] has no negative singular part, while its positive singular part satisfies and its absolutely continuous part satisfies ( [(1 t)u + tv]) + = (1 t)( u)+ + t( v)+, ( [(1 t)u + tv]) = (1 t)( u) + t( v). Thus φ((1 t)u + tv) = e [(1 t) u+t v] dx = [(1 t)e ( u) + te ( v) ] dx = (1 t)φ(u) + tφ(v), e [(1 t)( u) +t( v) ] dx 178 hence φ + ψ is convex. Lower semicontinuity. Consider a sequence u n u in H. We need to check (φ + ψ)(u) lim inf(φ + ψ)(u n ). n If u n / D(φ + ψ) for all large n, then lower semicontinuity is trivial. Without loss of generality, we can assume u n D(φ + ψ) for all n, and also lim inf(φ + ψ)(u n ) = lim(φ + ψ)(u n ). n n Since u n D(ψ), we have u n M() C, hence there exists v M() such that u n v. Since we also have u n u in H so v = u and we know u M() C. Then 0 = ψ(u n ) = ψ(u) and by Proposition 3, we have so the lower semicontinuity is proved. lim inf n φ(u n ) φ(u) Lemma 7 (τ 1 -convexity). For any u, v 0, v 1 D(φ + ψ), there exists a curve v : [0, 1] D(φ + ψ) such that v(0) = v 0, v(1) = v 1 and the functional satisfies Φ(τ, u; v) := (φ + ψ)(v) + 1 τ u v H (6) Φ(τ, u; v(t)) (1 t)φ(τ, u; v 0 ) + tφ(τ, u; v 1 ) 1 τ t(1 t) v 0 v 1 H (7) for all τ > 0. We remark that (7) is the so-called τ 1 -convexity [, Assumption 4.0.1].

11 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION 11 Proof. Let v(t) := (1 t)v 0 + tv 1. The proof follows from the simple identity (1 t)v 0 + tv 1 u H = (1 t) u v 0 H + t u v 1 H t(1 t) v 0 v 1 H. 185 The convexity of φ + ψ then gives Φ(τ, u; v(t)) = (φ + ψ)((1 t)v 0 + tv 1 ) + 1 τ u [(1 t)v 0 + tv 1 ] H (1 t)(φ + ψ)(v 0 ) + t(φ + ψ)(v 1 ) + 1 τ (1 t) u v 0 H + 1 τ t u v 1 H 1 τ t(1 t) v 0 v 1 H = (1 t)φ(τ, u; v 0 ) + tφ(τ, u; v 1 ) 1 τ t(1 t) v 0 v 1 H, and concludes the proof After above properties for functional φ + ψ, we state existence and uniqueness of the sequence {x τ n} chosen by Euler scheme (13). Proposition 8. Given parameter τ > 0, u 0 H, then for any n 1, there exists unique x τ n satisfying (13). Proof. Given n 1, we will prove this proposition by the direct method in calculus of variation. Let Φ(τ, x n 1 ; x) defined in (6) and A := inf x H Φ(τ, x n 1 ; x). Then there exist {x ni } D(Φ) such that Φ(τ, x n 1 ; x ni ) A as i + and Φ(τ, x n 1 ; x ni ) are uniformly bounded. Hence upon a subsequence, there exists x n H such that x ni x n in H. This, together with the uniform boundedness of x ni M() shows that x ni v = xn in M(). Then by Proposition 3 we have A = lim inf Φ(τ, x n 1; x ni ) Φ(τ, x n 1 ; x n ) A, i + which gives the existence of x n satisfying (13). The uniqueness of x n follows obviously by the convexity of φ and the strong convexity of H..5. Existence of variational inequality solution. After those preparations in Section.3 and Section.4, in this section we apply the convergence result in [, Theorem 4.0.4] to derive that the discrete solution u n obtained by Euler scheme 13 converges to the variational inequality solution defined in Definition. Denote φ (v) := lim sup w v max{φ(v) φ(w), 0} dist(v, w) the local slope. By Lemma 7 and [, Theorem.4.9] for λ = 0, the local slope coincides with the global slope max{φ(v) φ(w), 0} ι φ (v) := sup, v w v w H

12 1 YUAN GAO, JIAN-GUO LIU, AND XIN YANG LU i.e. φ (v) = ι φ (v). (8) We point out that with Lemma 6 and [, Theorem 1..5], we also know the global slope ι φ is a strong upper gradient for φ. Hence for ι φ, we recall [, Definition 1.3.] for curves of maximal slope. Definition 9. Given a functional φ : D(φ) R and the global slope ι φ, we say that a locally absolutely continuous map u : (0, T ) H is a curve of maximal slope for the functional φ with respect to ι φ if (φ(u(t))) 1 u t 1 ι φ(u) for a.e. t (0, T ). (9) Now the hypotheses of [, Theorem 4.0.4] are all satisfied: Lemma 6 gives convexity, lower semicontinuity and coercivity of φ + ψ [, (4.0.1)], while Lemma 7 gives τ 1 -convexity of φ + ψ with λ = 0 [, Assumption 4.0.1]. Thus we have: Theorem 10. Given u 0 D(φ + ψ) H, (i) (convergence and error estimate) for any t > 0, t = nτ, let u n in (14) be the solution obtained by Euler scheme (13), then there exists a local Lipschitz curve u(t) : [0, + ) H such that u n u(t) in L () (30) and if further u 0 D(φ + ψ), we have the error estimate u(t) u n H τ (φ + ψ) (u 0 ); (31) (ii) u : [0, + ) H is the unique variational inequality solution to (5), i.e., u is unique among all the locally absolutely continuous curves such that lim t 0 u(t) = u 0 in H and 1 d dt u(t) v (φ + ψ)(v) (φ + ψ)(u(t)), for a.e. t > 0, v D(φ + ψ); (3) (iii) u(t) is a locally Lipschitz curve of maximal slope of φ for t > 0 in the sense (iv) moreover, we have the following regularities ( (φ + ψ)(u(t)) ) 1 u t 1 ι φ(u) ; (33) (φ + ψ)(u(t)) (φ + ψ)(v) + 1 t v u0 H v D(φ + ψ), (34) 11 (φ + ψ) (u(t)) (φ + ψ) (v) + 1 t v u0 H v D( (φ + ψ) ), (35) (φ + ψ) (u(t)) u0 ū H t, (φ + ψ)(u(t)) (φ + ψ)(ū) u0 ū H, (36) t

13 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION 13 and t u(t) ū H is non-increasing, where ū is a minimum point for φ + ψ and is the local slope; φ (v) = lim sup w v (φ(v) φ(w)) + dist(v, w) = ι φ (v) (v) (L -contraction) let u 0, v 0 D(φ + ψ) H and u(t), v(t) be solutions to the variational inequality (3), then u(t) v(t) H u 0 v 0 H. (37) Proof. Since from Lemma 6 and Lemma 7, we are under the hypotheses of [, Theorem 4.0.4], we apply it with energy functional φ + ψ, and metric space (H, dist), dist(u, v) = u v H to obtain (30). Therefore the convergence result (i) comes from [, (4.0.11),(4.0.15)]. The variational inequality (3) follows from [, (4.0.13)]. [, Theorem (ii)] shows the result (iii) and (33) follows Definition 9 of maximal slope. Regularities (34) and (35) follow from [, (4.0.1)]. Asymptotic behavior (36) and monotonicity of t u(t) ū H follow from [, Corollary 4.0.6], which requires the same hypotheses of [, Theorem 4.0.4]. Finally, the contraction result (v) follows from [, (4.0.14)] Strong solution We will prove the variational inequality solution obtain in Theorem 10 is actually a strong solution in this section. Now we assume u : [0, + ) H is the unique solution of the variational inequality (3), i.e., 1 d dt u(t) v (φ + ψ)(v) (φ + ψ)(u(t)), for a.e. t > 0, v D(φ + ψ). (38) 3.1. Regularity of variational inequality solution. First we state the variational inequality solution has further regularities. Corollary 11. Given T > 0 and initial datum u 0 D( φ ), the solution obtained in Theorem 10 has the following regularities u L ([0, T ]; Ṽ ) C0 ([0, T ]; H), u t L ([0, T ]; H), ( u) L d for a.e. t [0, T ], where ( u) := min{0, u} is the negative part of u. Besides, we can rewrite the variational inequality (3) as u t (t), u(t) v H,H φ(v) φ(u(t)) for a.e. t > 0, v D(φ). (39)

14 14 YUAN GAO, JIAN-GUO LIU, AND XIN YANG LU 3 Proof. First, we claim the functional ψ can be taken off. Indeed, from (34) we have (φ + ψ)(u(t)) (φ + ψ)(v) + 1 t v u0 H v D(φ). (40) Then taking v = u 0 gives (φ + ψ)(u(t)) (φ + ψ)(u 0 ) < +, (41) which also implies φ(u(t)) φ(u 0 ) < + for a.e. t [0, T ]. (4) To make Section 1. rigorous, notice u Ṽ we have ϕ d( u) = u ϕ dx for any ϕ C b (). Particularly, taking ϕ 1 gives d( u) = 0, so we have ( u) + M() = ( u) M() = 1 u M(). 35 Since ( u) L 1 () = ( u) dx e ( u) dx e ( u)+ +( u) dx = φ(u) φ(u 0 ) we know ( u) L d for a.e. t [0, T ], u M() φ(u 0 ), so we can choose C := φ(u 0 ) + 1 in Definition (1) and ψ(u(t)) 0 ψ(u(t)). (43) Noticing also that if v / D(ψ) (38) still holds, we can rewrite (38) as 1 d dt u(t) v φ(v) φ(u(t)), for a.e. t > 0, v D(φ). Next, we need to show that u t L (0, T ; L ()). From Theorem 10 we know that t u(t) is locally Lipschitz in (0, T ), i.e. for any t 0 > 0 there exists L = L(t 0 ) > 0 such that u(t 0 + ε) u(t 0 ) L () L(t 0 )ε for all t [t 0, T ]. The key point is to obtain a uniform bound for L(t 0 ) for arbitrary t 0 0. variational solution satisfying (3), taking v = u(t 0 ) in (3) gives 1 d dt u(t 0) u(t) L () φ(u(t 0)) φ(u(t)) ξ, u(t 0 ) u(t) H,H for any ξ φ(u(t 0 )). In particular, by [, Proposition 1.4.4], we have φ (u(t 0 )) = min{ ξ H ; ξ φ(u(t 0 ))}. Since u(t) is the

15 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION Hence taking ξ as the elements of minimal dual norm in φ(u(t 0 )) implies 1 d dt u(t 0) u(t) L () φ(u(t 0)) φ(u(t)) ξ L () u(t 0) u(t) L () φ (u(t 0 )) u(t 0 ) u(t) L (). Furthermore, since t u(t 0 ) u(t) L () is locally Lipschitz, hence differentiable for a.e. t, we have d dt u(t 0) u(t) L () φ (u(t 0 )) φ (u 0 ) for a.e. t > 0, (44) where we have used (35) in the last inequality. Thus the function t u(t 0 ) u(t) L () is globally Lipschitz with Lipschitz constant less than φ (u 0 ), which is independent of t 0. Hence we know u(t 0 ) u(t 0 + ε) L () ε φ (u 0 ) Thus for a.e. t we have u ε := u(t + ε) u(t) ε L (), u ε L () φ (u 0 ), and the sequence of difference quotients (u ε ) ε is uniformly bounded in L (). there exists v L (), v L () φ (u 0 ), such that (upon subsequence) as ε 0. Thus v = u t (t), and Finally, from we obtain (39). 1 u ε v strongly in L () Consequently, u t L (0,T ;L ()) φ (u 0 ). (45) d dt u(t) v L () = u t(t), u(t) v H,H, 3.. Existence of strong solution. After establishing the regularity of variational inequality solution in Section 3.1, we start to prove the variational inequality solution is also a strong solution. We first clarify the definition of strong solution, which has a latent singularity. Definition 1. Given u 0 D( φ ), we call function u L ([0, T ]; Ṽ ) C0 ([0, T ]; H), u t L ([0, T ]; H) a strong solution to (5) if u satisfies u t = (e ( u) ) (46) for a.e. (t, h) [0, T ], where ( u) is the absolutely continuous part of u in the decomposition (10).

16 16 YUAN GAO, JIAN-GUO LIU, AND XIN YANG LU Let ϕ Cc () be given. We prove the sub-differential of functional φ is single-valued. The idea of proof is to test (39) with v := u ± εϕ and then take limit as ε 0. Let us state existence result for strong solution as follows. Theorem 13. Given T > 0, initial datum u 0 D( φ ), then the variational inequality solution u obtained in Theorem 10 is also a strong solution to (5), i.e., for a.e. (t, h) [0, T ]. Besides, we have and the dissipation inequality E(u(t)) := 1 u t = (e ( u) ) (47) (e ( u) ) L ([0, T ]; H) [ (e ( u) ) ] dx E(u 0 ), (48) where ( u) is the absolutely continuous part of u in the decomposition (10). Proof. Step 1. Integrability results. First from (4), we know Since ϕ C c () we also know for all sufficiently small ε. Step. Testing with v = u(t) ± εϕ. Plugging v = u(t) + εϕ in (39) gives Direct computation shows that φ(u(t) + εϕ) φ(u(t)) = = This, together with (51), gives u t (t), εϕ lim ε 0 e ( u(t)) L 1 (). (49) e ( u(t)) ε ϕ L 1 () (50) u t (t), εϕ + φ(u(t) + εϕ) φ(u(t)) 0. (51) [ ] e ( u(t)) ε ϕ e ( u(t)) dx e ( u(t)) ε ϕ ( 1 e ε ϕ) dx ( ) e ( u(t)) ε ϕ ε ϕ dx ( ) e ( u(t)) ε ϕ ε ϕ dx 0. (5) To take limit in (5), we claim e ( u(t)) ε ϕ ϕ dx = e ( u(t)) ϕ dx. (53)

17 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION 17 Proof of (53). First we have e ( u(t)) ε ϕ ϕ e ( u(t)) ϕ a.e. on. 60 Then by (50) we can see e ( u(t)) ε ϕ ϕ dx < +. Thus by dominated convergence theorem we infer (53). 61 Now we can divide by ε > 0 in (5) and take the limit ε 0 + to obtain u t (t), ϕ lim e ( u(t)) ε ϕ ϕ dx ε 0 + = u t (t), ϕ e ( u(t)) ϕ dx 0. Repeating the above arguments with v = u(t) εϕ gives u t (t), ϕ e ( u(t)) ϕ dx 0. Thus we finally have [ ] u t (t)ϕ e ( u(t)) ϕ dx = 0 ϕ Cc (). (54) 6 Therefore u t (t) e ( u(t)) Cc (), and u t (t) e ( u(t)) C c () = 0. The only zero 63 element of Cc () is the zero function. Thus u t = e ( u(t)) for a.e. (t, x) [0, T ]. 64 Finally, we turn to verify (48). Combining (47) and (45), we have the dissipation law E(u(t)) = 1 u t(t) H = 1 e ( u(t)) H 1 φ (u0 ), (55) where E(u(t)) = 1 [ e ( u(t)) ] 65 dx defined in (48). Hence the dissipation inequality (48) 66 holds and we completes the proof of Theorem Acknowledgment We would like to thank the support by the National Science Foundation under Grant No. DMS and KI-Net RNMS References [1] H. Al Hajj Shehadeh, R. V. Kohn and J. Weare, The evolution of a crystal surface: Analysis of a onedimensional step train connecting two facets in the adl regime, Physica D: Nonlinear Phenomena 40 (011), no. 1, [] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser Verlag, Basel, 008 [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New York, 010.

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19 GRADIENT FLOW APPROACH TO AN EXPONENTIAL THIN FILM EQUATION Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong address: maygao@ust.hk Department of Mathematics and Department of Physics, Duke University, Durham NC 7708, USA address: jliu@phy.duke.edu Department of Mathematics and Statistics, McGill University, Montreal QC H3A0B9, Canada address: xinyang.lu@mcgill.ca