Phase-field systems for grain boundary motions under isothermal solidifications
aa r X i v : . [ m a t h . A P ] F e b PHASE-FIELD SYSTEMS FOR GRAINBOUNDARY MOTIONS UNDERISOTHERMAL SOLIDIFICATIONS
Ken Shirakawa
Department of Mathematics, Faculty of Education, Chiba University1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan([email protected])
Hiroshi Watanabe
Department of General Education, Salesian Polytechnic4-6-8 Oyamagaoka, Machida, Tokyo 194-0215, Japan([email protected])
Noriaki Yamazaki
Department of Mathematics, Faculty of Engineering, Kanagawa University3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan([email protected])
Abstract.
Two main existence theorems are proved for two nonstandard systems ofparabolic initial-boundary value problems. The systems are based on the “ φ - η - θ model”proposed by Kobayashi [RIMS Kˆokyˆuroku, (2001), 68–77] as a phase-field modelof planar grain boundary motion under isothermal solidification. Although each of thesystems has specific characteristics and mathematical difficulties, the proofs of the maintheorems are based on the time discretization method by means of a common approx-imating problem. As a consequence, we provide a uniform solution method for a widescope of parabolic systems associated with the φ - η - θ model.———————————————————— This work is supported by Grants-in-Aid 24740099 (K. S.), 25800086 and 26400138 (H. W.), and 26400179(N. Y.) from the Japan Society for the Promotion of Science.AMS Subject Classification: 35K87, 35R06, 35K67 ntroduction Let 0 < T < ∞ be a constant, 1 < N ∈ N have a fixed value, and Ω ⊂ R N be abounded domain with a smooth boundary ∂ Ω. We denote by ν ∂ Ω the unit outer normalvector on ∂ Ω, and we set Q := (0 , T ) × Ω and Σ := (0 , T ) × ∂ Ω.Further, let ν ≥ u ∈ R be constants. In this paper, two themes, concerning twononstandard systems of parabolic variational inequalities, are addressed. In the first, weassume ν > ν .(S) ν : w t − ∆w + ∂γ ( w ) + g w ( w, η ) + α w ( w, η ) |∇ θ | + νβ w ( w, η ) |∇ θ | ∋ Q , ∇ w · ν ∂ Ω = 0 on Σ, w (0 , x ) = w ( x ), x ∈ Ω; (0.1) η t − ∆η + g η ( w, η ) + α η ( w, η ) |∇ θ | + νβ η ( w, η ) |∇ θ | = 0 in Q , ∇ η · ν ∂ Ω = 0 on Σ, η (0 , x ) = η ( x ), x ∈ Ω; (0.2) α ( w, η ) θ t − div (cid:18) α ( w, η ) ∇ θ |∇ θ | + 2 νβ ( w, η ) ∇ θ (cid:19) = 0 in Q , (cid:18) α ( w, η ) ∇ θ |∇ θ | + 2 νβ ( w, η ) ∇ θ (cid:19) · ν ∂ Ω = 0 on Σ, θ (0 , x ) = θ ( x ), x ∈ Ω. (0.3)Here, w = w ( x ), η = η ( x ), and θ = θ ( x ) are given initial data on Ω, ∂γ is thesubdifferential of a proper lower semi-continuous (l.s.c.) and convex function γ = γ ( w )on R , and g ( · ) = g ( w, η ), α = α ( w, η ), α = α ( w, η ), and β = β ( w, η ) are given real-valued functions. The subscripts “ w ” and “ η ” denote differentials with respect to thecorresponding variables.The system (S) ν is based on the “ φ - η - θ model” proposed by Kobayashi [25] as amathematical model of planar grain boundary motion under an isothermal solidification.Since this model was presented as an advanced version of the “Kobayashi-Warren-Cartermodel” of grain boundary motion, proposed by Kobayashi et al. [27, 28], our themes arerelated to the previous work (e.g., [16, 18–20, 24, 26–29, 34, 35, 39, 40]) associated withthe Kobayashi-Warren-Carter model. According to the modeling method of [25], (S) ν isderived as a gradient system of a governing free energy, defined as follows:[ w, η, θ ] ∈ [ H (Ω) ∩ L ∞ (Ω)] × [ H (Ω) ∩ L ∞ (Ω)] × H (Ω) F ν ( w, η, θ ) := 12 Z Ω |∇ w | dx + 12 Z Ω |∇ η | dx + Z Ω γ ( w ) dx (0.4)+ Z Ω g ( w, η ) dx + Z Ω α ( w, η ) |∇ θ | dx + ν Z Ω β ( w, η ) |∇ θ | dx.
2n this context, the constant u is the relative temperature with critical degree 0, andthe unknown w = w ( t, x ) is an order parameter to indicate the solidification order of thepolycrystal. The unknowns η = η ( t, x ) and θ = θ ( t, x ) are components of the vector field( t, x ) ∈ Q η ( t, x ) (cid:2) cos θ ( t, x ) , sin θ ( t, x ) (cid:3) ∈ R , which was adopted in [27, 28] as a vectorial phase field to reproduce the crystalline orien-tation in Q . Here, the components η and θ are order parameters to indicate, respectively,the orientation order and angle of the grain. In particular, w and η are taken to sat-isfy the constraints 0 ≤ w, η ≤ Q , and the cases [ w, η ] ≈ [1 ,
1] and [ w, η ] ≈ [0 , w, η ] ∈ R G ( w, η ) := γ ( w ) + g ( w, η ) ∈ ( −∞ , ∞ ]may have just two minimums, around [1 ,
1] and [0 , u is sufficiently lower than (resp. higher than) the critical degree, then thisfunction has a unique minimizer around [1 ,
1] (resp. [0 , γ ( w ) := 0 g ( w, η ) := c (cid:20) w ( w − − uw (cid:18) w − (cid:19)(cid:21) + 12 ( w − η ) for all w, η ∈ R ,and therefore G ( · ) = g ( · ) on R (cf. [1, 8, 14, 32, 38]);(g2) (logarithmic constraint) γ ( w ) := 12 (cid:0) w log w + (1 − w ) log(1 − w ) (cid:1) with γ (0) = γ (1) := 1 g ( w, η ) := − c (cid:18) w − u − (cid:19) + 12 ( w − η ) for all w, η ∈ R ,and therefore the range of w is constrained to the open interval (0 ,
1) (cf. [17, 37]);(g3) (non-smooth constraint) γ ( w ) := I [0 , ( w ) g ( w, η ) := − c (cid:18) w − u − (cid:19) + 12 ( w − η ) for all w, η ∈ R ,and therefore w is constrained to the compact interval [0 ,
1] (cf. [9, 10, 22, 33, 38]).3ere, c > K ⊂ R , I K denotes the indicator function on K ,i.e., τ ∈ R I K ( τ ) := ( , if τ ∈ K , ∞ , otherwise.Kobayashi [25] adopted a setting such that • the functions γ and g ( · ) are given in accordance with (g1), • α ( w, η ) = α ( w, η ) := η / β ( w, η ) := w / w, η ] ∈ R . (0.5)Applying this, the original profile of the φ - η - θ model in [25] is described by w t − ∆w + cw ( w − (cid:18) w − u − (cid:19) + ( w − η ) + νw |∇ θ | = 0 in Q , (0.6) η t − ∆η + ( η − w ) + η |∇ θ | = 0 in Q , (0.7) η θ t − div (cid:18) η ∇ θ |∇ θ | + 2 νw ∇ θ (cid:19) = 0 in Q , (0.8)with the initial-boundary conditions as in (0.1)–(0.3).From a mathematical point of view, there do not appear to be great differences be-tween (0.1) and (0.2). However, from the original profiles (0.6)–(0.8), it can be seen that(0.2) corresponds to the equation for the mobilities of grain boundaries (interfaces) as inKobayashi-Warren-Carter [27], while (0.1) is an Allen-Cahn type equation to reproduce“interfacial diffusions,” as in the models of phase transitions.Next, in our second theme, we consider a limiting system (S) ν as ν ց
0, similarly tothe case with ν = 0. This is denoted by (S) and formally described as follows.(S) : w t − ∆w + ∂γ ( w ) + g w ( w, η ) + α w ( w, η ) | Dθ | ∋ Q , ∇ w · ν ∂ Ω = 0 on Σ, w (0 , x ) = w ( x ), x ∈ Ω; (0.9) η t − ∆η + g η ( w, η ) + α η ( w, η ) | Dθ | = 0 in Q , ∇ η · ν ∂ Ω = 0 on Σ, η (0 , x ) = η ( x ), x ∈ Ω; (0.10) α ( w, η ) θ t − div (cid:18) α ( w, η ) Dθ | Dθ | (cid:19) = 0 in Q , α ( w, η ) Dθ | Dθ | · ν ∂ Ω = 0 on Σ, θ (0 , x ) = θ ( x ), x ∈ Ω. (0.11)4aking these together, we set the functional[ w, η, θ ] ∈ [ H (Ω) ∩ L ∞ (Ω)] × [ H (Ω) ∩ L ∞ (Ω)] × BV (Ω) F ( w, η, θ ) := 12 Z Ω |∇ w | dx + 12 Z Ω |∇ η | dx + Z Ω γ ( w ) dx (0.12)+ Z Ω g ( w, η ) dx + Z Ω α ( w, η ) | Dθ | as the corresponding free energy in (S) .Because of the absence of the term νβ ( w, η ), the limiting system (S) may appearto be a simplified version of (S) ν when ν >
0. However, it must be noted that thedefinition (0.12) of the limiting free energy F in this system includes a nontrivial term R Ω α ( w, η ) | Dθ | . The spatial gradients Dθ in (0.9)–(0.11) necessitate much more delicatemathematical treatment than ∇ θ in (0.1)–(0.3).Now, the objective of this paper is to establish a uniform solution method for thesystems (S) ν for all ν ≥
0, including their relaxed versions. We here adopt an analyticalapproach based on time discretization and set the goal to prove two main theorems:
Main Theorem 1.
There exists a solution to (S) ν for each fixed ν > Main Theorem 2.
There exists a solution to (S) .The plan of this paper is as follows: In the next section, we set forth some specificnotation. In Section 2, we state our two main theorems with proper definitions of thesolutions to the respective systems. In Section 3, we present approximating problems forour systems and supply some auxiliary lemmas aimed at the method of obtaining theapproximating solutions. The approximating problems are provided in the forms of timediscretization of (S) ν for ν >
0, and the existence and uniqueness of the approximatingsolutions are proved in the following Section 4. Sections 5 and 6 are devoted to theproofs of Main Theorems 1 and 2, respectively. Finally, we add an Appendix to makesupplementary statements for some preliminary facts and the solutions to our systems.
First we elaborate the notation used throughout.
Notation 1 (real analysis)
For arbitrary a , b ∈ [ −∞ , ∞ ], we define a ∨ b := max { a , b } and a ∧ b := min { a , b } , and for arbitrary −∞ ≤ a ≤ b ≤ ∞ , we define the truncation function (operator) T ba : R → [ a, b ] by letting r ∈ R T ba r := a ∨ ( b ∧ r ) ∈ [ a, b ] . d ∈ N take any fixed value. We denote by | x | and x · y the Euclidean norm of x ∈ R d and the standard scalar product of x, y ∈ R d , respectively, as usual, i.e., | x | := p x + · · · + x d and x · y := x y + · · · + x d y d for all x = [ x , . . . , x d ] , y = [ y , . . . , y d ] ∈ R d .For arbitrary x = [ x , . . . , x d ] ∈ R d and y = [ y , . . . , y d ] ∈ R d , we say that x ≤ y or y ≥ x if x k ≤ y k , for k = 1 , . . . , d .The d -dimensional Lebesgue measure is denoted by L d . Also, unless otherwise speci-fied, the measure-theoretic phrases such as “a.e.,” “ dt ,” “ dx ”, and so on, are with respectto the Lebesgue measure in each corresponding dimension. For a (Lebesgue) measurablefunction f : B → [ −∞ , ∞ ] on a Borel subset B ⊂ R d , we denote by [ f ] + and [ f ] − ,respectively, the positive and negative parts of f , i.e.,[ f ] + ( x ) := T ∞ f ( x ) and [ f ] − ( x ) := − T −∞ f ( x ) , a.e. x ∈ B . Notation 2 (abstract functional analysis)
For an abstract Banach space X , we de-note by | · | X the norm of X , and when X is a Hilbert space, we denote by ( · , · ) X itsinner product. For a subset A of a Banach space X , we denote by int( A ) and A theinterior and the closure of A , respectively.Fix 1 < d ∈ N . Then, for a Banach space X the topology of the product Banach space X d := d times z }| { X × · · · × X has the norm | z | X d := d X k =1 | z k | X , for z = [ z , . . . , z d ] ∈ X d .However, if X is a Hilbert space, then the topology of the product Hilbert space X d hasthe inner product( z, ˜ z ) X d := d X k =1 ( z k , ˜ z k ) X , for z = [ z , . . . , z d ] ∈ X d and ˜ z = [˜ z , . . . , ˜ z d ] ∈ X d ,and, hence the norm in this case is provided by | ζ | X := p ( z, z ) X d = (cid:18) d X k =1 | z k | X (cid:19) / , for z = [ z , . . . , z d ] ∈ X d .For a Banach space X , we denote the dual space by X ∗ . For a single-valued operator A : X → X ∗ , we write A z = [ A z , . . . , A z d ] ∈ [ X ∗ ] d for any z = [ z , . . . , z d ] ∈ X d .For any proper lower semi-continuous (l.s.c. hereafter) and convex function Ψ definedon a Hilbert space X , we denote by D (Ψ) its effective domain and by ∂ Ψ its subdifferential.6he subdifferential ∂ Ψ is a set-valued map corresponding to a weak differential of Ψ, andit has a maximal monotone graph in the product Hilbert space X . More precisely, foreach z ∈ X , the value ∂ Ψ( z ) is defined as the set of all elements z ∗ ∈ X that satisfy thevariational inequality( z ∗ , z − z ) X ≤ Ψ( z ) − Ψ( z ) for any z ∈ D (Ψ),and the set D ( ∂ Ψ) := { z ∈ X | ∂ Ψ( z ) = ∅} is called the domain of ∂ Ψ. We often usethe notation “[ z , z ∗ ] ∈ ∂ Ψ in X ” to mean “ z ∗ ∈ ∂ Ψ( z ) in X with z ∈ D ( ∂ Ψ),” byidentifying the operator ∂ Ψ with its graph in X . Remark 1.1
It is often useful to consider the subdifferentials under time-dependent set-tings. In this regard, several general theories have been established by previous researchers(e.g., Kenmochi [21], and ˆOtani [31]). From these (e.g., [21, Chapter 2]), one can see thefollowing fact: (Fact 1)
Let E be a convex subset in a Hilbert space X , let I ⊂ [0 , ∞ ) be a timeinterval, and for any t ∈ I , let Ψ t : X → ( −∞ , ∞ ] be a proper l.s.c. and convexfunction such that D (Ψ t ) = E for all t ∈ I . Based on this, define a convex functionΨ I : L ( I ; X ) → ( −∞ , ∞ ], by setting ζ ∈ L ( I ; X ) Ψ I ( ζ ) := Z I Ψ t ( ζ ( t )) dt, if Ψ ( · ) ( ζ ) ∈ L ( I ), ∞ , otherwise.Here, if E ⊂ D (Ψ I ), and the function t ∈ I Ψ t ( z ) is integrable for any z ∈ E ,then the following holds:[ ζ , ζ ∗ ] ∈ ∂ Ψ I in L ( I ; X ) if and only if ζ ∈ D (Ψ I ) and [ ζ ( t ) , ζ ∗ ( t )] ∈ ∂ Ψ t in X , a.e. t ∈ I . Notation 3 (basic elliptic operators)
Let F : H (Ω) → H (Ω) ∗ be the duality map-ping, defined as h F ϕ, ψ i ∗ := ( ϕ, ψ ) H (Ω) = ( ϕ, ψ ) L (Ω) + ( ∇ ϕ, ∇ ψ ) L (Ω) N for all ϕ, ψ ∈ H (Ω),where h · , · i ∗ is the duality pairing between H (Ω) and its dual H (Ω) ∗ .Let ∆ N be the Laplacian operator subject to the zero Neumann boundary condition,i.e., ∆ N : z ∈ D N := (cid:8) z ∈ H (Ω) ∇ z · ν ∂ Ω = 0 in L ( ∂ Ω) (cid:9) ⊂ L (Ω) ∆z ∈ L (Ω) . As is well known,
F z = − ∆ N z + z in L (Ω) if z ∈ D N . (1.1)7 emark 1.2 We here show a representative example of the subdifferential. Let d ∈ N befixed, and let V d D : L (Ω) d → [0 , ∞ ] be a proper l.s.c. and convex function of the so-calledDirichlet type integral, i.e., z ∈ L (Ω) d V d D ( z ) := Z Ω |∇ z | R d × N dx, if z ∈ H (Ω) d , ∞ , otherwise. (1.2)Then, with regard to the subdifferential ∂V d D ⊂ [ L (Ω) d ] , it is known (see, e.g., [5] or [7])that z ∈ L (Ω) d ∂V d D ( z ) = ( {− ∆ N z } , if z ∈ D d N , ∅ , otherwise. (1.3)In this light, ∂V d D and − ∆ N are identified as the maximal monotone graphs in [ L (Ω) d ] . Notation 4 (BV theory; cf. [3, 4, 13, 15])
Let d ∈ N , and let U ⊂ R d be an openset. We denote by M ( U ) the space of all finite Radon measures on U . The space M ( U )is known as the dual space of the Banach space C ( U ), i.e., M ( U ) = C ( U ) ∗ , where C ( U ) denotes the closure of the space C c ( U ) of all continuous functions having compactsupports, in the topology of C ( U ).A function z ∈ L ( U ) is called a function of bounded variation on U (or simply z ∈ BV ( U )) if and only if its distributional gradient Dz is a finite Radon measure on U ,namely, Dz ∈ M ( U ) d . Here, for any z ∈ BV ( U ) the Radon measure Dz is called the variation measure of z , and its total variation | Dz | is similarly the total variation measure of z . Additionally, | Dz | ( U ) = sup (cid:26) Z U z div ϕ dx ϕ ∈ C ( U ) N and | ϕ | ≤ U (cid:27) . The space BV ( U ) is a Banach space, with the norm | z | BV ( U ) := | z | L ( U ) + | Dz | ( U ) for any z ∈ BV ( U ).Additionally, we say that z n → z “weakly- ∗ ” in BV ( U ) if z ∈ BV ( U ), { z n | n ∈ N } ⊂ BV ( U ), z n → z in L ( U ), and Dz n → Dz weakly- ∗ in M ( U ) as n → ∞ .The space BV ( U ) has another topology, called “strict topology,” which has the fol-lowing distance (cf. [3, Definition 3.14]):[ ϕ, ψ ] ∈ BV ( U )
7→ | ϕ − ψ | L ( U ) + (cid:12)(cid:12) | Dϕ | ( U ) − | Dψ | ( U ) (cid:12)(cid:12) . In this regard, we say that z n → z strictly in BV ( U ) if z ∈ BV ( U ), { z n | n ∈ N } ⊂ BV ( U ), z n → z in L ( U ), and | Dz n | ( U ) → | Dz | ( U ) as n → ∞ .Specifically, when the boundary ∂U is Lipschitz, the Banach space BV ( U ) is con-tinuously embedded into L d/ ( d − ( U ) and compactly embedded into L p ( U ) for any 1 ≤ p < d/ ( d −
1) (cf. [3, Corollary 3.49] or [4, Theorems 10.1.3–10.1.4]). Additionally, if1 ≤ q < ∞ , then the space C ∞ ( U ) is dense in BV ( U ) ∩ L q ( U ) for the intermediate con-vergence (cf. [4, Definition 10.1.3 and Theorem 10.1.2]), i.e., for any z ∈ BV ( U ) ∩ L q ( U )there exists a sequence { z n | n ∈ N } ⊂ C ∞ ( U ) such that z n → z in L q ( U ) and strictly in BV ( U ) as n → ∞ . 8 otation 5 (weighted total variation; cf. [2, 3]) In this paper, we define X c (Ω) := { ̟ ∈ L ∞ (Ω) N | div ̟ ∈ L (Ω) and supp ̟ is compact in Ω } ,W (Ω) := (cid:8) ̺ ∈ H (Ω) ∩ L ∞ (Ω) ̺ ≥ (cid:9) ,W c (Ω) := (cid:26) ̺ ∈ H (Ω) ∩ L ∞ (Ω) there exists c ̺ > ̺ ≥ c ̺ a.e. in Ω (cid:27) , (1.4)and for any ̺ ∈ W (Ω) and any z ∈ L (Ω), we call the value Var ̺ ( z ) ∈ [0 , ∞ ], defined as,Var ̺ ( v ) := sup (cid:26) Z Ω v div ̟ dx ̟ ∈ X c and | ̟ | ≤ ̺ a.e. in Ω (cid:27) ∈ [0 , ∞ ] , “the total variation of v weighted by ̺ ,” or the “weighted total variation” for short. Remark 1.3
Referring to the general theories (e.g., [2, 3, 6]), we can confirm the followingfacts associated with the weighted total variations: (Fact 2) (cf. [6, Theorem 5]) For any ̺ ∈ W (Ω), the functional z ∈ L (Ω) Var ̺ ( z ) ∈ [0 , ∞ ] is a proper l.s.c. and convex function that coincides with the lower semi-continuous envelope of z ∈ W , (Ω) ∩ L (Ω) Z Ω ̺ |∇ z | dx ∈ [0 , ∞ ) . (Fact 3) (cf. [2, Theorem 4.3] and [3, Proposition 5.48]) If ̺ ∈ W (Ω) and z ∈ BV (Ω) ∩ L (Ω), then there exists a Radon measure | Dz | ̺ ∈ M (Ω) such that | Dz | ̺ (Ω) = Z Ω d | Dz | ̺ = Var ̺ ( z )and | Dz | ̺ ( A ) ≤ | ̺ | L ∞ (Ω) | Dz | ( A ), | Dz | ̺ ( A ) = inf lim inf n →∞ Z A ̺ |∇ ˜ z n | dx { ˜ z n | n ∈ N } ⊂ W , ( A ) ∩ L ( A )such that ˜ z n → z in L ( A ) as n → ∞ (1.5)for any open set A ⊂ Ω. (Fact 4) If ̺ ∈ W c (Ω) and z ∈ BV (Ω) ∩ L (Ω), then for any open set A ⊂ Ω, it followsthat | Dv | ̺ ( A ) ≥ c ̺ | Dz | ( A ) for any open set A ⊂ Ω, D (Var ̺ ) = BV (Ω) ∩ L (Ω), andVar ̺ ( z ) = sup (cid:26) Z Ω z div ( ̺ϕ ) dx ϕ ∈ X c (Ω) and | ϕ | ≤ (cid:27) , (1.6)where c ̺ is a constant as in (1.4). 9oreover, the following properties can be inferred from (1.5)–(1.6): • | Dz | c = c | Dz | in M (Ω) for any constant c ≥ z ∈ BV (Ω) ∩ L (Ω); • | Dz | ̺ = ̺ |∇ z | L N in M (Ω), if ̺ ∈ W (Ω) and z ∈ W , (Ω) ∩ L (Ω). Notation 6 (generalized weighted total variation; cf. [29, Section 2])
For any ̺ ∈ H (Ω) ∩ L ∞ (Ω) and any z ∈ BV (Ω) ∩ L (Ω) we define a real-valued Radon measure[ ̺ | Dz | ] ∈ M (Ω), as follows:[ ̺ | Dz | ]( B ) := | Dz | [ ̺ ] + ( B ) − | Dz | [ ̺ ] − ( B ) for any Borel set B ⊂ Ω.Note that [ ̺ |∇ z | ](Ω) can be thought of as a generalized version of the total variation of z ∈ BV (Ω) ∩ L (Ω) weighted by the possibly sign-changing weight ̺ ∈ H (Ω) ∩ L ∞ (Ω).So, hereafter, we simply refer to [ ̺ | Dz | ](Ω) as the generalized weighted total variation . Remark 1.4
With regard to the generalized weighted total variations, the following factsare verified in [29, Section 2]: (Fact 5) (strict approximation) Let ̺ ∈ H (Ω) ∩ L ∞ (Ω) and z ∈ BV (Ω) ∩ L (Ω) bearbitrary fixed functions, and let { z n | n ∈ N } ⊂ C ∞ (Ω) be any sequence such that z n → z in L (Ω) and strictly in BV (Ω) as n → ∞ .Then Z Ω ̺ |∇ z n | dx → Z Ω d [ ̺ | Dz | ] as n → ∞ . (Fact 6) For any z ∈ BV (Ω) ∩ L (Ω), the mapping ̺ ∈ H (Ω) ∩ L ∞ (Ω) Z Ω d [ ̺ | Dz | ] ∈ R is a linear functional, and moreover, if ϕ ∈ H (Ω) ∩ C (Ω) and ̺ ∈ H (Ω) ∩ L ∞ (Ω),then Z Ω d [ ϕ̺ | Dz | ] = Z Ω ϕ d [ ̺ | Dz | ] . Notation 7 (specific classes of functions)
Let X be a Banach space, defined as X := H (Ω) × H (Ω) × BV (Ω) . For any 1 ≤ p ≤ ∞ and any open interval I ⊂ R , we set L p ( I ; BV (Ω)) := (cid:26) ˜ θ ˜ θ : I → BV (Ω) is measurable for the stricttopology of BV (Ω), and | ˜ θ ( · ) | BV (Ω) ∈ L p ( I ) (cid:27) and L p ( I ; X ) := (cid:8) [ ˜ w, ˜ η, ˜ θ ] ˜ w, ˜ η ∈ L p ( I ; H (Ω)) and ˜ θ ∈ L p ( I ; BV (Ω)) (cid:9) . ≤ p ≤ ∞ and any open interval I ⊂ R , we note that L p ( I ; BV (Ω)) and L p ( I ; X ) are normed spaces, with | ˜ θ | L p ( I ; BV (Ω)) := (cid:12)(cid:12) | ˜ θ ( · ) | BV (Ω) (cid:12)(cid:12) L p ( I ) , for ˜ θ ∈ L p ( I ; BV (Ω))and | [ ˜ w, ˜ η, ˜ θ ] | L p ( I ; X ) := | ˜ w | L p ( I ; H (Ω)) + | ˜ η | L p ( I ; H (Ω)) + | ˜ θ | L p ( I ; BV (Ω)) for [ ˜ w, ˜ η, ˜ θ ] ∈ L p ( I ; X ),respectively.Finally, we mention the notion of functional convergences. Definition 1.1 (Mosco convergence; cf. [30])
Let X be an abstract Hilbert space.Let Ψ : X → ( −∞ , ∞ ] be a proper l.s.c. and convex function, and let { Ψ n | n ∈ N } be asequence of proper l.s.c. and convex functions Ψ n : X → ( −∞ , ∞ ], n ∈ N . We say thatΨ n → Ψ on X , in the sense of Mosco [30], as n → ∞ , if and only if the following twoconditions are fulfilled:(m1) (lower bound) lim inf n →∞ Ψ n ( z † n ) ≥ Ψ( z † ) if z † ∈ X , { z † n | n ∈ N } ⊂ X , and z † n → z † weakly in X as n → ∞ ;(m2) (optimality) for any z ‡ ∈ D (Ψ), there exists a sequence { z ‡ n | n ∈ N } ⊂ X such that z ‡ n → z ‡ in X and Ψ n ( z ‡ n ) → Ψ( z ‡ ) as n → ∞ . Definition 1.2 ( Γ -convergence; cf. [11]) Let X be an abstract Hilbert space, Ψ : X → ( −∞ , ∞ ] be a proper functional, and { Ψ n | n ∈ N } be a sequence of proper function-als Ψ n : X → ( −∞ , ∞ ], n ∈ N . We say that Ψ n → Ψ on X , in the sense of Γ-convergence[11], as n → ∞ if and only if the following two conditions are fulfilled:( γ
1) (lower bound) lim inf n →∞ Ψ n ( z † n ) ≥ Ψ( z † ) if z † ∈ X , { z † n | n ∈ N } ⊂ X , and z † n → z † (strongly) in X as n → ∞ ;( γ
2) (optimality) for any z ‡ ∈ D (Ψ), there exists a sequence { z ‡ n | n ∈ N } ⊂ X such that z ‡ n → z ‡ in X and Ψ n ( z ‡ n ) → Ψ( z ‡ ) as n → ∞ . Remark 1.5
Note that if the functionals are convex, then Mosco convergence implies Γ-convergence, i.e., the Γ-convergence of convex functions can be regarded as a weak versionof Mosco convergence. Additionally, in the Γ-convergence of convex functions, we can seethe following: (Fact 7)
Let Ψ : X → ( −∞ , ∞ ] and Ψ n : X → ( −∞ , ∞ ] be proper l.s.c. and convex func-tions on a Hilbert space X such that Ψ n → Ψ on X , in the sense of Γ-convergence,as n → ∞ . Assume that ( [ z, z ∗ ] ∈ X , [ z n , z ∗ n ] ∈ ∂ Ψ n in X , n ∈ N , z n → z in X and z ∗ n → z ∗ weakly in X , as n → ∞ .It then holds that [ z, z ∗ ] ∈ ∂ Ψ in X and Ψ n ( z n ) → Ψ( z ) as n → ∞ .11 Statement of the main theorems
We begin by reiterating the assumptions in this paper.(A1) γ : R → [0 , ∞ ] is a proper l.s.c. and convex function such that • K γ := D ( γ ) ⊂ R is a closed interval, and K γ ⊃ (0 , • γ ∈ C (int( K γ )), so that the subdifferential ∂γ coincideswith the usual differential γ ′ on int( K γ ).(A2) g ( · , · ) ∈ C ( R ), and there exists a constant c ∗ ∈ R such that γ ( ˜ w ) + g ( ˜ w, ˜ η ) ≥ c ∗ for all ˜ w, ˜ η ∈ [0 , .(A3) α ∈ W , ∞ loc ( R ) is a positive-valued function, and α, β ∈ C ( R ) ∩ C ([0 , ) arenonnegative-valued convex functions.(A4) There exist two constants o ∗ , ι ∗ ∈ R such that { o ∗ , ι ∗ } ⊂ D ( ∂γ ) and 0 ≤ o ∗ < ι ∗ ≤ , and the subdifferential ∂γ and the partial differentials g w ( · , · ) = ∂∂w g ( · , · ), g η ( · , · ) = ∂∂η g ( · , · ) fulfill that \ ˜ η ∈ [0 , (cid:0) ∂γ ( o ∗ ) + g w ( o ∗ , ˜ η ) (cid:1) ∩ ( −∞ , = ∅ and min ˜ w ∈ [0 , g η ( ˜ w, ≤ \ ˜ w ∈ [0 , (cid:0) ∂γ ( ι ∗ ) + g w ( ι ∗ , ˜ w ) (cid:1) ∩ [0 , ∞ ) = ∅ and max ˜ w ∈ [0 , g η ( ˜ w, ≥
0. (2.1)Furthermore, the partial differentials α w = ∂α∂w , α η = ∂α∂η , β w = ∂β∂w and β η = ∂β∂η fulfill that sup [ ˜ w, ˜ η ] ∈ R (cid:8) α w ( o ∗ , ˜ η ) , α η ( ˜ w, , β w ( o ∗ , ˜ η ) , β η ( ˜ w, (cid:9) ≤ [ ˜ w, ˜ η ] ∈ R (cid:8) α w ( ι ∗ , ˜ η ) , α η ( ˜ w, , β w ( ι ∗ , ˜ η ) , β η ( ˜ w, (cid:9) ≥
0. (2.2)
Remark 2.1
Assumptions (A1)–(A4) cover all of the settings presented in (g1)–(g3) and(0.5). In particular, we note that assumptions (A3)–(A4) encompass more interactionalfunctions, such as[ ˜ w, ˜ η ] ∈ R (cid:0) | ˜ w − o ∗ | p + | ˜ η | q (cid:1) r with constants p, q, r > α and β .Next, for descriptive convenience, we introduce the following abbreviations.12 otation 8 For any ˜ v = [ ˜ w, ˜ η ] ∈ R , we abbreviate α ( ˜ w, ˜ η ), α ( ˜ w, ˜ η ), and β ( ˜ w, ˜ η ) by α (˜ v ), α (˜ v ), and β (˜ v ), respectively, and we set [ ∇ g ](˜ v ) = [ ∇ g ]( ˜ w, ˜ η ) := [ g w ( ˜ w, ˜ η ) , g η ( ˜ w, ˜ η )][ ∇ α ](˜ v ) = [ ∇ α ]( ˜ w, ˜ η ) := [ α w ( ˜ w, ˜ η ) , α η ( ˜ w, ˜ η )][ ∇ β ](˜ v ) = [ ∇ β ]( ˜ w, ˜ η ) := [ β w ( ˜ w, ˜ η ) , β η ( ˜ w, ˜ η )] for ˜ v = [ ˜ w, ˜ η ] ∈ R .For any ˜ v = [ ˜ w, ˜ η ] ∈ [ H (Ω) ∩ L ∞ (Ω)] , let Φ ν (˜ v ; · ) = Φ ν ( ˜ w, ˜ η ; · ) be a proper l.s.c. andconvex function on L (Ω) defined as ϑ ∈ L (Ω) Φ ν (˜ v ; ϑ ) = Φ ν ( ˜ w, ˜ η ; ϑ ):= Z Ω α (˜ v ) |∇ ϑ | dx + ν Z Ω β (˜ v ) |∇ ϑ | dx, if ν > ϑ ∈ H (Ω), Z Ω d [ α (˜ v ) | Dϑ | ] , if ν = 0 and ϑ ∈ BV (Ω), ∞ , otherwise,and let ∂ Φ ν (˜ v ; · ) = ∂ Φ ν ( ˜ w, ˜ η ; · ) be the L -subdifferential of Φ ν (˜ v ; · ) = Φ ν ( ˜ w, ˜ η ; · ). Remark 2.2
By virtue of Notation 8, we can uniformly provide proper definitions of thefree energies in (0.4) and (0.12) by assigning[˜ v, ˜ θ ] = [ ˜ w, ˜ η, ˜ θ ] ∈ L (Ω) F ν (˜ v, ˜ θ ) = F ν ( ˜ w, ˜ η, ˜ θ ):= V ( ˜ w, ˜ η ) + Γ(˜ v ) + G (˜ v ) + Φ ν (˜ v ; ˜ θ ) for all ν ≥
0. (2.3)In this context,– V is the functional given in (1.2) for the case d = 2;– Γ is a proper l.s.c. and convex function on L (Ω) , defined as˜ v = [ ˜ w, ˜ η ] ∈ L (Ω) Γ(˜ v ) = Γ( ˜ w, ˜ η ) := Z Ω γ ( ˜ w ) dx ∈ ( −∞ , ∞ ]; (2.4)– G ( · ) is a functional on L (Ω) given by˜ v = [ ˜ w, ˜ η ] ∈ L (Ω) G (˜ v ) = G ( ˜ w, ˜ η ) := Z Ω g ( ˜ w, ˜ η ) dx ∈ R . The main theorems can now be stated.
Main Theorem 1 (existence for (S) ν when ν > Fix the constant ν > and as-sume (A1)–(A4). Additionally, assume that δ := inf (cid:8) α ( ˜ w, ˜ η ) , β ( ˜ w, ˜ η ) [ ˜ w, ˜ η ] ∈ R (cid:9) > nd [ w , η , θ ] ∈ D := [ ˜ w, ˜ η, ˜ θ ] ∈ H (Ω) o ∗ ≤ ˜ w ≤ ι ∗ a.e. in Ω , ≤ ˜ η ≤ a.e. in Ω ,and ˜ θ ∈ L ∞ (Ω) . (2.6) The system (S) ν then admits at least one solution [ w, η, θ ] , defined by the following con-ditions.(S0) ν [ w, η, θ ] ∈ W , (0 , T ; L (Ω) ) ∩ L ∞ (0 , T ; H (Ω) ) ∩ L ∞ ( Q ) , o ∗ ≤ w ≤ ι ∗ , ≤ η ≤ ,and | θ | ≤ | θ | L ∞ (Ω) , a.e. in Q .(S1) ν w solves (0.1) in the following variational sense: Z Ω (cid:0) w t ( t ) + g w ( w, η )( t ) (cid:1) ( w ( t ) − ϕ ) dx + Z Ω ∇ w ( t ) · ∇ ( w ( t ) − ϕ ) dx + Z Ω (cid:0) α w ( w, η )( t ) |∇ θ ( t ) | + νβ w ( w, η )( t ) |∇ θ ( t ) | (cid:1) ( w ( t ) − ϕ ) dx + Z Ω γ ( w ( t )) dx ≤ Z Ω γ ( ϕ ) dx for any ϕ ∈ H (Ω) ∩ L ∞ (Ω) and a.e. t ∈ (0 , T ) , (2.7) with the initial condition w (0) = w in L (Ω) .(S2) ν η solves (0.2) in the following variational sense: Z Ω (cid:0) η t ( t ) + g η ( w, η )( t ) (cid:1) ψ dx + Z Ω ∇ η ( t ) · ∇ ψ dx + Z Ω (cid:0) α η ( w, η )( t ) |∇ θ ( t ) | + νβ η ( w, η )( t ) |∇ θ ( t ) | (cid:1) ψ dx = 0 for any ψ ∈ H (Ω) ∩ L ∞ (Ω) and a.e. t ∈ (0 , T ) , (2.8) with the initial condition η (0) = η in L (Ω) .(S3) ν θ solves (0.3) in the following variational sense: Z Ω α ( w, η )( t ) θ t ( t ) ( θ ( t ) − ω ) dx + 2 ν Z Ω β ( w, η )( t ) ∇ θ ( t ) · ∇ ( θ ( t ) − ω ) dx + Z Ω α ( w, η )( t ) |∇ θ ( t ) | dx ≤ Z Ω α ( w, η )( t ) |∇ ω | dx for any ω ∈ H (Ω) and a.e. t ∈ (0 , T ) , (2.9) with the initial condition θ (0) = θ in L (Ω) . ain Theorem 2 (existence for (S) ) Assume (A1)–(A4), and in addition, assumethat δ := inf (cid:8) α ( ˜ w, ˜ η ) , α ( ˜ w, ˜ η ) [ ˜ w, ˜ η ] ∈ R (cid:9) > and [ w , η , θ ] ∈ D := [ ˜ w, ˜ η, ˜ θ ] ∈ X o ∗ ≤ ˜ w ≤ ι ∗ a.e. in Ω , ≤ ˜ η ≤ a.e. in Ω ,and ˜ θ ∈ L ∞ (Ω) . (2.11) The system (S) then admits at least one solution [ w, η, θ ] , defined by the following con-ditions.(S0) [ w, η, θ ] ∈ W , (0 , T ; L (Ω) ) ∩ L ∞ (0 , T ; X ) ∩ L ∞ ( Q ) , o ∗ ≤ w ≤ ι ∗ , ≤ η ≤ ,and | θ | ≤ | θ | L ∞ (Ω) , a.e. in Q .(S1) w solves (0.9) in the following variational sense: Z Ω (cid:0) w t ( t ) + g w ( w, η )( t ) (cid:1) ( w ( t ) − ϕ ) dx + Z Ω ∇ w ( t ) · ∇ ( w ( t ) − ϕ ) dx + Z Ω d (cid:2) ( w ( t ) − ϕ ) α w ( w, η )( t ) | Dθ ( t ) | (cid:3) + Z Ω γ ( w ( t )) dx ≤ Z Ω γ ( ϕ ) dx for any ϕ ∈ H (Ω) ∩ L ∞ (Ω) and a.e. t ∈ (0 , T ) , (2.12) with the initial condition w (0) = w in L (Ω) .(S2) η solves (0.10) in the following variational sense: Z Ω (cid:0) η t ( t ) + g η ( w, η )( t ) (cid:1) ψ dx + Z Ω ∇ η ( t ) · ∇ ψ dx + Z Ω d (cid:2) ψα η ( w, η )( t ) | Dθ ( t ) | (cid:3) = 0 for any ψ ∈ H (Ω) ∩ L ∞ (Ω) and a.e. t ∈ (0 , T ) , (2.13) with the initial condition η (0) = η in L (Ω) .(S3) θ solves (0.11) in the following variational sense: Z Ω α ( w, η )( t ) θ t ( t ) ( θ ( t ) − ω ) dx + Z Ω d [ α ( w, η )( t ) | Dθ ( t ) | ] ≤ Z Ω d [ α ( w, η )( t ) | Dω | ] for any ω ∈ BV (Ω) ∩ L (Ω) and a.e. t ∈ (0 , T ) , (2.14) with the initial condition θ (0) = θ in L (Ω) . emark 2.3 Hereafter, whenever ν ≥
0, we set v := [ w, η ] in C ([0 , T ]; L (Ω) ), for the solution [ w, η, θ ] to (S) ν .Thus, if ν >
0, then the variational inequalities (2.7)–(2.8) for v = [ w, η ] can be unifiedas follows:( v t ( t ) , v ( t ) − ̟ ) L (Ω) + ([ ∇ g ]( v ( t )) , v ( t ) − ̟ ) L (Ω) + ( ∇ w ( t ) , ∇ ( w ( t ) − ϕ )) L (Ω) N + ( ∇ η ( t ) , ∇ ( η ( t ) − ψ )) L (Ω) N + Z Ω (cid:0) |∇ θ ( t ) | [ ∇ α ]( v ( t )) + ν |∇ θ ( t ) | [ ∇ β ]( v ( t )) (cid:1) · ( v ( t ) − ̟ ) dx +Γ( v ( t )) ≤ Γ( ̟ ) for any ̟ = [ ϕ, ψ ] ∈ [ H (Ω) ∩ L ∞ (Ω)] . (2.15)Meanwhile, if ν = 0, then the corresponding variational inequalities (2.12)–(2.13) can beunified as ( v t ( t ) , v ( t ) − ̟ ) L (Ω) + ([ ∇ g ]( v ( t )) , v ( t ) − ̟ ) L (Ω) + ( ∇ w ( t ) , ∇ ( w ( t ) − ϕ )) L (Ω) N + ( ∇ η ( t ) , ∇ ( η ( t ) − ψ )) L (Ω) N + Z Ω d (cid:2)(cid:0) ( v ( t ) − ̟ ) · [ ∇ α ]( v ( t )) (cid:1) | Dθ ( t ) | (cid:3) +Γ( v ( t )) ≤ Γ( ̟ ) for any ̟ = [ ϕ, ψ ] ∈ [ H (Ω) ∩ L ∞ (Ω)] . (2.16)Moreover, for every ν ≥ θ , i.e., (2.9) when ν > ν = 0, uniformly reduce to the following form of an evolution equation: α ( v ( t )) θ t ( t ) + ∂ Φ ν ( v ( t ); θ ( t )) ∋ L (Ω), a.e. t ∈ (0 , T ), (2.17)governed by the subdifferentials ∂ Φ ν ( v ( t ); · ) of unknown-dependent convex functionsΦ ν ( v ( t ); · ) for t ∈ [0 , T ]. Remark 2.4
In fact, reductions similar to (2.17) are available for the variational inequal-ities for v = [ w, η ] in some special cases. For instance, if K γ = ( −∞ , ∞ ), then by (A1) thefunctional Γ becomes locally Lipschitz on L (Ω) , and the subdifferential ∂ Γ ⊂ [ L (Ω) ] coincides with the Fr´echet differential Γ ′ . Therefore, with (1.1), (1.3), (Fact 6) in Remark1.4 and [7, Chapter 2] in mind, the variational inequalities (2.7)–(2.8) and (2.12)–(2.13)can be reduced to the following evolution equation form: v t ( t ) + (cid:0) F v ( t ) − v ( t ) (cid:1) +Γ ′ ( v ( t )) + [ ∇ g ]( v ( t )) + [ | Dθ ( t ) | [ ∇ α ]( v ( t ))]+ |∇ ( νθ )( t ) | [ ∇ β ]( v ( t )) = 0 in [ H s (Ω) ∗ ] , a.e. t ∈ (0 , T ),where s > N/ H s (Ω) ⊂ C (Ω), and for any σ = [ ξ, ζ ] ∈ [ H (Ω) ∩ L ∞ (Ω)] and any ϑ ∈ BV (Ω) ∩ L (Ω), [ | Dϑ | σ ] denotes a (vectorial)Radon measure, defined as ̟ := [ ϕ, ψ ] ∈ C (Ω) Z Ω ̟ d [ | Dϑ | σ ] := Z Ω ϕ d [ ξ | Dϑ | ] + Z Ω ψ d [ ζ | Dϑ | ] ∈ R . However, it must be noted that such reductions may not be valid for general instances ofthe convex function γ . 16 Approximating problems and auxiliary lemmas
In this section, the approximating problems for our systems are presented along withrelaxed versions. As mentioned in the introduction, these problems are formulated astime discretization systems for (S) ν when ν >
0. Hence, throughout the descriptionof the approximating problems, we fix ν as a positive constant and assume (2.5) as inMain Theorem 1. Additionally, for any time step 0 < h < νh theapproximating problem for (S) ν , prescribed as follows:(AP) νh For any initial value[ v ν , θ ν ] = [ w ν , η ν , θ ν ] ∈ D with v ν = [ w ν , η ν ] ∈ H (Ω) , (3.1)find a sequence of triplets { [ v νi , θ νi ] = [ w νi , η νi , θ νi ] | i ∈ N } ⊂ D with v νi = [ w νi , η νi ] ∈ H (Ω) , i ∈ N ,such that1 h (cid:0) v νi − v νi − , v νi − ̟ (cid:1) L (Ω) + ([ ∇ g ]( v νi ) , v νi − ̟ ) L (Ω) + (cid:0) ∇ w νi , ∇ ( w νi − ϕ ) (cid:1) L (Ω) N + (cid:0) ∇ η νi , ∇ ( η νi − ψ ) (cid:1) L (Ω) N + Z Ω (cid:0) |∇ θ νi − | [ ∇ α ]( v νi ) + ν |∇ θ νi − | [ ∇ β ]( v νi ) (cid:1) · ( v νi − ̟ ) dx +Γ( v νi ) ≤ Γ( ̟ ) , for any ̟ = [ ϕ, ψ ] ∈ [ H (Ω) ∩ L ∞ (Ω)] ∩ D (Γ), (3.2)1 h (cid:0) α ( v νi )( θ νi − θ νi − ) , θ νi − ω (cid:1) L (Ω) + Φ ν ( v νi ; θ νi ) ≤ Φ ν ( v νi ; ω ) , for any ω ∈ H (Ω), (3.3)and | θ νi | ≤ | θ νi − | L ∞ (Ω) , a.e. in Ω, (3.4)for i = 1 , , , . . . .We call the sequence { [ v νi , θ νi ] = [ w νi , η νi , θ νi ] | i ∈ N } ⊂ D the solution to (AP) νh , or theapproximating solution for short.Let us fix the time step 0 < h <
1. Then, our immediate task is to demonstrate thefollowing theorem about the solvability of the problem (AP) νh . Theorem 1 (solvability of the approximating problem)
There exists a small con-stant h †∗ ∈ (0 , such that for < h < h †∗ , the approximating problem (AP) νh admitsa unique solution { [ v νi , θ νi ] = [ w νi , η νi , θ νi ] | i ∈ N } ⊂ D , starting from any given initialvalue [ v ν , θ ν ] = [ w ν , η ν , θ ν ] ∈ D . Moreover, for < h < h †∗ , the approximating solution { [ v νi , θ νi ] } fulfills the following inequality of energy dissipation: h | v νi − v νi − | L (Ω) + 1 h | p α ( v νi )( θ νi − θ νi − ) | L (Ω) + F ν ( v νi , θ νi ) ≤ F ν ( v νi − , θ νi − ) , i = 1 , , , . . . . (3.5)17he proof of Theorem 1 will be quite extended, because some regularizations will beneeded to relax the L -terms ν |∇ θ νi − | [ ∇ β ]( v νi ), i ∈ N , in (3.2). In view of this, weintroduce one more relaxation index 0 < ε <
1, and we fix a large number M ∈ N satisfying M > ( N + 2) /
2. Additionally, for any ˜ v = [ ˜ w, ˜ η ] ∈ [ H (Ω) ∩ L ∞ (Ω)] , we definea relaxed convex function Ψ νε (˜ v ; · ) on L (Ω) by setting ϑ ∈ L (Ω) Ψ νε (˜ v ; ϑ ) = Ψ νε ( ˜ w, ˜ η ; ϑ ) := Φ ν ( ˜ w, ˜ η ; ϑ ) + ε | ϑ | H M (Ω) ∈ [0 , ∞ ] . Also, for any 0 < ε < v ∈ [ H (Ω) ∩ L ∞ (Ω)] , we denote by ∂ Ψ νε (˜ v ; · ) thesubdifferential of Ψ νε (˜ v ; · ) in the topology of L (Ω). As can easily be verified, Ψ νε (˜ v ; · ) isproper l.s.c and convex on L (Ω), and D (Ψ νε (˜ v ; · )) = H M (Ω) ⊂ W , ∞ (Ω) (3.6)for all 0 < ε < v ∈ L ∞ (Ω) . Based on this, we define a relaxed energy functional E νε on L (Ω) , by setting[˜ v, ˜ θ ] = [ ˜ w, ˜ η, ˜ θ ] ∈ L (Ω) E νε (˜ v, ˜ θ ) = E νε ( ˜ w, ˜ η, ˜ θ ):= V (˜ v ) + Γ(˜ v ) + G (˜ v ) + Ψ νε (˜ v ; ˜ θ ) for any 0 < ε < < ε <
1, we denote by (RX ε ) νh the relaxed system for (AP) νh prescribedas follows.(RX ε ) νh : For any initial value[ v νε, , θ ν ] = [ w νε, , η νε, , θ νε, ] ∈ D M := D ∩ [ H (Ω) × H M (Ω)]with v νε, = [ w νε, , η νε, ] ∈ H (Ω) ,find a sequence of triplets { [ v νε,i , θ νε,i ] = [ w νε,i , η νε,i , θ νε,i ] | i ∈ N } ⊂ D M with v νε,i = [ w νε,i , η νε,i ] ∈ H (Ω) for i = 1 , , , . . . such that1 h ( v νε,i − v νε,i − ) − ∆ N v νε,i + ∂ Γ( v ε,i ) + [ ∇ g ]( v νε,i )+ |∇ θ νε,i − | [ ∇ α ]( v νε,i ) + ν |∇ θ νε,i − | [ ∇ β ]( v νε,i ) ∋ L (Ω) (3.7)and 1 h α ( v νε,i )( θ νε,i − θ νε,i − ) + ∂ Ψ νε ( v νε,i ; θ νε,i ) ∋ L (Ω), (3.8)for i = 1 , , , . . . . 18 emark 3.1 In the relaxed system (RX ε ) νh , we note that the inclusions (3.7) and (3.8)are relaxed versions of the variational inequalities (3.2) and (3.3), respectively, and theseare expressed in the reduced forms by means of L -subdifferentials. As mentioned inRemark 2.4, such reductions may not be available for (3.7). However, in light of (3.6), weobserve that θ νε,i − ∈ H M (Ω) ⊂ W , ∞ (Ω) for i ∈ N ,and this enables us to suppose that |∇ θ νε,i − | ∈ L ∞ (Ω) and ν |∇ θ νε,i − | [ ∇ β ]( v νi ) ∈ L (Ω) N for i ∈ N .Hence, the relaxed system (RX ε ) νh will be in the applicable scope of the general theoriesof L -subdifferentials.Now we fix 0 < ε < ε ) νh . Lemma 3.1
For arbitrary θ † ∈ W , ∞ (Ω) and v † ∈ H (Ω) , consider an auxiliary inclu-sion v − v † h − ∆ N v + ∂ Γ( v ) + [ ∇ g ]( T v )+ |∇ θ † | [ ∇ α ]( v ) + ν |∇ θ † | [ ∇ β ]( v ) ∋ in L (Ω) . (3.9) There exists a small constant h † ∈ (0 , , depending only on | g | C ([0 , ) , and for any < h < h † , the inclusion (3.9) admits a unique solution v ∈ H (Ω) . Proof.
Let S † h : H (Ω) → H (Ω) be an operator that maps any v † ∈ H (Ω) to a uniqueminimizer S † h v † ∈ H (Ω) of a proper l.s.c. and strictly convex function on H (Ω) , definedas ̟ ∈ H (Ω) h | ̟ − v † | L (Ω) + V ( ̟ ) + Γ( ̟ ) + ([ ∇ g ]( T v † ) , ̟ ) L (Ω) + Z Ω (cid:16) |∇ θ † | α ( ̟ ) + ν |∇ θ † | β ( ̟ ) (cid:17) dx ∈ ( −∞ , ∞ ] . On this basis, let us take two functions v † k ∈ H (Ω) ∩ D (Γ), k = 1 ,
2, and consider thesmallness condition on h for S † h to become contractive. From the definition of S † h , thefunctions v k := S † h v † k ∈ H (Ω) , k = 1 ,
2, fulfill v k − v † h − ∆ N v k + ∂ Γ( v k ) + [ ∇ g ]( T v † k )+ |∇ θ † | [ ∇ α ]( v k ) + ν |∇ θ † | [ ∇ β ]( v k ) ∋ L (Ω) , k = 1 ,
2, (3.10)respectively. Here, taking differences between two inclusions in (3.10) and multiplyingboth sides of the result by v − v , we infer from (A1)–(A3) that1 h | v − v | L (Ω) + |∇ ( v − v ) | L (Ω) × N ≤ | g | C ([0 , ) | v † − v † | L (Ω) | v − v | L (Ω) . h | v − v | L (Ω) + |∇ ( v − v ) | L (Ω) × N ≤ h | g | C ([0 , ) | v † − v † | L (Ω) . (3.11)So if we assume that 0 < h < h † := 12(1 ∨ | g | C ([0 , ) ) , (3.12)then it can be seen from (3.11) that S † h becomes a contraction mapping from H (Ω) into itself. Therefore, applying Banach’s fixed-point theorem, we find a unique fixedpoint v †∗ ∈ H (Ω) of S † h under (3.12). The identity v †∗ = S † h v †∗ implies that v †∗ solves theauxiliary inclusion (3.9). Lemma 3.2
Let us assume < h < h † with the constant h † ∈ (0 , as in Lemma 3.1. Let θ † ∈ W , ∞ (Ω) and v † = [ w † , η † ] ∈ H (Ω) be fixed functions, and let v = [ w, η ] ∈ H (Ω) be the unique solution to the auxiliary inclusion (3.9). Let o ∗ , ι ∗ ∈ D ( ∂γ ) be constants asin (A4), and let ˇ r, ˆ r ∈ R be two constant vectors given by ˇ r := [ o ∗ , and ˆ r := [ ι ∗ , . If ˇ r ≤ v † ≤ ˆ r, i.e. v † ∈ [ o ∗ , ι ∗ ] × [0 , , a.e. in Ω , (3.13) then the following ordering property is preserved: ˇ r ≤ v ≤ ˆ r, i.e. v ∈ [ o ∗ , ι ∗ ] × [0 , , a.e. in Ω . Proof.
By (2.1) in (A4), we find two elements ˇ o ∗ ∈ ∂γ ( o ∗ ) and ˆ ι ∗ ∈ ∂γ ( ι ∗ ) such thatˇ o ∗ + g w ( o ∗ , ˜ η ) ≤ ι ∗ + g w ( ι ∗ , ˜ η ) ≥ , for any ˜ η ∈ [0 , r ∗ := [ˇ o ∗ ,
0] and ˆ r ∗ := [ˆ ι ∗ ,
0] in R ,it follows from (2.1)–(2.2) in (A4) and (3.13)–(3.14) that[ˇ r, ˇ r ∗ ] ∈ ∂ Γ and [ˆ r, ˆ r ∗ ] ∈ ∂ Γ in L (Ω) (3.15)and ˇ r − v † h − ∆ N ˇ r + ˇ r ∗ + (cid:20) g w ( o ∗ , T η ) g η ( T w, (cid:21) + |∇ θ † | (cid:20) α w ( o ∗ , η ) α η ( w, (cid:21) + ν |∇ θ † | (cid:20) β w ( o ∗ , η ) β η ( w, (cid:21) ≤ (cid:20) (cid:21) , a.e. in Q , (3.16)ˆ r − v † h − ∆ N ˆ r + ˆ r ∗ + (cid:20) g w ( ι ∗ , T η ) g η ( T w, (cid:21) + |∇ θ † | (cid:20) α w ( ι ∗ , η ) α η ( w, (cid:21) + ν |∇ θ † | (cid:20) β w ( ι ∗ , η ) β η ( w, (cid:21) ≥ (cid:20) (cid:21) , a.e. in Q . (3.17)20ake the difference from (3.16) to (3.9) and multiply both sides of the result by [ˇ r − v ] + .Then, by virtue of (A1)–(A4), (2.4), (3.14)–(3.15),1 h | [ˇ r − v ] + | L (Ω) + |∇ [ˇ r − v ] + | L (Ω) × N ≤ | g | C ([0 , ) | [ˇ r − v ] + | L (Ω) . (3.18)As well as, we also have:1 h | [ v − ˆ r ] + | L (Ω) + |∇ [ v − ˆ r ] + | L (Ω) × N ≤ | g | C ([0 , ) | [ v − ˆ r ] + | L (Ω) , (3.19)by taking the difference from (3.9) to (3.17) and multiplying both sides of the result by[ v − ˆ r ] + .On account of (3.18) (resp. (3.19)), the inequalityˇ r ≤ v a.e. in Ω (resp. v ≤ ˆ r a.e. in Ω)can be inferred by using the assumption h ∈ (0 , h † ). Lemma 3.3
Let h † ∈ (0 , be a constant as in Lemma 3.1, and let o ∗ , ι ∗ ∈ R be constantsas in (A4). Let θ † ∈ W , ∞ (Ω) and v † = [ w † , η † ] ∈ H (Ω) be fixed functions. If < h This lemma is immediately deduced by combining the conclusions of Lemmas 3.1and 3.2. Lemma 3.4 Let v † ∈ H (Ω) and θ † ∈ H M (Ω) be fixed functions. Then the inclusion α ( v † ) θ − θ † h + ∂ Ψ νε ( v † ; θ ) ∋ in L (Ω) (3.20) admits a unique solution θ ∈ H M (Ω) . Proof. The inclusion (3.20) corresponds to the Euler-Lagrange equation for the followingproper l.s.c. and convex function on L (Ω): θ ∈ L (Ω) h | p α ( v † )( θ − θ † ) | L (Ω) + Ψ νε ( v † ; θ ) ∈ [0 , ∞ ] . Since this function is coercive and strictly convex on L (Ω), the lemma is a direct conse-quence of the general theory of convex analysis (cf. [12, Chapter II]).21 emma 3.5 (solvability of the relaxed system) Assume < h < h † := h † / withconstant h † ∈ (0 , as in Lemma 3.1. Then, the relaxed system (RX ε ) νh admits aunique solution { [ v hε,i , θ hε,i ] = [ w hε,i , η hε,i , θ hε,i ] | i ∈ N } ⊂ D M , starting from any initial value [ v hε, , θ hε, ] = [ w hε, , η hε, , θ hε, ] ∈ D M , and moreover, h | v hε,i − v hε,i − | L (Ω) + 1 h (cid:12)(cid:12)(cid:12)q α ( v hε,i )( θ ε,i − θ hε,i − ) (cid:12)(cid:12)(cid:12) L (Ω) + E νε ( v hε,i , θ hε,i ) ≤ E νε ( v hε,i − , θ hε,i − ) , for i = 1 , , , . . . . (3.21) Proof. On the basis of Lemmas 3.3–3.4, we can obtain a unique solution { [ v hε,i , θ hε,i ] =[ w hε,i , η hε,i , θ hε,i ] | i ∈ N } ⊂ H (Ω) × H M (Ω) of the relaxed system (RX ε ) νh as follows:(Step 0) Let i = 1 and fix [ v hε, , θ hε, ] = [ w hε, , η hε, , θ hε, ] ∈ D M .(Step 1) Obtain a unique solution v hε,i = [ w hε,i , η hε,i ] ∈ H (Ω) to (3.7) with the rangeconstraint v ε,i ∈ [ o ∗ , ι ∗ ] × [0 , 1] a.e. in Ω by applying Lemma 3.3 as the case when θ † = θ hε,i − , v † = v hε,i − , and v = v hε,i .(Step 2) Obtain a unique solution θ hε,i ∈ H M (Ω) to (3.20) by applying Lemma 3.4 as thecase when v † = v hε,i , θ † = θ hε,i − , and θ = θ νε,i .(Step 3) Iterate the value of i , i.e., i ← i + 1, and return to step 1.Next we verify the inequality (3.21). Multiply both sides of (3.7) by v hε,i − v hε,i − . Byusing (A1), (2.4), and Young’s inequality, we have1 h | v hε,i − v hε,i − | L (Ω) + 12 |∇ v hε,i | L (Ω) × N − |∇ v hε,i − | L (Ω) × N + Z Ω [ ∇ g ]( v hε,i ) · ( v hε,i − v hε,i − ) dx + Z Ω |∇ θ hε,i − | [ ∇ α ]( θ hε,i ) · ( v hε,i − v hε,i − ) dx + ν Z Ω |∇ θ hε,i − | [ ∇ β ]( θ hε,i ) · ( v hε,i − v hε,i − ) dx +Γ( v hε,i ) − Γ( v hε,i − ) ≤ , for i = 1 , , , . . . . (3.22)Invoking (A2)–(A3), we compute that Z Ω [ ∇ g ]( v hε,i ) · ( v hε,i − v hε,i − ) dx ≥ Z Ω g ( v hε,i ) dx − Z Ω g ( v hε,i − ) dx + Z Ω (cid:0) [ ∇ g ]( v hε,i ) − [ ∇ g ]( v hε,i − ) (cid:1) · ( v hε,i − v hε,i − ) dx − | g | C ([0 , ) | v hε,i − v hε,i − | L (Ω) ≥ G ( v hε,i ) − G ( v hε,i − ) − | g | C ([0 , ) | v hε,i − v hε,i − | L (Ω) (3.23)22nd Z Ω |∇ θ hε,i − | [ ∇ α ]( v hε,i ) · ( v hε,i − v hε,i − ) dx + ν Z Ω |∇ θ hε,i − | [ ∇ β ]( v hε,i ) · ( v hε,i − v hε,i − ) dx ≥ Z Ω α ( v hε,i ) |∇ θ hε,i − | dx − Z Ω α ( v hε,i − ) |∇ θ hε,i − | dx + ν Z Ω β ( v hε,i ) |∇ θ hε,i − | dx − ν Z Ω β ( v hε,i − ) |∇ θ hε,i − | dx (3.24)for i = 1 , , , . . . . On the basis of (3.22)–(3.24), it is deduced that (cid:18) − | g | C ([0 , ) h (cid:19) h | v hε,i − v hε,i − | L (Ω) + V ( v hε,i ) + Γ( v hε,i ) + G ( v hε,i )+ Z Ω α ( v hε,i ) |∇ θ hε,i − | dx + ν Z Ω β ( v hε,i ) |∇ θ hε,i − | dx ≤ V ( v hε,i − ) + Γ( v hε,i − ) + G ( v hε,i − )+ Z Ω α ( v hε,i − ) |∇ θ hε,i − | dx + ν Z Ω β ( v hε,i − ) |∇ θ hε,i − | dx, for i = 1 , , , . . . . (3.25)Meanwhile, by multiplying both sides of (3.8) by θ hε,i − θ hε,i − , it can seen that1 h | q α ( v hε,i )( θ hε,i − θ hε,i − ) | L (Ω) + Z Ω α ( v hε,i ) |∇ θ hε,i | dx + ν Z Ω β ( v hε,i ) |∇ θ hε,i | dx − Z Ω α ( v hε,i ) |∇ θ hε,i − | dx − ν Z Ω β ( v hε,i ) |∇ θ hε,i − | dx + ε | θ hε,i | H M (Ω) ≤ ε | θ hε,i − | H M (Ω) , for i = 1 , , , . . . . (3.26)Since 1 − | g | C ([0 , ) h > , if 0 < h < h † < ∨ | g | C ([0 , ) ) , (3.27)the required inequality (3.21) is obtained by taking the sum of (3.25) and (3.26) andapplying (3.27). In this section, we fix ν > < h < h † with the constant as in Lemma 3.5and prove Theorem 1 concerning the approximating problem (AP) νh . The proof of thistheorem is divided into two parts, which respectively concerned with “the existence” and“the uniqueness and energy dissipation.” 23 xistence of approximating solutions First, we prepare some lemmas for the limiting observations of the relaxed systems(RX ε ) νh as ε ց Lemma 4.1 Assume v † ∈ [ H (Ω) ∩ L ∞ (Ω))] , { v † ε | < ε < } ⊂ [ H (Ω) ∩ L ∞ (Ω))] ,and ( v † ε → v † in the pointwise sense a.e. in Ω as ε ց , { v † ε | < ε < } is bounded in L ∞ (Ω) . (4.1) Then, for the sequence of convex functions { Ψ νε ( v † ε ; · ) | < ε < } , it holds that Ψ νε ( v † ε ; · ) → Φ ν ( v † ; · ) on L (Ω) in the sense of Mosco [30] as ε ց , i.e.,(m1) νε (lower bound) lim inf ε ց Ψ νε ( v † ε ; θ † ε ) ≥ Φ ν ( v † ; θ † ) if θ † ∈ L (Ω) , { θ † ε | < ε < } ⊂ L (Ω) , and θ † ε → θ † weakly in L (Ω) as ε ց ;(m2) νε (optimality) for any θ ‡ ∈ H (Ω) , there exists a sequence { θ ‡ ε | < ε < } ⊂ H M (Ω) such that θ ‡ ε → θ ‡ in L (Ω) and Ψ νε ( v † ε ; θ ‡ ε ) → Φ ν ( v † ; θ ‡ ) as ε ց .Additionally, in light of Remark 1.5, the above Mosco convergence implies Γ -convergenceon L (Ω) as ε ց . Proof. To verify condition (m1) νε , it is enough to consider only the case in whichlim inf ε ց Ψ νε ( v † ε ; θ † ε ) < ∞ , because the other case is trivial. On this basis, we supposethat > ε † > · · · > ε † m ց θ † ε † m → θ † weakly in H (Ω) as m → ∞ ,lim inf ε ց Ψ νε ( v † ε ; θ † ε ) = lim m →∞ Ψ νε † m ( v † ε † m ; θ † ε † m ). (4.2)Then, as a result of (A3), (4.1)–(4.2), and Lebesgue’s dominated convergence theorem, itis inferred that α ( v † ε † m ) ∇ θ † ε † m → α ( v † ) ∇ θ † q β ( v † ε † m ) ∇ θ † ε † m → p β ( v † ) ∇ θ † weakly in L (Ω) N as m → ∞ .Therefore, keeping in mind the lower semi-continuity of the norms, it can be seen thatlim inf ε ց Ψ νε ( v † ε ; θ † ε ) = lim m →∞ Ψ νε † m ( v † ε † m ; θ † ε † m ) ≥ lim inf m →∞ (cid:12)(cid:12) α ( v † ε † m ) ∇ θ † ε † m (cid:12)(cid:12) L (Ω; R N ) + ν lim inf m →∞ (cid:12)(cid:12)q β ( v † ε † m ) ∇ θ † ε † m (cid:12)(cid:12) L (Ω) N ≥ (cid:12)(cid:12) α ( v † ) ∇ θ † (cid:12)(cid:12) L (Ω; R N ) + ν (cid:12)(cid:12)p β ( v † ) ∇ θ † (cid:12)(cid:12) L (Ω) N = Φ ν ( v † ; θ † ) . Thus, we have verified the condition (m1) νε .24ext we prove (m2) νε . For any θ ‡ ∈ H (Ω), let { ˜ θ ‡ m | m ∈ N } ⊂ C ∞ (Ω) be the standardapproximating sequence of θ ‡ , such that˜ θ ‡ m → θ ‡ in H (Ω) as m → ∞ .Also, let us take a sequence { ε ‡ m | m ∈ Z , m ≥ } such that ε ‡ > ε ‡ > · · · > ε ‡ m ց m → ∞ , ε (cid:12)(cid:12) ˜ θ ‡ m (cid:12)(cid:12) H M (Ω) < − m for all 0 < ε < ε ‡ m and m = 1 , , , . . . . (4.3)Considering (4.1), (4.3), and Lebesgue’s dominated convergence theorem, it can be ob-served that α ( v † ε ‡ m ) → α ( v † ) in L (Ω) β ( v † ε ‡ m ) ∇ ˜ θ ‡ m → β ( v † ) ∇ θ ‡ in L (Ω) N as m → ∞ and Φ ν ( v † ε ‡ m ; ˜ θ ‡ m ) = (cid:0) α ( v † ε ‡ m ) , |∇ ˜ θ ‡ m | (cid:1) L (Ω) + ν (cid:0) ∇ ˜ θ ‡ m , β ( v † ε ‡ m ) ∇ ˜ θ ‡ m (cid:1) L (Ω) N → (cid:0) α ( v † ) , |∇ θ ‡ | (cid:1) L (Ω) + ν (cid:0) ∇ θ ‡ , β ( v † ) ∇ θ ‡ (cid:1) L (Ω) N = Φ ν ( v † ; θ ‡ ) , as m → ∞ . (4.4)Now, based on (4.3)–(4.4), the required sequence { θ ‡ ε | < ε < } ⊂ H M (Ω) is con-structed as follows: θ ‡ ε := ˜ θ ‡ m in H M (Ω) if ε ‡ m +1 ≤ ε < ε ‡ m for some m ∈ Z with m ≥ Lemma 4.2 Assume that σ † ∈ [ H (Ω) ∩ L ∞ (Ω)] , { σ † m | m ∈ N } ⊂ [ H (Ω) ∩ L ∞ (Ω)] , { σ † m | m ∈ N } is bounded in L ∞ (Ω) , σ † m → σ † in the pointwise sense, a.e. in Ω , as m → ∞ , (4.5) and ( ω † ∈ H (Ω) , { ω † m | m ∈ N } ⊂ H (Ω) , ω † m → ω † in L (Ω) and Φ ν ( σ † m ; ω † m ) → Φ ν ( σ † ; ω † ) , as m → ∞ . (4.6) Then ω † m → ω † in H (Ω) as m → ∞ . Proof. In light of (A3), (2.5) and (4.6), we may suppose that ω † m → ω † weakly in H (Ω) as m → ∞ (4.7)25y taking a subsequence if necessary. Here, keeping in mind (2.5), (4.5), (4.7), andLebesgue’s dominated convergence theorem, we infer α ( σ † m ) ∇ ω † m → α ( σ † ) ∇ ω † q β ( σ † m ) ∇ ω † m → p β ( σ † ) ∇ ω † √ β ( σ † m ) ∇ ω † m → √ β ( σ † ) ∇ ω † weakly in L (Ω) N as m → ∞ . (4.8)Additionally, it follows from (4.6) thatlim sup m →∞ (cid:12)(cid:12)(cid:12)q β ( σ † m ) ∇ ω † m (cid:12)(cid:12)(cid:12) L (Ω) N = 1 ν h lim m →∞ Φ ν ( σ † m ; ω † m ) − lim inf m →∞ (cid:12)(cid:12) α ( σ † m ) ∇ ω † m (cid:12)(cid:12) L (Ω; R N ) i (4.9) ≤ ν h Φ ν ( σ † ; ω † ) − (cid:12)(cid:12) α ( σ † ) ∇ ω † (cid:12)(cid:12) L (Ω; R N ) i = (cid:12)(cid:12)p β ( σ † ) ∇ ω † (cid:12)(cid:12) L (Ω) N . By virtue of (4.8)–(4.9) and the uniform convexity of the L -topology, it can be seen that q β ( σ † m ) ∇ ω † m → p β ( σ † ) ∇ ω † in L (Ω) N as m → ∞ , (4.10)and hence |∇ ω † m | L (Ω) N = (cid:18)q β ( σ † m ) ∇ ω † m , √ β ( σ † m ) ∇ ω † m (cid:19) L (Ω) N → (cid:18)p β ( σ † ) ∇ ω † , √ β ( σ † ) ∇ ω † (cid:19) L (Ω) N = |∇ ω † | L (Ω) N as m → ∞ . (4.11)The strong convergence of { ω † m } in H (Ω) is demonstrated by taking into account (4.6)–(4.7), (4.11) and the uniform convexity of the L -topology. Remark 4.1 Under the same notation as in Lemma 4.1, let us assume (4.1). Then, forthe sequence { θ ‡ ε | < ε < } as in (m2) νε , we deduce that θ ‡ ε → θ ‡ in H (Ω) and √ εθ ‡ ε → H M (Ω) as ε ց L ∞ -estimate (3.4) as in (AP) νh . Lemma 4.3 (T-monotonicity) Let v † ∈ H (Ω) be a fixed function. Then ( ω ∗ − ω ∗ , [ ω − ω ] + ) L (Ω) ≥ if [ ω k , ω ∗ k ] ∈ ∂ Φ ν ( v † ; · ) in L (Ω) , k = 1 , . (4.12)26 roof. This lemma can be proved by applying the theory of T-monotonicity (cf. [7, 23]).According to the general theory, we need to start by checking thatΦ ν ( v † ; ˜ ω ∧ ˜ ω ) + Φ ν ( v † ; ˜ ω ∨ ˜ ω )= Z Ω α ( v † ) |∇ (˜ ω ∧ ˜ ω ) | dx + ν Z Ω β ( v † ) |∇ (˜ ω ∧ ˜ ω ) | dx + Z Ω α ( v † ) |∇ (˜ ω ∨ ˜ ω ) | dx + ν Z Ω β ( v † ) |∇ (˜ ω ∨ ˜ ω ) | dx = X k =1 (cid:20)Z Ω α ( v † ) |∇ ˜ ω k | dx + ν Z Ω β ( v † ) |∇ ˜ ω k | dx (cid:21) =Φ ν ( v † ; ˜ ω ) + Φ ν ( v † ; ˜ ω ) for all ˜ ω k ∈ H (Ω), k = 1 , ω ∗ − ω ∗ , [ ω − ω ] + ) L (Ω) = ( ω ∗ , ω − ω ∧ ω ) L (Ω) + ( ω ∗ , ω − ω ∨ ω ) L (Ω) ≥ Φ ν ( v † ; ω ) + Φ ν ( v † ; ω ) − (cid:0) Φ ν ( v † ; ω ∧ ω ) + Φ ν ( v † ; ω ∨ ω ) (cid:1) = 0 . Lemma 4.4 Let v † ∈ H (Ω) ∩ L ∞ (Ω) and ˇ θ † , ˆ θ † ∈ H (Ω) be fixed functions, and let [ˇ θ, ˇ θ ∗ ] , [ˆ θ, ˆ θ ∗ ] ∈ L (Ω) be pairs of functions such that [ˇ θ, ˇ θ ∗ ] ∈ ∂ Φ ν ( v † ; · ) in L (Ω) and h α ( v † )(ˇ θ − ˇ θ † ) + ˇ θ ∗ ≤ a.e. in Ω , [ˆ θ, ˆ θ ∗ ] ∈ ∂ Φ ν ( v † ; · ) in L (Ω) and h α ( v † )(ˆ θ − ˆ θ † ) + ˆ θ ∗ ≥ a.e. in Ω , (4.13) respectively. Then | p α ( v † )[ˇ θ − ˆ θ ] + | L (Ω) ≤ | p α ( v † )[ˇ θ † − ˆ θ † ] + | L (Ω) . Moreover, by (2.5), if ˇ θ † ≤ ˆ θ † a.e. in Ω , then ˇ θ ≤ ˆ θ a.e. in Ω . Proof. This lemma is obtained by taking the difference between the inequalities in(4.13), multiplying both sides of the result by [ˇ θ − ˆ θ ] + , and applying Lemma 4.3. Proof of Theorem 1 (existence). Fix the initial value [ v ν , θ ν ] = [ w ν , η ν , θ ν ] ∈ D as in (3.1). Then, in light of Lemmas 4.1-4.2 and Remark 4.1, we can take a sequence { ˜ θ νε, | < ε < } ⊂ H M (Ω) such that˜ θ νε, → θ ν in H (Ω) and Ψ νε ( v ν ; ˜ θ νε, ) → Φ ν ( v ν ; θ ν ) as ε ց 0. (4.14)Based on this, assume0 < h < h †∗ := h † (cid:18) = 14(1 ∨ | g | C ([0 , ) ) (cid:19) (4.15)27nd denote by { [˜ v νε,i , ˜ θ νε,i ] = [ ˜ w νε,i , ˜ η νε,i , ˜ θ νε,i ] | i ∈ N } ⊂ D M the solution to (RX ε ) νh , startingfrom the initial value [ v νε, , θ νε, ] = [ v ν , ˜ θ νε, ]. Then, from (4.14) and Lemma 3.5, it can beseen that { [˜ v νε,i , ˜ θ νε,i ] | i ∈ N } ⊂ D M and this sequence is bounded in H (Ω) . By applyingSobolev’s embedding theorem, we find a sequence { ε n | n ∈ N } ⊂ (0 , 1) and a sequence oftriplets { [ v νi , θ νi ] = [ w νi , η νi , θ νi ] | i ∈ N } ⊂ H (Ω) with v νi = [ w νi , η νi ], i ∈ N such that1 > ε > · · · > ε n ց n → ∞ , ˜ v νn,i = [ ˜ w νn,i , ˜ η νn,i ] := ˜ v νε n ,i → v νi in L (Ω) , weakly in H (Ω) ,weakly- ∗ in L ∞ (Ω) , andin the pointwise sense a.e. in Ω˜ θ νn,i := ˜ θ νε n ,i → θ νi in L (Ω), weakly in H (Ω), andin the pointwise sense a.e. in Ω as n → ∞ , (4.16)and v νi = [ w νi , η νi ] ∈ [ o ∗ , ι ∗ ] × [0 , , a.e. in Ω, (4.17)for any 0 ≤ i ∈ Z .Subsequently, from (3.8), (4.16), Lemmas 4.1–4.2 and (Fact 7) in Remark 1.5, weobserve that ( (cid:2) θ νi , − h α ( v νi )( θ νi − θ νi − ) (cid:3) ∈ ∂ Φ ν ( v νi ; · ) in L (Ω) ,Ψ νε n (˜ v νn,i ; ˜ θ νn,i ) → Φ ν ( v νi ; θ νi ) and ˜ θ νn,i → θ νi in H (Ω) as n → ∞ (4.18)for any i ∈ N . Furthermore, since[ c, ∈ ∂ Φ ν ( v νi ; · ) in L (Ω) for any constant c ∈ R and any 0 ≤ i ∈ Z ,it is inductively observed that θ νi − ∈ L ∞ (Ω) and θ νi ≤ | θ νi − | L ∞ (Ω) a.e. in Ω(resp. θ νi ≥ −| θ νi − | L ∞ (Ω) a.e. in Ω) for any i ∈ N , (4.19)by applying Lemma 4.4 as the case in which v † = v νi ,ˇ θ † = θ νi − , ˆ θ † = | θ νi − | L ∞ (Ω) (resp. ˇ θ † = −| θ νi − | L ∞ (Ω) , ˆ θ † = θ νi − ),[ˇ θ, ˇ θ ∗ ] = (cid:2) θ νi , − h α ( v νi )( θ νi − θ νi − ) (cid:3) (resp. [ˇ θ, ˇ θ ∗ ] = [ −| θ νi − | L ∞ (Ω) , θ, ˆ θ ∗ ] = [ | θ νi − | L ∞ (Ω) , 0] (resp. [ˆ θ, ˆ θ ∗ ] = (cid:2) θ νi , − h α ( v νi )( θ νi − θ νi − ) (cid:3) ), for i ∈ N .By invoking (A1)–(A3), (3.7), (4.16)–(4.18), and Lebesgue’s dominated convergence28heorem and taking further subsequences if necessary, we observe that1 h ( v νi − v νi − , v νi − ̟ ) L (Ω) + ([ ∇ g ]( v νi ) , v νi − ̟ ) L (Ω) + Z Ω (cid:0) |∇ θ νi − | [ ∇ α ]( v νi ) + ν |∇ θ νi − | [ ∇ β ]( v νi ) (cid:1) · ( v νi − ̟ ) dx +( ∇ v νi , ∇ ( v νi − ̟ )) L (Ω) × N + Γ( v νi ) ≤ lim inf n →∞ (cid:20) h (˜ v νn,i − ˜ v νn,i − , ˜ v νn,i − ̟ ) L (Ω) + ([ ∇ g ](˜ v νn,i ) , ˜ v νn,i − ̟ ) L (Ω) (cid:21) + lim n →∞ Z Ω (cid:16) |∇ ˜ θ νn,i − | [ ∇ α ](˜ v νn,i ) + ν |∇ ˜ θ νn,i − | [ ∇ β ](˜ v νn,i ) (cid:17) · (˜ v νn,i − ̟ ) dx + lim inf n →∞ (cid:2) ( ∇ ˜ v νn,i , ∇ (˜ v νn,i − ̟ )) L (Ω) × N + Γ(˜ v νn,i ) (cid:3) ≤ Γ( ̟ ) for any ̟ ∈ [ H (Ω) ∩ L ∞ (Ω)] . (4.20)With (4.17)–(4.20) in mind, we conclude that the limiting sequence { [ v νi , θ νi ] | i ∈ N } mustbe the solution to the approximating problem (AP) νh . Uniqueness and energy dissipation of approximating solutions We start with auxiliary lemmas to demonstrate uniqueness. Lemma 4.5 Assume < h < h †∗ with the constant h †∗ as in (4.15). Let θ † ∈ H (Ω) and v † ,k ∈ [ H (Ω) ∩ L ∞ (Ω)] , k = 1 , , be fixed functions, and let v k ∈ [ H (Ω) ∩ L ∞ (Ω)] , k = 1 , , be functions such that v k ∈ [ o ∗ , ι ∗ ] × [0 , a.e. in Ω (4.21) and h ( v k − v † ,k , v k − ̟ ) L (Ω) + ( ∇ v k , ∇ ( v k − ̟ )) L (Ω) × N +([ ∇ g ]( v k ) , v k − ̟ ) L (Ω) + Γ( v k )+ Z Ω (cid:16) |∇ θ † | [ ∇ α ]( v k ) + ν |∇ θ † | [ ∇ β ]( v k ) (cid:17) · ( v k − ̟ ) dx ≤ Γ( ̟ ) , for all ̟ ∈ [ H (Ω) ∩ L ∞ (Ω)] and k = 1 , . (4.22) Then | v − v | L (Ω) ≤ | v † , − v † , | L (Ω) . (4.23) Proof. We prepare two inequalities by setting k ⊥ := ( k mod 2) + 1 and letting ̟ = v k ⊥ in (4.22), for k = 1 , 2. By taking the sum of these, (4.23) follows by virtue of (A2)–(A3),(4.15), (4.21), and Young’s inequality. Lemma 4.6 Let v † ∈ [ H (Ω) ∩ L ∞ (Ω)] and θ † ,k ∈ H (Ω) , k = 1 , , be fixed functions,and let θ k ∈ H (Ω) , k = 1 , , be functions such that h α ( v † )( θ k − θ † ,k ) ∈ ∂ Φ ν ( v † ; θ k ) in L (Ω) , k = 1 , . hen | p α ( v † )( θ − θ ) | L (Ω) ≤ | p α ( v † )( θ , − θ , ) | L (Ω) . Proof. This lemma is obtained by applying the standard analytic method for theuniqueness of inclusions governed by subdifferentials (see, e.g., [7, 21]). Proof of Theorem 1 (uniqueness and energy dissipation). Assume 0 < h < h †∗ with the constant as in (4.15). Then, on the basis of the foregoing lemmas, the uniquenessfor (AP) νh is verified through the following steps:(Step 0) Let i = 1 and fix [ v ν , θ ν ] = [ w ν , η ν , θ ν ] ∈ D .(Step 1) Confirm the uniqueness of the component v νi = [ w νi , η νi ] of the approximatingsolution by applying Lemma 4.5 as the case in which θ † = θ νi − and v † , = v † , = v νi − .(Step 2) Confirm the uniqueness of the component θ νi of the approximating solution bykeeping in mind (2.5) and applying Lemma 4.6 as the case in which v † = v νi and θ † , = θ † , = θ νi − .(Step 3) Iterate the value of i , i.e., i ← i + 1 and return to step 1.We next verify the inequality (3.5) of energy dissipation. Set ̟ = v νi − in (3.2). Then,by applying a similar derivation method as for (3.25) and using (3.27), it can be seen that12 h | v νi − v νi − | L (Ω) + V ( v νi ) + G ( v νi ) + Γ( v νi )+ Z Ω α ( v νi ) |∇ θ νi − | dx + ν Z Ω β ( v νi ) |∇ θ νi − | dx ≤ V ( v νi − ) + G ( v νi − ) + Γ( v νi − )+ Z Ω α ( v νi − ) |∇ θ νi − | dx + ν Z Ω β ( v νi − ) |∇ θ νi − | dx, i = 1 , , , . . . . (4.24)Conversely, consider ω = θ νi − in (3.3). Then1 h | p α ( v νi )( θ νi − θ νi − ) | L (Ω) + Z Ω α ( v νi ) |∇ θ νi | dx + ν Z Ω β ( v νi ) |∇ θ νi | dx − Z Ω α ( v νi ) |∇ θ νi − | dx − ν Z Ω β ( v νi ) |∇ θ νi − | dx ≤ , for i = 1 , , , . . . . (4.25)The inequality (3.5) of energy dissipation is obtained by taking the sum of (4.24) and(4.25). 30 Proof of Main Theorem 1 Throughout this section, we fix the constant ν > < h < h †∗ with the constant as in (4.15), and wedenote by { [ v νi , θ νi ] = [ w νi , η νi , θ νi ] | i ∈ N } ⊂ D the solution to the approximating problem(AP) νh under the initial condition[ v ν , θ ν ] = [ w ν , η ν , θ ν ] = [ w , η , θ ] ∈ D . On this basis, we define three kinds of time interpolation: [ v νh , θ νh ] = [ w νh , η νh , θ νh ] ∈ L ([0 , ∞ ); L (Ω) ), [ v νh , θ νh ] = [ w νh , η νh , θ νh ] ∈ L ([0 , ∞ ); L (Ω) ), and [ b v νh , b θ νh ] = [ b w νh , b η νh , b θ νh ] ∈ L ([0 , ∞ ); L (Ω) ) with the shorthand v νh = [ w νh , η νh ], v νh = [ w νh , η νh ], and b v νh = [ b w νh , b η νh ],as • [ v νh ( t ) , θ νh ( t )] = [ w νh ( t ) , η νh ( t ) , θ νh ( t )] := [ v νi , θ νi ] = [ w νi , η νi , θ νi ] in L (Ω) if t ∈ (( i − h, ih ] for some i ∈ N , • [ v νh ( t ) , θ νh ( t )] = [ w νh ( t ) , η νh ( t ) , θ νh ( t )] := [ v νi − , θ νi − ] = [ w νi − , η νi − , θ νi − ]in L (Ω) if t ∈ [( i − h, ih ) for some i ∈ N , • [ b v νh ( t ) , b θ νh ( t )] = [ b w νh ( t ) , b η νh ( t ) , b θ νh ( t )] := [ v νi , θ νi ]+ (cid:0) th − i (cid:1) [ v νi − v νi − , θ νi − θ νi − ]in L (Ω) if t ∈ [( i − h, ih ) for some i ∈ N , (5.1)for all t ≥ 0. By using these interpolations, the inequality (3.5) of energy dissipation leadsto 12 Z ts | ( b v νh ) t ( τ ) | L (Ω) dτ + Z ts | p α ( v νh )( τ )( b θ νh ) t ( τ ) | L (Ω) dτ + F ν ( v νh ( t ) , θ νh ( t )) ≤ F ν ( v νh ( s ) , θ νh ( s )) for all 0 ≤ s ≤ t < ∞ and ν > | F ν ( v νh ( t ) , θ νh ( t )) | ≤ F ν ∗ := | F ν ( v ν , θ ν ) | + c ∗ L N (Ω) for all t > ν > 0, (5.3)where c ∗ > { w νh ( t, x ) , w νh ( t, x ) , b w νh ( t, x ) | < h < h †∗ } ⊂ [ o ∗ , ι ∗ ], { η νh ( t, x ) , η νh ( t, x ) , b η νh ( t, x ) | < h < h †∗ } ⊂ [0 , { θ νh ( t, x ) , θ νh ( t, x ) , b θ νh ( t, x ) | < h < h †∗ } ⊂ [ −| θ ν | L ∞ (Ω) , | θ ν | L ∞ (Ω) ],a.e. x ∈ Ω and any t ∈ [0 , T ]; (5.4)and • { [ b v νh , b θ νh ] = [ b w νh , b η νh , b θ νh ] | < h < h †∗ } is bounded in W , (0 , T ; L (Ω) ) ∩ L ∞ (0 , T ; H (Ω) ) ∩ L ∞ ( Q ) ; • { [ v νh , θ νh ] = [ w νh , η νh , θ νh ] | < h < h †∗ } and { [ v νh , θ νh ] = [ w νh , η νh , θ νh ] | < h < h †∗ } are boundedin L ∞ (0 , T ; H (Ω) ) ∩ L ∞ ( Q ) . (5.5)31aking into account (5.4)–(5.5) and Aubin-type compactness theory (see [36]), we find asequence h †∗ > h ν > · · · > h νn ց n → ∞ and a triplet of functions [ v ν , θ ν ] = [ w ν , η ν , θ ν ] ∈ L (0 , T ; L (Ω) ) such that ( v ν ∈ W , (0 , T ; L (Ω) ) ∩ L ∞ (0 , T ; H (Ω) ), θ ν ∈ W , (0 , T ; L (Ω)) ∩ L ∞ (0 , T ; H (Ω)), (5.6) ( v ν ( t, x ) = [ w ν ( t, x ) , η ν ( t, x )] ∈ [ o ∗ , ι ∗ ] × [0 , θ ν ( t, x ) ∈ [ −| θ ν | L ∞ (Ω) , | θ ν | L ∞ (Ω) ],a.e. x ∈ Ω and any t ∈ [0 , T ], (5.7)and b v νn = [ b w νn , b η νn ] := b v νh νn → v ν in C ( I ; L (Ω) ),weakly in W , ( I ; L (Ω) ), weakly- ∗ in L ∞ ( I ; H (Ω) ),and weakly- ∗ in L ∞ ( I × Ω) , b θ νn := b θ νh νn → θ ν in C ( I ; L (Ω)),weakly in W , ( I ; L (Ω)), weakly- ∗ in L ∞ ( I ; H (Ω)),and weakly- ∗ in L ∞ ( I × Ω),as n → ∞ , for any open interval I ⊂ (0 , T ). (5.8)Additionally, noting that | v νh − b v νh | L ∞ (0 ,T ; L (Ω) ) ∨ | v νh − b v νh | L ∞ (0 ,T ; L (Ω) ) ≤ max i ∈ Z , ih ∈ [0 ,T ] Z ( i +1) hih | ( b v νh ) t ( t ) | L (Ω) dt ≤ √ h | ( b v νh ) t | L (0 ,T ; L (Ω) ) , | p α ( v νh )( θ νh − b θ νh ) | L ∞ (0 ,T ; L (Ω)) ∨ | p α ( v νh )( θ νh − b θ νh ) | L ∞ (0 ,T ; L (Ω)) ≤ max i ∈ Z , ih ∈ [0 ,T ] Z ( i +1) hih | p α ( v νh )( b θ νh ) t ( t ) | L (Ω) dt ≤ √ h | p α ( v νh )( b θ νh ) t | L (0 ,T ; L (Ω)) , (5.9)we also have v νn = [ w νn , η νn ] := v νh νn → v ν and v νn = [ w νn , η νn ] := v νh νn → v ν in L ∞ ( I ; L (Ω) ), weakly- ∗ in L ∞ ( I ; H (Ω) ),and weakly- ∗ in L ∞ ( I × Ω) , θ νn := θ νh νn → θ ν and θ νn := θ νh νn → θ ν in L ∞ ( I ; L (Ω)),weakly- ∗ in L ∞ ( I ; H (Ω)) and weakly- ∗ in L ∞ ( I × Ω),as n → ∞ , for any open interval I ⊂ (0 , T ), (5.10)and, in particular, v νn ( t ) → v ν ( t ), v νn ( t ) → v ν ( t ) and b v νn ( t ) → v ν ( t )in L (Ω) and weakly in H (Ω), θ νn ( t ) → θ ν ( t ), θ νn ( t ) → θ ν ( t ) and b θ νn ( t ) → θ ν ( t )in L (Ω) and weakly in H (Ω),as n → ∞ , a.e. t ∈ (0 , T ). (5.11)32ased on these considerations, we next demonstrate the following auxiliary lemmas. Lemma 5.1 (Mosco convergence) Let I ⊂ (0 , T ) be any open interval, and let Φ Iν : L ( I ; L (Ω)) → [0 , ∞ ] and Φ Iν,n : L ( I ; L (Ω)) → [0 , ∞ ] , n ∈ N , be functionals defined as ζ ∈ L ( I ; L (Ω)) Φ Iν ( ζ ) := Z I Φ ν ( v ν ( t ); ζ ( t )) dt, (5.12) and ζ ∈ L ( I ; L (Ω)) Φ Iν,n ( ζ ) := Z I Φ ν ( v νn ( t ); ζ ( t )) dt, n ∈ N , (5.13) by using the functions v ν = [ w ν , η ν ] ∈ L ∞ (0 , T ; H (Ω) ) ∩ L ∞ ( Q ) and v νn = [ w νn , η νn ] ∈ L ∞ (0 , T ; H (Ω) ) ∩ L ∞ ( Q ) , n ∈ N , as in (5.6)–(5.11). Then, the following two statementshold:(I-1) Φ Iν and Φ ν,n , n ∈ N , are proper l.s.c and convex functions on L ( I ; L (Ω)) such that D (Φ Iν ) = D (Φ Iν,n ) = L ( I ; H (Ω)) for all n ∈ N .(I-2) Φ Iν,n → Φ Iν on L ( I ; L (Ω)) , in the sense of Mosco [30], as n → ∞ . Proof. Since (I-1) is a straightforward consequence of Notation 8, (A3), (2.5), and(5.5)–(5.7), it is enough to prove only (I-2).To verify the lower bound condition, let us take a function ζ † ∈ L ( I ; L (Ω)), a se-quence { ζ † n | n ∈ N } ⊂ L ( I ; L (Ω)), and a subsequence { ζ † n j | j ∈ N } ⊂ { ζ † n } to posit thefollowing nontrivial situation: ζ † n → ζ † weakly in L ( I ; L (Ω)) as n → ∞ ,lim inf n →∞ Φ Iν,n ( ζ † n ) = lim j →∞ Φ Iν,n j ( ζ † n j ) < ∞ . (5.14)By virtue of (2.5) and (5.14), the subsequence { ζ † n j } must be bounded in L ( I ; H (Ω)).So, taking a subsequence if necessary, we may also suppose that ζ † n j → ζ † weakly in L ( I ; H (Ω)) as j → ∞ .Furthermore, with (A3) and (5.4)–(5.11) in mind, we have α ( v νn j ) ∇ ζ † n j → α ( v ν ) ∇ ζ † q β ( v νn j ) ∇ ζ † n j → p β ( v ν ) ∇ ζ † weakly in L ( I ; L (Ω) N ) as j → ∞ .From the above convergence, satisfaction of the lower bound condition is confirmed asfollows:lim inf n →∞ Φ Iν,n ( ζ † n ) = lim j →∞ Φ Iν,n j ( ζ † n j )= lim inf j →∞ (cid:18)(cid:12)(cid:12)(cid:12) α ( v νn j ) ∇ ζ † n j (cid:12)(cid:12)(cid:12) L ( I ; L (Ω; R N )) + ν (cid:12)(cid:12)(cid:12)q β ( v νn j ) ∇ ζ † n j (cid:12)(cid:12)(cid:12) L ( I ; L (Ω) N ) (cid:19) ≥ (cid:12)(cid:12) α ( v ν ) ∇ ζ † (cid:12)(cid:12) L ( I ; L (Ω; R N )) + ν (cid:12)(cid:12)(cid:12)p β ( v ν ) ∇ ζ † (cid:12)(cid:12)(cid:12) L ( I ; L (Ω) N ) = Φ Iν ( ζ † ) . | Φ Iν,n ( ζ ‡ ) − Φ Iν ( ζ ‡ ) |≤ Z I Z Ω | α ( η νn ) − α ( η ν ) ||∇ ζ ‡ | dx dt + ν Z I Z Ω | β ( w νn ) − β ( w ν ) ||∇ ζ ‡ | dx dt → n → ∞ , for any ζ ‡ ∈ D (Φ Iν ) = L ( I ; H (Ω)).This implies the validity of the condition of optimality, for Mosco convergence Φ Iν,n → Φ Iν on L ( I ; L (Ω)) as n → ∞ . Lemma 5.2 In addition to the assumptions and notations of Lemma 5.1, assume that ζ ‡ ∈ L ( I ; H (Ω)) , { ζ ‡ n | n ∈ N } ⊂ L ( I ; H (Ω)) and Φ Iν,n ( ζ ‡ n ) → Φ Iν ( ζ ‡ ) as n → ∞ . (5.15) Then ζ ‡ n → ζ ‡ in L ( I ; H (Ω)) as n → ∞ . Proof. The proof of this lemma is a slight modification of that of Lemma 4.2. Indeed,from (A3), (2.5), (5.4)–(5.11) and (5.15), we infer that α ( v νn ) ∇ ζ ‡ n → α ( v ν ) ∇ ζ ‡ p β ( v νn ) ∇ ζ ‡ n → p β ( v ν ) ∇ ζ ‡ √ β ( v νn ) ∇ ζ ‡ n → √ β ( v ν ) ∇ ζ ‡ weakly in L ( I ; L (Ω) N ) as n → ∞ . (5.16)Therefore, as in the derivations of (4.10)–(4.11), convergences (5.15)–(5.16) show that p β ( v νn ) ∇ ζ ‡ n → p β ( v ν ) ∇ ζ ‡ in L ( I ; L (Ω) N ) as n → ∞ , (5.17)and |∇ ζ ‡ n | L ( I ; L (Ω) N ) → |∇ ζ ‡ | L ( I ; L (Ω) N ) as n → ∞ . (5.18)Thus, strong convergence of { ζ ‡ n } in L ( I ; H (Ω)) follows from (5.17), (5.18), and theuniform convexity of the L -topology. Proof of Main Theorem 1. From (5.6)–(5.7), we see that the limiting triplet [ v ν , θ ν ] =[ w ν , η ν , θ ν ] fulfills the condition (S0) ν . Hence, all we have to do is verify the compatibilityof [ v ν , θ ν ] with conditions (S1) ν –(S3) ν .Fix any open interval I ⊂ (0 , T ). Then, from Remark 1.1, (3.2)–(3.3), and (5.1), thefunctions [ v νn , θ νn ], [ v νn , θ νn ], [ b v νn , b θ νn ], n ∈ N , must satisfy Z I (cid:0) ( b v νn ) t ( t ) , v νn ( t ) − ̟ (cid:1) L (Ω) dt + Z I (cid:0) [ ∇ g ]( v νn ( t )) , v νn − ̟ (cid:1) L (Ω) dt + Z I (cid:0) ∇ v νn ( t ) , ∇ ( v νn ( t ) − ̟ ) (cid:1) L (Ω) × N dt + Z I (cid:0) |∇ θ νn ( t ) | [ ∇ α ]( v νn ( t )) + ν |∇ θ νn | [ ∇ β ]( v νn ( t )) (cid:1) · ( v νn ( t ) − ̟ ) dt + Z I Γ( v νn ( t )) dt ≤ Z I Γ( ̟ ) dt (cid:0) = Γ( ̟ ) · L ( I ) (cid:1) for any ̟ ∈ [ H (Ω) ∩ L ∞ (Ω)] and any n ∈ N , (5.19)34nd [ θ νn , − α ( v νn )( b θ νn ) t ] ∈ ∂ Φ Iν,n in L ( I ; L (Ω)) for any n ∈ N . (5.20)Here, from (5.8)–(5.13), (5.20), (I-2) of Lemma 5.1, and (Fact 7) in Remark 1.5, it followsthat [ θ ν , − α ( v ν )( θ ν ) t ] ∈ ∂ Φ Iν in L ( I ; L (Ω)) (5.21)and Φ Iν,n ( θ νn ) → Φ Iν ( θ ν ) as n → ∞ . (5.22)Subsequently, in light of (5.21), (I-1) of Lemma 5.1, and (Fact 1) in Remark 1.1, we candeduce that the triplet [ v ν , θ ν ] = [ w ν , η ν , θ ν ] fulfills condition (S3) ν .Next, from (5.8)–(5.10), (5.22), and Lemma 5.2, it follows that θ νn → θ ν in L ( I ; H (Ω)) as n → ∞ . (5.23)Meanwhile, from (2.5), (5.2)–(5.3), and (5.5), it can be seen that (cid:12)(cid:12)(cid:12)(cid:12)Z I Z Ω |∇ θ νh | dx dt − Z I Z Ω |∇ θ νh | dx dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ F ν ∗ δ · hν for all 0 < h < h †∗ and ν > 0. (5.24)Bearing in mind (5.10), (5.23), and (5.24), it further follows that θ νn → θ ν and b θ νn → θ ν in L ( I ; H (Ω)), as n → ∞ .Now, on account of (5.8)–(5.10), (5.23)–(5.24), and Lebesgue’s dominated convergencetheorem, allowing n → ∞ in (5.19) yields Z I (cid:0) ( v ν ) t ( t ) , v ν ( t ) − ̟ (cid:1) L (Ω) dt + Z I (cid:0) [ ∇ g ]( v ν ( t )) , v ν ( t ) − ̟ (cid:1) L (Ω) dt + Z I (cid:0) ∇ v ν ( t ) , ∇ ( v ν ( t ) − ̟ ) (cid:1) L (Ω) × N dt + Z I (cid:0) |∇ θ ν ( t ) | [ ∇ α ]( v ν ( t )) + ν |∇ θ ν ( t ) | [ ∇ β ]( v ν ( t )) (cid:1) · ( v ν ( t ) − ̟ ) dt + Z I Γ( v ν ( t )) dt ≤ Z I Γ( ̟ ) dt (cid:0) = Γ( ̟ ) · L ( I ) (cid:1) for any ̟ ∈ [ H (Ω) ∩ L ∞ (Ω)] .Since the choice of the open interval I ⊂ (0 , T ) is arbitrary, we can verify the remainingconditions (S1) ν –(S2) ν on the basis of this inequality and the reformulation (2.15) inRemark 2.3. 35 Proof of Main Theorem 2 Assume (2.11) for the initial value [ w , η , θ ] ∈ D of the system (S) . Then, roughlyspeaking, our second main theorem is proved through some limiting observations for (AP) νh as h, ν ց 0. With this scenario in mind, we rely upon [29, Section 2] for the followingauxiliary lemmas. Lemma 6.1 Let δ ∗ > be a constant and let I ⊂ (0 , T ) be any open interval. Assumethat ̺ ∈ C ( I ; L (Ω)) ∩ L ∞ ( I ; H (Ω)) ∩ L ∞ ( I × Ω) , { ̺ n | n ∈ N } ⊂ L ( I ; L (Ω)) , ̺ ≥ δ ∗ and ̺ n ≥ δ ∗ , a.e. in I × Ω , for all n ∈ N , ̺ n ( t ) → ̺ ( t ) in L (Ω) and weakly in H (Ω) , as n → ∞ , a.e. t ∈ I ,and ( ζ ∈ C ( I ; L (Ω)) ∩ L ( I ; BV (Ω)) , { ζ n | n ∈ N } ⊂ L ( I ; H (Ω)) , ζ n ( t ) → ζ ( t ) in L (Ω) as n → ∞ , a.e. t ∈ I .Then the functions t ∈ I Z Ω d [ ̺ ( t ) | Dζ ( t ) | ] and t ∈ I Z Ω ̺ n ( t ) |∇ ζ n ( t ) | dx, n ∈ N , are integrable. Moreover, if Z I Z Ω ̺ n ( t ) |∇ ζ n ( t ) | dx dt → Z I Z Ω d [ ̺ ( t ) | Dζ ( t ) | ] as n → ∞ and ω ∈ C ( I ; L (Ω)) ∩ L ∞ ( I ; H (Ω)) ∩ L ∞ ( I × Ω) and { ω n | n ∈ N } ⊂ L ( I ; L (Ω)) , { ω n | n ∈ N } is a bounded sequence in L ∞ ( I × Ω) , ω n ( t ) → ω ( t ) in L (Ω) and weakly in H (Ω) as n → ∞ , a.e. t ∈ I ,then Z I Z Ω ω n ( t ) |∇ ζ n ( t ) | dx dt → Z I Z Ω d [ ω ( t ) | Dζ ( t ) | ] as n → ∞ . Proof. This lemma is a straightforward consequence of [29, Lemmas 4 and 7]. Lemma 6.2 (Γ-convergence) Assume v ‡ ∈ [ H (Ω) ∩ L ∞ (Ω)] , { v ‡ ν | ν > } ⊂ [ H (Ω) ∩ L ∞ (Ω)] , and ( v ‡ ν → v ‡ in the pointwise sense, a.e. in Ω , as ν ց , { v ‡ ν | ν > } is bounded in L ∞ (Ω) . (6.1) Then, for the sequence of convex functions { Φ ν ( v ‡ ν ; · ) } , it holds that Φ ν ( v ‡ ν ; · ) → Φ ( v ‡ ; · ) on L (Ω) , in the sense of Γ -convergence [11], as ν ց , i.e.: γ ν (lower bound) lim inf ν ց Φ ν ( v ‡ ν ; θ ‡ ν ) ≥ Φ ν ( v ‡ ; θ ‡ ) if θ ‡ ∈ L (Ω) , { θ ‡ ν | ν > } ⊂ L (Ω) ,and θ ‡ ν → θ ‡ in L (Ω) as ν ց ;( γ ν (optimality) for any θ ‡‡ ∈ H (Ω) , there exists a sequence { θ ‡‡ ν | ν > } ⊂ H (Ω) such that θ ‡‡ ν → θ ‡‡ in L (Ω) and Φ ν ( v ‡ ν ; θ ‡‡ ν ) → Φ ( v ‡ ; θ ‡‡ ) as ν ց . Proof. We start by confirming that R := sup ν> | v ‡ ν | L ∞ (Ω) < ∞ , α ( v ‡ ν ) → α ( v ‡ ) in L (Ω) and weakly in H (Ω), as ν ց 0. (6.2)This is easily confirmed from assumption (A3) and (6.1).Next, for any ν > 0, any δ > 0, and any ˜ v ∈ [ H (Ω) ∩ L ∞ (Ω)] , let ˜Φ ν,δ (˜ v ; · ) be aproper l.s.c. and convex function on L (Ω), defined as ϑ ∈ L (Ω) ˜Φ ν,δ (˜ v ; ϑ ) := Φ (˜ v ; ϑ ) + νδV ( ϑ ) ∈ [0 , ∞ ] , (6.3)where V is the convex function V d D given in (1.2), in the case that d = 1. Then, setting δ ∗ ( R ) := | α | C ([ − R,R ] ) + | β | C ([ − R,R ] ) for any R > 0, (6.4)it immediately follows from (A3), (2.5), and (6.2) that˜Φ ν,δ ( v ‡ ν ; z ) ≤ Φ ν ( v ‡ ν ; z ) ≤ ˜Φ ν,δ ∗ ( R ) ( v ‡ ν ; z ) for any ν > z ∈ L (Ω). (6.5)In addition, we can apply [29, Lemma 3] to see that˜Φ ν,δ ( v ‡ ν ; · ) → Φ ( v ‡ ; · ) and ˜Φ ν,δ ∗ ( R ) ( v ‡ ν ; · ) → Φ ( v ‡ ; · ) on L (Ω),in the sense of Γ-convergence, as ν ց 0. (6.6)On the basis of these facts, Γ-convergence for the sequence { Φ ν ( v ‡ ν ; · ) | ν > } can beverified as follows.First, to verify the lower bound, let us take arbitrary θ ‡ ∈ L (Ω) and { θ ‡ ν | ν > } ⊂ L (Ω) such that θ ‡ ν → θ ‡ in L (Ω) as ν ց 0. Then, using (6.5)–(6.6), we deduce thefollowing inequality:Φ ( v ‡ ; θ ‡ ) ≤ lim inf ν ց ˜Φ ν,δ ( v ‡ ν ; θ ‡ ν ) ≤ lim inf ν ց Φ ν ( v ‡ ν ; θ ‡ ν ) . Thus, the lower bound for { Φ ν ( v ‡ ν ; · ) } is verified.Second, to verify optimality, we take any θ ‡‡ ∈ BV (Ω) ∩ L (Ω) (= D (Φ ( v ‡ ; · ))) anduse the fact (6.6) to take a sequence { θ ‡‡ ν | ν > } ⊂ H (Ω) such that θ ‡‡ ν → θ ‡‡ in L (Ω) and ˜Φ ν,δ ∗ ( R ) ( v ‡ ν ; θ ‡‡ ν ) → Φ ( v ‡ ; θ ‡‡ ) as ν ց { Φ ν ( v ‡ ν ; · ) } , we observe thatlim sup ν ց Φ ν ( v ‡ ν ; θ ‡‡ ν ) ≤ lim ν ց ˜Φ ν,δ ∗ ( R ) ( v ‡ ν ; θ ‡‡ ν ) = Φ ( v ‡ ; θ ‡‡ ν ) ≤ lim inf ν ց Φ ν ( v ‡ ν ; θ ‡‡ ν ) . This implies optimality for { Φ ν ( v ‡ ν ; · ) } . 37 emark 6.1 In previous work [29, Lemma 3], it was reported that strong L -convergenceas in ( γ ν was necessary to obtain convergence as in (6.6). Hence, we still have notsucceeded in generalizing the result of Lemma 6.2 by means of Mosco convergence.In light of Lemma 6.2 and [29, Remark 2], we find a sequence { ˜ θ ν | ν > } ⊂ H (Ω)such that ( | ˜ θ ν | L ∞ (Ω) ≤ | θ | L ∞ (Ω) for all ν > θ ν → θ and Φ ν ( v ; ˜ θ ν ) → Φ ( v ; θ ) as ν ց 0. (6.7)Now, for arbitrary 0 < h < h †∗ and ν > 0, let us denote by [ v νh , θ νh ] = [ w νh , η νh , θ νh ] ∈ L ([0 , ∞ ); L (Ω) ), [ v νh , θ νh ] = [ w νh , η νh , θ νh ] ∈ L ([0 , ∞ ); L (Ω) ), and [ b v νh , b θ νh ] = [ b w νh , b η νh , b θ νh ] ∈ L ([0 , ∞ ); L (Ω) ) the three time interpolations, as in (5.1), consisting of the solution { [ v νi , θ νi ] = [ w νi , η νi , θ νi ] | i ∈ N } ⊂ D to the approximating problem (AP) νh , under theassumption (2.5) and the following initial condition:[ v ν , θ ν ] = [ w ν , η ν , θ ν ] := [ v ν , ˜ θ ν ] = [ w , η , ˜ θ ν ] ∈ D . (6.8)Then, from (A1)–(A3), (5.2)–(5.3), (5.5)–(5.6), and (6.7)–(6.8), it is deduced that F ∗ := sup <ν<ν †∗ | F ν ( v ν , θ ν ) | + c ∗ L N (Ω)= sup <ν<ν †∗ | F ν ( w , η , ˜ θ ν ) | + c ∗ L N (Ω) < ∞ for some ν †∗ > 0, (6.9) { w νh ( t, x ) , w νh ( t, x ) , b w νh ( t, x ) | < h < h †∗ , < ν < ν †∗ } ⊂ [ o ∗ , ι ∗ ], { η νh ( t, x ) , η νh ( t, x ) , b η νh ( t, x ) | < h < h †∗ , < ν < ν †∗ } ⊂ [0 , { θ νh ( t, x ) , θ νh ( t, x ) , b θ νh ( t, x ) | < h < h †∗ , < ν < ν †∗ } ⊂ [ −| θ | L ∞ (Ω) , | θ | L ∞ (Ω) ],a.e. x ∈ Ω and any t ∈ [0 , T ], (6.10) | ( b v νh ) t | L (0 ,T ; L (Ω)) + | p α ( v νh )( b θ νh ) t | L (0 ,T ; L (Ω)) ≤ X i ∈ Z , ih ∈ [0 ,T ] (cid:18) h | v νi − v νi − | L (Ω) + 1 h | p α ( v νi )( θ νi − θ νi − ) | L (Ω) (cid:19) ≤ F ∗ , sup t ∈ [0 ,T ] | F ν ( v νh , θ νh ) | ∨ sup t ∈ [0 ,T ] | F ν ( v νh , θ νh ) | ≤ max i ∈ Z , ih ∈ [0 ,T ] | F ν ( v νi , θ νi ) | ≤ F ∗ , for any 0 < h < h †∗ and any 0 < ν < ν †∗ , (6.11)and, therefore, • { [ b v νh , b θ νh ] = [ b w νh , b η νh , b θ νh ] | < h < h †∗ , < ν < ν †∗ } is boundedin W , (0 , T ; L (Ω) ) ∩ L ∞ (0 , T ; X ) ∩ L ∞ ( Q ) ; • { [ v νh , θ νh ] = [ w νh , η νh , θ νh ] | < h < h †∗ , < ν < ν †∗ } and { [ v νh , θ νh ] = [ w νh , η νh , θ νh ] | < h < h †∗ , < ν < ν †∗ } are boundedin L ∞ (0 , T ; L (Ω) ) ∩ L ∞ (0 , T ; X ) ∩ L ∞ ( Q ) . (6.12)Taking into account (6.7)–(6.12) and Aubin-type compactness theory (see [36]), we findsequences h †∗ > h > · · · > h n ց ν †∗ > ν > · · · > ν n ց n → ∞ v, θ ] = [ w, η, θ ] ∈ L (0 , T ; L (Ω) ) such that ( v ∈ W , (0 , T ; L (Ω) ) ∩ L ∞ (0 , T ; H (Ω) ), θ ∈ W , (0 , T ; L (Ω)) ∩ L ∞ (0 , T ; BV (Ω)), (6.13) ( v ( t, x ) = [ w ( t, x ) , η ( t, x )] ∈ [ o ∗ , ι ∗ ] × [0 , θ ( t, x ) ∈ [ −| θ | L ∞ (Ω) , | θ | L ∞ (Ω) ],a.e. x ∈ Ω and any t ∈ [0 , T ], (6.14)and b v n = [ b w n , b η n ] := b v ν n h n → v in C ( I ; L (Ω) ),weakly in W , ( I ; L (Ω) ), weakly- ∗ in L ∞ ( I ; H (Ω) ) , and weakly- ∗ in L ∞ ( I × Ω) , b θ n := b θ ν n h νn → θ in C ( I ; L (Ω)),weakly in W , ( I ; L (Ω)), and weakly- ∗ in L ∞ ( I × Ω),as n → ∞ , for any open interval I ⊂ (0 , T ). (6.15)Additionally, from (2.10) and (5.9), we deduced that v n = [ w n , η n ] := v ν n h n → v and v n = [ w n , η n ] := v ν n h n → v in L ∞ ( I ; L (Ω) ), weakly- ∗ in L ∞ ( I ; H (Ω) ), andweakly- ∗ in L ∞ ( I × Ω) , θ n := θ ν n h n → θ and θ n := θ ν n h n → θ in L ∞ ( I ; L (Ω)), andweakly- ∗ in L ∞ ( I × Ω),as n → ∞ , for any open interval I ⊂ (0 , T ), (6.16)and, in particular, v n ( t ) → v ( t ), v n ( t ) → v ( t ) and b v n ( t ) → v ( t )in L (Ω) and weakly in H (Ω) , θ n ( t ) → θ ( t ), θ n ( t ) → θ ( t ) and b θ n ( t ) → θ ( t )in L (Ω) and weakly- ∗ in BV (Ω),as n → ∞ , a.e. t ∈ (0 , T ). (6.17)Based on these, we next check the following lemmas. Lemma 6.3 (cf. [29, Section 2.2]) Let v = [ w, η ] and v n = [ w n , η n ] , n ∈ N , be pairs offunctions as in (6.13)–(6.17). Also, for any open interval I ⊂ (0 , T ) , let Φ I : L ( I ; L (Ω)) → [0 , ∞ ] and Φ In : L ( I ; L (Ω)) → [0 , ∞ ] , n ∈ N , be functionals defined as ζ ∈ L ( I ; L (Ω)) Φ I ( ζ ) := Z I Φ ( v ( t ); ζ ( t )) dt, if Φ ( v ( · ); ζ ( · )) ∈ L ( I ) , ∞ , otherwise, nd ζ ∈ L ( I ; L (Ω)) Φ In ( ζ ) := Z I Φ ν n ( v n ; ζ ( t )) dt, n ∈ N , respectively. Then, for any open interval I ⊂ (0 , T ) , the following three statements hold:(II-1) The restriction Φ I | C ( I ; L (Ω)) is a proper l.s.c. and convex function on C ( I ; L (Ω)) such that D (Φ I | C ( I ; L (Ω)) ) ⊃ C ( I ; L (Ω)) ∩ L ( I ; BV (Ω)) , and also the Φ In , n ∈ N , areproper l.s.c. and convex functions on L ( I ; L (Ω)) such that D (Φ In ) = L ( I ; H (Ω)) , n ∈ N .(II-2) If ζ †† ∈ C ( I ; L (Ω)) , { ζ †† n | n ∈ N } ⊂ L ( I ; H (Ω)) , and ζ †† n ( t ) → ζ †† ( t ) in L (Ω) ,a.e. t ∈ I , then lim inf n →∞ Φ In ( ζ †† n ) ≥ lim inf n →∞ Φ I ( ζ †† n ) ≥ Φ I ( ζ †† ) .(II-3) For any ζ ‡‡ ∈ C ( I ; L (Ω)) ∩ L ( I ; BV (Ω)) , there exists a sequence { ζ ‡‡ n | n ∈ N } ⊂ C ∞ ( I × Ω) such that ζ ‡‡ n → ζ ‡‡ in L ( I ; L (Ω)) and Φ In ( ζ ‡‡ n ) → Φ I ( ζ ‡‡ ) as n → ∞ . Proof. From (A3), (2.5), (2.10), and (6.13)–(6.15), we easily have • { α ( v ) , β ( v ) } ⊂ C ( I ; L (Ω)) ∩ L ∞ ( I ; H (Ω)) ∩ L ∞ ( Q ), δ ≤ α ( v ) ≤ δ ∗ (1), and δ ≤ β ( v ) ≤ δ ∗ (1) a.e. in Q , • { α ( v n ) , β ( v n ) | n ∈ N } ⊂ L ∞ ( I ; H (Ω)) ∩ L ∞ ( Q ), δ ≤ α ( v n ) ≤ δ ∗ (1), and δ ≤ β ( v n ) ≤ δ ∗ (1) a.e.in Q for n ∈ N , • α ( v n ( t )) → α ( v ( t )) in L (Ω) and weakly in H (Ω) as n → ∞ , (6.18)where for any R > δ ∗ ( R ) is the constant given in (6.4). From (6.18), item (II-1) is astraightforward consequence of [29, Lemma 4].Next, let us take ζ †† ∈ C ( I ; L (Ω)), { ζ †† n | n ∈ N } ⊂ L ( I ; H (Ω)), ζ ‡‡ ∈ C ( I ; L (Ω)) ∩ L ( I ; BV (Ω)), and { ζ ‡‡ n | n ∈ N } ⊂ C ∞ ( I × Ω), fulfilling the assumptions in (II-2) and(II-3). By using the functional given in (6.3), we set ζ ∈ L ( I ; L (Ω)) ˜Φ In,δ ( ζ ) := Z I ˜Φ ν n ,δ ( v n ( t ); ζ ( t )) dt ∈ [0 , ∞ ]for any n ∈ N and any δ > In,δ ( ζ ) ≤ Φ In ( ζ ) ≤ ˜Φ In,δ ∗ (1) ( ζ ) for any z ∈ L ( I ; L (Ω)). (6.19)Furthermore, by virtue of (A3), (2.10), (6.5), (6.10), (6.14), (6.17), and [29, Lemma 8],we find a sequence { ζ ‡‡ n | n ∈ N } ⊂ C ∞ ( I × Ω) such that ζ ‡‡ n → ζ ‡‡ in L ( I ; L (Ω)) and ˜Φ n,δ ( ζ ‡‡ n ) → Φ ( ζ ‡‡ ) as n → ∞ for any δ > 0. (6.20)40ow, taking into account (6.19)–(6.20) and applying [29, Lemma 6], we can verify theremaining items (II-2) and (II-3), as follows: lim inf n →∞ Φ In ( ζ †† n ) ≥ lim inf n →∞ ˜Φ In,δ ( ζ †† n ) ≥ Φ ( ζ †† ),lim sup n →∞ Φ In ( ζ ‡‡ n ) ≤ lim n →∞ ˜Φ In,δ ∗ (1) ( ζ ‡‡ n ) = Φ I ( ζ ‡‡ ) ≤ lim inf n →∞ Φ In ( ζ ‡‡ n ). Proof of Main Theorem 2. From (6.13)–(6.14), it can be seen that the limitingtriplet [ v, θ ] = [ w, η, θ ] fulfills the condition (S0) . Hence, all we have to do is verify thecompatibility of [ v, θ ] with conditions (S1) –(S3) .Fix any open interval I ⊂ (0 , T ). Then, from (3.2)–(3.3) and (5.1), the functions[ v n , θ n ], [ v n , θ n ], and [ b v n , b θ n ], n ∈ N , must fulfill Z I (cid:0) ( b v n ) t ( t ) , v n ( t ) − ̟ (cid:1) L (Ω) dt + Z I (cid:0) [ ∇ g ]( v n ( t )) , v n ( t ) − ̟ (cid:1) L (Ω) dt + Z I (cid:0) ∇ v n ( t ) , ∇ ( v n ( t ) − ̟ ) (cid:1) L (Ω) × N dt + Z I Z Ω (cid:0) |∇ θ n ( t ) | [ ∇ α ]( v n ( t )) + ν |∇ θ n | [ ∇ β ]( v n ( t )) (cid:1) · ( v n ( t ) − ̟ ) dx dt + Z I Γ( v n ( t )) dt ≤ Z I Γ( ̟ ) dt for any ̟ ∈ [ H (Ω) ∩ L ∞ (Ω)] and any n ∈ N (6.21)and Z I (cid:0) α ( v n ( t ))( b θ n ) t ( t ) , θ n ( t ) − ζ ( t ) (cid:1) L (Ω) dt + Φ In ( θ n ) ≤ Φ In ( ζ )for any ζ ∈ L ( I ; H (Ω)) and n ∈ N . (6.22)On this basis, we next take the limit of (6.22) as n → ∞ . Then, invoking (A3),(6.15)–(6.17), and (II-2) of Lemma 6.3, we calculate that Z I (cid:0) α ( v ( t )) θ t ( t ) , θ ( t ) − ζ ( t ) (cid:1) L (Ω) dt + Φ I ( θ ) ≤ lim n →∞ Z I (cid:0) α ( v n )( b θ n ) t ( t ) , θ n ( t ) − ζ ( t ) (cid:1) L (Ω) dt + lim inf n →∞ Φ In ( θ n ) ≤ lim n →∞ Φ In ( ζ ) = Φ I ( ζ ) for any ζ ∈ L ( I ; H (Ω)).Since the choice of the open interval I ⊂ (0 , T ) is arbitrary, this inequality implies that (cid:0) α ( v ( t )) θ t ( t ) , θ ( t ) − ω (cid:1) L (Ω) + Φ ( v ( t ); θ ( t )) ≤ Φ ( v ( t ); ω )for any ω ∈ H (Ω). (6.23)Moreover, by virtue of the strict approximation given by (Fact 5) in Remark 1.4, we canverify that (6.23) is valid for any ω ∈ BV (Ω) ∩ L (Ω), and the triplet [ v, θ ] = [ w, η, θ ] iscompatible with condition (S3) . 41ext, with (6.13) in mind, we can apply (II-3) of Lemma 6.3 for the case in which ζ ‡‡ = θ and take a sequence { ˜ θ n | n ∈ N } ⊂ C ∞ ( I × Ω) such that˜ θ n → θ in L ( I ; L (Ω)) and Φ In (˜ θ n ) → Φ I ( θ ) as n → ∞ . (6.24)Setting ζ = ˜ θ n in (6.22), from (6.15)–(6.17), (6.18), (6.24), and (II-2) of Lemma 6.3, wehave Φ I ( θ ) ≤ lim inf n →∞ Φ I ( θ n ) ≤ lim inf n →∞ Φ In ( θ n ) ≤ lim sup n →∞ Φ In ( θ n ) ≤ lim n →∞ (cid:20) Φ In (˜ θ n ) − Z I (cid:0) α ( v n ( t ))( b θ n ) t ( t ) , θ n ( t ) − ˜ θ n ( t ) (cid:1) L (Ω) dt (cid:21) = Φ I ( θ ) , and therefore 0 ≤ lim inf n →∞ ν n Z I Z Ω β ( v n ( t )) |∇ θ n ( t ) | dx dt ≤ lim sup n →∞ ν n Z I Z Ω β ( v n ( t )) |∇ θ n ( t ) | dx dt ≤ lim n →∞ Φ In ( θ n ) − lim inf n →∞ Φ I ( θ n ) = 0 (6.25)and lim n →∞ Z I Z Ω α ( v n ( t )) |∇ θ n ( t ) | dx dt = Z I Z Ω d [ α ( v ( t )) | Dθ ( t ) | ] . (6.26)Again, keeping in mind (6.15)–(6.17), (6.18), and (6.26), we apply Lemma 6.1 for the casein which ̺ = α ( v ), { ̺ n } = { α ( v n ) } , ζ = θ , { ζ n } = { θ n } , ω = 1, and { ω n } = { } .We then have Z I Z Ω |∇ θ n ( t ) | dx dt → Z I Z Ω | Dθ ( t ) | dt as n → ∞ .Conversely, as a result of (2.10), (5.1) and (6.11), | θ n − θ n | L ∞ (0 ,T ; L (Ω)) ≤ p h n | ( b θ n ) t | L (0 ,T ; L (Ω)) → (cid:12)(cid:12)(cid:12)(cid:12)Z I Z Ω |∇ θ n | dx dt − Z I Z Ω |∇ θ n | dx dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ F ∗ δ h n → n → ∞ .Therefore, taking any ̟ ∈ [ H (Ω) ∩ L ∞ (Ω)] and applying Lemma 6.1 for the case inwhich ̺ = 1, { ̺ n } = { } , ζ = θ , { ζ n } = { θ n } , ω = ̟ · [ ∇ α ]( v ), and { ω n } = { ̟ · [ ∇ α ]( v n ) } ,42e obtainlim n →∞ Z I Z Ω ̟ · [ ∇ α ]( v n ( t )) |∇ θ n ( t ) | dx dt = Z I Z Ω d [ ̟ · [ ∇ α ]( v ( t )) | Dθ ( t ) | ]for any ̟ ∈ [ H (Ω) ∩ L ∞ (Ω)] . (6.27)Furthermore, from (2.5), (5.24), and (6.25), we can compute the following:0 ≤ lim inf n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ν n Z I Z Ω ̟ · [ ∇ β ]( v n ( t )) |∇ θ n ( t ) | dx dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ν n Z I Z Ω ̟ · [ ∇ β ]( v n ( t )) |∇ θ n ( t ) | dx dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ̟ | L ∞ (Ω) | β | C ([0 , ) lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ν n Z I Z Ω |∇ θ n ( t ) | dx dt (cid:12)(cid:12)(cid:12)(cid:12) (6.28)= | ̟ | L ∞ (Ω) | β | C ([0 , ) lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ν n Z I Z Ω |∇ θ n ( t ) | dx dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ̟ | L ∞ (Ω) | β | C ([0 , ) δ lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ν n Z I Z Ω β ( v n ( t )) |∇ θ n ( t ) | dx dt (cid:12)(cid:12)(cid:12)(cid:12) = 0 for any ̟ ∈ [ H (Ω) ∩ L ∞ (Ω)] .Thanks to (2.4), (6.15)–(6.17), and (6.27)–(6.28), allowing n → ∞ in (6.21) yields Z I (cid:0) v t ( t ) , v ( t ) − ̟ (cid:1) L (Ω) dt + Z I (cid:0) [ ∇ g ]( v ( t )) , v ( t ) − ̟ (cid:1) L (Ω) dt + Z I (cid:0) ∇ v ( t ) , ∇ ( v ( t ) − ̟ ) (cid:1) L (Ω) × N dt + Z I Z Ω d (cid:2) ( v ( t ) − ̟ ) · [ ∇ α ]( v ( t )) | Dθ ( t ) | (cid:3) dt + Z I Γ( v ( t )) dt ≤ Z I Γ( ̟ ) dt for any ̟ ∈ [ H (Ω) ∩ L ∞ (Ω)] . (6.29)Since the open interval I ⊂ (0 , T ) is arbitrary, from (6.29) and the reformulation (2.16)in Remark 2.3, the triplet [ v, θ ] = [ w, η, θ ] fulfills conditions (S1) –(S2) . Remark 6.2 From the proof of Main Theorem 2, it can be said that the convex function β ∈ C ( R ) ∩ C ([0 , ) can be one of the approximation components for (S) . In this Appendix, we make supplementary statements for some preliminary facts as inSection 1, and the solutions to our systems.First, we show the proof of (Fact 1) in Remark 1.1, because the results in the reference[21, Chapter 2] were discussed under quite general settings, and these might not be to-the-point under our simplified setting. 43 roof of (Fact 1). Let us assume that [ ζ , ζ ∗ ] ∈ ∂ Ψ I in L ( I ; X ) , i.e.: Z I (cid:0) ζ ∗ ( t ) , ξ ( t ) − ζ ( t ) (cid:1) X dt ≤ Ψ I ( ξ ) − Ψ I ( ζ ) , for any ξ ∈ L ( I ; X ). (7.1)Also, let us take any open interval A ⊂ I , and denote by χ A : R → { , } the characteristicfunction of A . On this basis, we take any z ∈ E , and set ξ ( t ) := ζ ( t ) + χ A ( t )( z − ζ ( t )) in X , for a.e. t ∈ I ,as the test function ξ ∈ L ( I ; X ) in (7.1). Then, it is computed that Z A (cid:0) ζ ∗ ( t ) , z − ζ ( t ) (cid:1) X dt ≤ Ψ I ( ξ ) − Ψ I ( ζ ) ≤ Z A Ψ t ( z ) dt + Z I \ A Ψ t ( ζ ( t )) − Z I Ψ t ( ζ ( t )) dt = Z A Ψ t ( z ) dt − Z A Ψ t ( ζ ( t )) dt, for any z ∈ E and any open interval A ⊂ I . (7.2)Now, on account of the assumptions for E (= D (Ψ t )), we can see from (7.2) that[ ζ ( t ) , ζ ∗ ( t )] ∈ ∂ Ψ t in X , a.e. t ∈ I . (7.3)Conversely, if we suppose (7.3) for a pair [ ζ , ζ ∗ ] ∈ D (Ψ I ) × L ( I ; X ), then we imme-diately derive (7.1) as a straightforward consequence of the definition of subdifferential.Next, we briefly see the demonstration scenario of (Fact 7) in Remark 1.5. Proof of (Fact 7). First, let us take any ˜ z ∈ D (Ψ) with a sequence { ˜ z n | n ∈ N } ⊂ X such that ˜ z n ∈ D (Ψ n ) for any n ∈ N , and ˜ z n → ˜ z in X and Ψ n (˜ z n ) → Ψ(˜ z ) as n → ∞ .Then, with the assumptions in mind, we compute that:( z ∗ , ˜ z − z ) X + Ψ( z ) ≤ lim n →∞ ( z ∗ n , ˜ z n − z n ) X + lim inf n →∞ Ψ n ( z n ) ≤ lim sup n →∞ (cid:2) ( z ∗ n , ˜ z n − z n ) X + Ψ n ( z n ) (cid:3) ≤ lim n →∞ Ψ n (˜ z n ) = Ψ(˜ z ) , for any ˜ z ∈ D (Ψ). (7.4)Hence, [ z, z ∗ ] ∈ ∂ Ψ in X . Moreover, if we consider (7.4) anew, by setting ˜ z = z , then wecan show that lim sup n →∞ Ψ n ( z n ) ≤ lim n →∞ (cid:2) Ψ(˜ z n ) − ( z ∗ n , ˜ z n − z n ) X (cid:3) = Ψ( z ) . (7.5)The lower bound condition and (7.5) lead to the convergence lim n →∞ Ψ n ( z n ) = Ψ( z ),immediately.Finally, we leave the following remark as a further observation for the future works.44 emark 7.1 (Energy estimate) For any ν > 0, let [ v ν , θ ν ] = [ w ν , η ν , θ ν ] be the solutionto (S) ν obtained as the limits as in (5.6)-(5.11). On this basis, let us consider the inequality(5.2) in the cases when s = 0 and h νn , for n ∈ N , and take the limit as n → ∞ . Then, itis derived from (5.6)-(5.11) and Lemma 5.1 that12 Z t | ( v ν ) t ( τ ) | L (Ω) dτ + Z t | p α ( v ν ( τ ))( θ ν ) t ( τ ) | L (Ω) dτ + F ν ( v ν ( t ) , θ ν ( t )) ≤ F ν ( v ν , θ ν ) , for all t ∈ [0 , T ]. 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