Periodic solutions of a phase-field model with hysteresis
aa r X i v : . [ m a t h . A P ] J u l Periodic solutions of a phase-field model withhysteresis ∗ Chen Bin † and Sergey A. Timoshin ‡ July 21, 2020
Abstract
In the present paper we consider a partial differential system de-scribing a phase-field model with temperature dependent constraintfor the order parameter. The system consists of an energy balanceequation with a fairly general nonlinear heat source term and a phasedynamics equation which takes into account the hysteretic characterof the process. The existence of a periodic solution for this system isproved under a minimal set of assumptions on the curves defining thecorresponding hysteresis region.
Keywords : evolution system, hysteresis, phase transitions, periodic solutions.
In the space-time cylinder Q := [0 , T ] × Ω, where Ω ⊂ R N ( N ≥
1) is a boundeddomain with smooth boundary ∂ Ω and
T > ∗ Supported by the Program for Innovative Research Team in Science and Technologyin Fujian Province University, and Quanzhou High-Level Talents Support Plan underGrant 2017ZT012, and by the Scientific Research Funds of Huaqiao University nos. 605-50Y19017, 605-50Y14040. The research of the second author was also supported in partby RFBR grant no. 18-01-00026 † Fujian Province University Key Laboratory of Computational Science, Schoolof Mathematical Sciences, Huaqiao University, Quanzhou 362021, China, E-mail: [email protected] . ‡ Fujian Province University Key Laboratory of Computational Science, School of Math-ematical Sciences, Huaqiao University, Quanzhou 362021, China – and – Matrosov Insti-tute for System Dynamics and Control Theory, Russian Academy of Sciences, Lermontovstr. 134, 664033 Irkutsk, Russia, E-mail: [email protected] . u t − ∆ u = h ( u, v ) in Q, (1.1) v t − κ ∆ v + ∂I ( u ; v ) ∋ g ( u, v ) in Q, (1.2) u = v = 0 on (0 , T ) × ∂ Ω , (1.3) u ( x,
0) = u ( x, T ) , v ( x,
0) = v ( x, T ) on Ω . (1.4)Here, I ( u ; · ) is the indicator function of the interval [ f ∗ ( u ) , f ∗ ( u )], ∂I ( u ; · ) is itssubdifferential in the sense of convex analysis, h, g, f ∗ , f ∗ are given functions withthe properties specified in the next section, κ > . .
4) by ( P ). System ( P ) can be regardedas a dynamical model of a phase transition process between two distinct phases(such as solid-liquid) placed in the container Ω. The state variables u = u ( t, x ) and v = v ( t, x ) are then interpreted as the relative temperature and the order parameter(phase fraction of an individual phase), respectively. Eq. (1 .
2) with g ≡ v = f ∗ ( u ) and v = f ∗ ( u ), see [1–3] for details. The introduction of the latteroperator to the model accounts for hysteretic relationship between u and v , playingin this case the roles of the input and output functions, respectively.Recent years have seen a considerable amount of works on partial differentialequations with hysteresis. In particular, the questions on existence, uniquenessand large time behaviour of solutions to Cauchy problems for systems with state-dependent constraints, related to (1 . .
3) were addressed in a number of papers(see, e.g., [4–11], and references therein). At the same time, periodic problemsfor systems with hysteresis have received much less attention. In this respect,we mention the works [12–16]. While the first references in this list deal withdiscontinuous hysteresis operators of relay type, the last one addresses the system(1 . .
4) with h ( u, v ) = v , g ≡
0, and κ = 0. We note, however, that therequirements on the functions f ∗ and f ∗ in [16]: f ∗ , f ∗ ∈ C ( R ) ∩ L ∞ ( R ) are non-decreasing Lipschitz continuous, f ∗ ( u ) = f ∗ ( u ) for u ∈ ( −∞ , a ] ∪ [ b, + ∞ ) with a < < b , f ∗ is convex on ( −∞ , b ) and f ∗ is concave on ( a, + ∞ ), which are indispensablefor the proof in [16], appear to be too demanding. In our paper, we dispense withthese assumptions on the functions f ∗ and f ∗ retaining only the Lipschitz continuityand convexity. In this respect, as a byproduct of our analysis, we also improve theresults on the existence of a solution to the Cauchy problem from [7] by removingthe assumptions of smoothness, monotonicity and boundedness on the functions f ∗ and f ∗ . Moreover, the convergences of approximate solutions which the authorsobtain in [16, The proof of Theorem 2.1] would not be sufficient to treat generalnonlinear right-hand sides as in (1 . . v assuming the coefficient of the interfacial energy κ to be nonzero.Here, we would like to mention that hysteresis curves encountered in the prac-tice of physical measurements, stress-strain hysteresis loops in shape memory alloywires, load-displacement hysteresis curves in composite structures, magnetic hys-teresis curves of nano-minerals may genuinely occur to lack the smoothness. On the eriodic solutions of a phase-field model with hysteresis other hand, the convexity of hysteresis region is a common feature in mechanicaltransmission processes of motor-harmonic drives, plastic deformation behavior ofhigh-strength steel, damage models of welded joints in steel structures, dampingof martensitic shape memory alloys at low strain ranges and others. Also, we addthat for the completed relay operator and the truncated play operators employedto approximate the former (see [3]) the hysteresis regions are convex and the curvesdescribing them are piecewise linear but nonsmooth.The purpose of the present paper is to prove the existence of a solution tosystem ( P ) with sufficiently general h , g , f ∗ , and f ∗ . In our approach to establishthe existence for problem ( P ) we follow in part the idea from [17]. The originalproblem ( P ) being not easily amenable to a direct application of the Poincar´e maptechnique usually applied when dealing with periodic problems, we construct a fam-ily of suitable approximate problems based on the Yosida regularization ∂I λ ( u ; · ), λ >
0, of the subdifferential ∂I ( u ; · ). We then further regularize the nonsmoothfunctions f ∗ and f ∗ describing ∂I λ ( u ; · ) by sequences of mollifiers depending on aregularizing parameter ε > P ) λ,ε . Employing the Schauder fixed point argument we finda fixed point of this map which provides us with a solution to the approximateproblem ( P ) λ,ε . Next, we establish a priori estimates independent of ε for solutionsof the approximate problems and performing a limiting procedure as ε → P ) λ depending now on the parame-ter λ only. Deriving uniform estimates with respect to λ for the latter system, wefinally prove the existence of a solution to problem ( P ) through the passage-to-the-limit procedure when λ →
0. We note that in order to get suitable compactnessproperties and, thus, to legitimate this passage-to-the-limit we exploit essentiallythe properties derived from the specific structure of the approximate equations for(1 . . . In this section, we recall some notions which we use in the paper and posit assump-tions on the data describing Problem ( P ).Throughout the paper, we denote by H the Hilbert space L (Ω) with thestandard inner product h· , ·i H , and by V the Sobolev space H (Ω). The innerproduct in H × H we denote by h· , ·i H × H and | · | ∞ stands for L ∞ norms in variousspaces.Given a Hilbert space X with the inner product h· , ·i X and a convex, lowersemicontinuous function ϕ : X → R ∪ { + ∞} which is not identically + ∞ , thesubdifferential ∂ϕ ( x ) of ϕ at a point x ∈ X is, in general, a set defined by the rule ∂ϕ ( x ) = { h ∈ X ; h h, y − x i X ≤ ϕ ( y ) − ϕ ( x ) ∀ y ∈ X } . Let f ∗ , f ∗ be two Lipschitz continuous functions defined on R . Then, the subdifferential of the indicator function I ( u ; · ), u ∈ R , I ( u ; v ) := (cid:26) f ∗ ( u ) ≤ v ≤ f ∗ ( u ) , + ∞ otherwise , of the interval [ f ∗ ( u ) , f ∗ ( u )] has the form: ∂I ( u ; v ) = ∅ if v / ∈ K ( u ) , [0 , + ∞ ) if v = f ∗ ( u ) > f ∗ ( u ) , { } if f ∗ ( u ) < v < f ∗ ( u ) , ( −∞ ,
0] if v = f ∗ ( u ) < f ∗ ( u ) , ( −∞ , + ∞ ) if v = f ∗ ( u ) = f ∗ ( u ) . (2.1)For λ >
0, the Yosida regularization of ∂I ( u ; v ) is the function ∂I λ ( u ; v ) = 1 λ [ v − f ∗ ( u )] + − λ [ f ∗ ( u ) − v ] + , u, v ∈ R . (2.2)Let f : R → R be a Lipschitz continuous function. For ε >
0, denote by f ε ( u ), u ∈ R , the following regularization of the function f ε ( u ): f ε ( u ) := Z R f ( s ) ρ ε ( u − s ) ds = Z R f ( u − εs ) ρ ( s ) ds, (2.3)where ρ ∈ C ∞ ( R ) is such that ρ ≥ ρ ( s ) = 0 when | s | ≥ ρ ( s ) = ρ ( − s ), R R ρ ( s ) ds = 1, ρ ε ( s ) := ε − ρ (cid:0) sε (cid:1) .The lemma below follows directly from the definition of f ε ( u ) and the propertiesof ρ ε ( s ). Lemma 2.1.
The function f ε ( u ) possesses the following properties: (1) f ε ( u ) ∈ C ∞ ( R );(2) f ε ( u ) is Lipschitz continuous with the same Lipschitz constant as f ( u );(3) f ε ( u ) → f ( u ) as ε → uniformly on R ;(4) | f ε | ∞ ≤ | f | ∞ . Problem (1 . .
4) is considered under the following hypotheses: (H1) the functions h, g : R → R are bounded and Lipschitz continuous (with acommon Lipschitz constant L > (H2) the functions f ∗ , f ∗ : R → R are Lipschitz continuous, f ∗ ( u ) ≤ f ∗ ( u ) forall u ∈ R , and there exist constants a, b , a < b , such that f ∗ ( u ) = f ∗ ( u ) for u ∈ R \ ( a, b ), f ∗ is convex on ( −∞ , b ), f ∗ is concave on ( a, + ∞ ). In addition,we assume the following compatibility condition f ∗ (0) ≤ ≤ f ∗ (0).Next, we define a notion of solution to our Problem ( P ). eriodic solutions of a phase-field model with hysteresis Definition 2.1.
A pair { u, v } is called a solution of system (1 . – (1 . if ( i ) u, v ∈ W , (0 , T ; H ) ∩ L ∞ (0 , T ; V ) ∩ L (0 , T ]; H (Ω)) ; ( ii ) u ′ − ∆ u = h ( u, v ) in H a.e. on [0 , T ] ; ( iii ) v ′ − κ ∆ v + ∂I ( u ; v ) ∋ g ( u, v ) in H a.e. on [0 , T ] ; ( iv ) u = v = 0 on ∂ Ω ( in the sense of traces ) a.e. on [0 , T ] ; ( iv ) u (0) = u ( T ) , v (0) = v ( T ) in H ,where the prime denotes the derivative with respect to t . In order to prove the existence of a solution to our Problem ( P ), we approximate thelatter by a family of suitable problems depending on two approximation parameterswhich we introduce next.Let ˜ f ∗ , ˜ f ∗ be two Lipschitz continuous functions (with a common Lipschitzconstant L >
1) such that ˜ f ∗ is convex on R , ˜ f ∗ ( u ) = f ∗ ( u ) for u ∈ ( −∞ , b ), ˜ f ∗ isconcave on R , ˜ f ∗ ( u ) = f ∗ ( u ) for u ∈ ( a, + ∞ ). Further, let ˜ f ∗ ε ( u ) and ˜ f ∗ ε ( u ) be theregularizations as in (2 .
3) of the functions ˜ f ∗ ( u ) and ˜ f ∗ ( u ), respectively.For λ, ε >
0, we consider the following approximate periodic problem denotedby ( P ) λ,ε : u ′ − ∆ u = h ( u, v ) in H a.e. on [0 , T ] , (3.1) v ′ − κ ∆ v + ∂ ˜ I λε ( u ; v ) = g ( u, v ) in H a.e. on [0 , T ] , (3.2) u = v = 0 on ∂ Ω a.e. on [0 , T ] , (3.3) u (0) = u ( T ) , v (0) = v ( T ) in H, (3.4)where ∂ ˜ I λε ( u ; v ) is defined as ∂I λ ( u ; v ) in (2 .
2) with f ∗ and f ∗ replaced by ˜ f ∗ ε and˜ f ∗ ε , respectively.A pair of functions { u, v } is called a solution to ( P ) λ,ε if u, v ∈ W , (0 , T ; H ) ∩ L ∞ (0 , T ; V ) ∩ L (0 , T ; H (Ω)) and (3 . .
4) hold.In this section, we prove the existence of solutions for problems ( P ) λ,ε , λ, ε > P ) λ,ε , define the so-called Poincar´e map which with theinitial data of the Cauchy problem associates the values at the final time T of itsunique solution and establish the continuity of this map. Then, in the second step,we show that the Poincar´e map is a self-mapping. Finally, we use these propertiesto construct a solution to ( P ) λ,ε by the Schauder fixed point argument.To this aim, consider the following approximate Cauchy problem stemmingfrom ( P ) λ,ε which we denote by ( C ) λ,ε : u ′ − ∆ u = h ( u, v ) in H a.e. on [0 , T ] , (3.5) v ′ − κ ∆ v + ∂ ˜ I λε ( u ; v ) = g ( u, v ) in H a.e. on [0 , T ] , (3.6) u = v = 0 on ∂ Ω a.e. on [0 , T ] , (3.7) u (0) = u , v (0) = v in H, (3.8)where u , v are given functions such that u , v ∈ L ∞ (Ω) ∩ V . By the results of [18]for any λ, ε >
0, there exists a unique solution { u λε , v λε } to Problem ( C ) λ,ε .For any λ, ε >
0, define a single-valued mapping P λε : H × H → H × H by P λε : ( u , v ) ( u λε ( T ) , v λε ( T )) , where { u λε , v λε } is the unique solution of ( C ) λ,ε with the initial data ( u , v ). Theorem 3.1 (existence of approximate solutions) . There exists a constant κ > such that for any ε, λ > , κ ∈ (0 , κ ] Problem ( P ) λ,ε admits a solution { u λε , v λε } . In order to prove this theorem, we invoke Schauder’s fixed point theorem showingthat P λε is a continuous self-mapping on a compact convex set which we introducein the next section. P λε in H × H Proposition 3.1.
The mapping P λε is continuous in H × H .Proof. Let u ,n , v ,n , u , v ∈ H be such that ( u ,n , u ,n ) → ( u , v ) in H × H strongly and let { u n , v n } and { u, v } be the unique solutions of ( C ) λ,ε with the initialdata ( u ,n , v ,n ) and ( u , v ), respectively. Subtracting (3.5), (3.6) for { u n , v n } fromthat for { u, v } we have( u n − u ) ′ − ∆( u n − u ) = h ( u n , v n ) − h ( u, v ) , (3.9)( v n − v ) ′ − κ ∆( v n − v ) + (cid:0) ∂I λε ( u n ; v n ) − ∂I λε ( u ; v ) (cid:1) = g ( u n , v n ) − g ( u, v ) , (3.10)Testing (3.9) by u n − u and (3.10) by v n − v , then summing up the results andusing the Lipschitz continuity of h, g and Young’s inequality we obtain12 ddt (cid:0) | u n − u | H + | v n − v | H (cid:1) + |∇ ( u n − u ) | H + κ |∇ ( v n − v ) | H ≤ L (cid:0) | u n − u | H + | v n − v | H (cid:1) + Z Ω (cid:16) ∂ ˜ I λε ( u n ; v n ) − ∂ ˜ I λε ( u ; v ) (cid:17) ( v − v n ) dx. (3.11)We denote the last integral by S and estimate it as follows. S = 1 λ Z Ω (cid:16) [ v n − ˜ f ∗ ε ( u n )] + − [ v − ˜ f ∗ ε ( u )] + (cid:17) ( v − v n ) dx + 1 λ Z Ω (cid:16) [ ˜ f ∗ ε ( u ) − v ] + − [ ˜ f ∗ ε ( u n ) − v n )] + (cid:17) ( v − v n ) dx =: S ∗ + S ∗ (3.12) eriodic solutions of a phase-field model with hysteresis S ∗ = 1 λ Z Ω (cid:16) [ v n − v + v − ˜ f ∗ ε ( u ) + ˜ f ∗ ε ( u ) − ˜ f ∗ ε ( u n )] + − [ v − ˜ f ∗ ε ( u )] + (cid:17) ( v − v n ) dx ≤ λ Z Ω (cid:16) | v n − v | + (cid:12)(cid:12)(cid:12) ˜ f ∗ ε ( u ) − ˜ f ∗ ε ( u n ) (cid:12)(cid:12)(cid:12) (cid:17) | v n − v | dx ≤ L + 1 λ (cid:0) | u n − u | H + | v n − v | H (cid:1) . (3.13)Here, we have used the Lipschitz continuity of ˜ f ∗ ε and the following inequality validfor arbitrary α, β, γ, δ ∈ R : (cid:0) [ α + β + γ ] + − [ β ] + (cid:1) δ ≤ ( | α | + | γ | ) | δ | . (3.14)If δ >
0, (3.14) follows from the inequalities [ α + β ] + ≤ [ α ] + + [ β ] + and [ α ] + ≤ | α | .If δ <
0, we have (cid:0) [ β ] + − [ α + β + γ ] + (cid:1) ( − δ ) ≤ [ − ( α + γ )] + ( − δ ) ≤ | α + γ || δ | and (3.14) follows again. Similarly, we have S ∗ ≤ L + 1 λ (cid:0) | u n − u | H + | v n − v | H (cid:1) . (3.15)Hence, since the second and the third terms on the left-hand side of (3.11) arenonnegative, invoking Gronwall’s lemma from (3.11)–(3.13), (3.15) we infer that | u n ( T ) − u ( T ) | H + | v n ( T ) − v ( T ) | H ≤ C (cid:0) | u n, − u | H + | v n, − v | H (cid:1) → n → ∞ , where C = C ( λ, L, L ) is a positive constant. Therefore,( u n ( T ) , v n ( T )) → ( u ( T ) , v ( T ))strongly in H × H as n → ∞ , and, hence, P λε is continuous in H × H . (cid:3) P λε is a self-mapping First, we recall the following result.
Lemma 3.1 ( [7, Lemma 4.1]) . Let ( u, v ) be a solution of (3 . , (3 . . Then, thefunction t ˜ I λε ( u ; v ) is absolutely continuous on [0 , T ] and ddt ˜ I λε ( u ; v ) ≤ h ∂ ˜ I λε ( u ; v ) , v ′ i H + L | u ′ | H | ∂ ˜ I λε ( u ; v ) | H a.e. in (0 , T ) . Define on H × H the function φ λε ( u, v ) = (cid:26) L |∇ u | H + κ |∇ v | H + ˜ I λε ( u ; v ) if ( u, v ) ∈ V × V, + ∞ otherwise , where ˜ I λε ( u ; v ) = 12 λ (cid:12)(cid:12)(cid:12) [ v − ˜ f ∗ ε ( u )] + (cid:12)(cid:12)(cid:12) H + 12 λ (cid:12)(cid:12)(cid:12) [ ˜ f ∗ ε ( u ) − v ] + (cid:12)(cid:12)(cid:12) H . Proposition 3.2.
There exist constants κ , R > such that P λε maps the set B R := { ( u, v ) ∈ V × V : φ λε ( u, v ) ≤ R } into itself for κ ∈ (0 , κ ] .Proof. From the definition of subdifferential in view of the compatibility conditionof ( H
2) we obtain φ λε ( u λε , v λε ) ≤ φ λε (0 ,
0) + h ∂φ λε ( u λε , v λε ) , ( u λε , v λε ) i H × H = h ∂ ˜ I λε ( u λε ; v λε ) − κ ∆ v λε , v λε i H + h− L ∆ u λε , u λε i H − (cid:28) λ [ v λε − ˜ f ∗ ε ( u λε )] + ( ˜ f ∗ ε ) ′ ( u λε ) , u λε (cid:29) H + (cid:28) λ [ ˜ f ∗ ε ( u λε ) − v λε ] + ( ˜ f ∗ ε ) ′ ( u λε ) , u λε (cid:29) H ≤ h g ( u λε , v λε ) − v ′ λε , v λε i H + 8 L h h ( u λε , v λε ) − u ′ λε , u λε i H + L (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H | u λε | H . Using Young’s and the Poincar´e inequalities we further obtain φ λε ( u λε , v λε ) ≤ L γ | u ′ λε | H + 12 γ | v ′ λε | H + γC P (cid:0) |∇ u λε | H + |∇ v λε | H (cid:1) + 3 L γ (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + C ≤ L γ | u ′ λε | H + 12 γ | v ′ λε | H + 2 γC P κ φ λε ( u λε , v λε )+ 3 L γ (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + C , where C := L γ | Ω | ( | g | ∞ + | h | ∞ ), | Ω | is the Lebesgue measure of Ω, and C P = C (Ω) > γ ≤ κ C P from thelast inequality we have φ λε ( u λε , v λε ) ≤ L γ | u ′ λε | H + 1 γ | v ′ λε | H + 3 L γ (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + 2 C . (3.16) eriodic solutions of a phase-field model with hysteresis Testing Eq. (3.5) by u ′ λε and applying Young’s inequality we obtain | u ′ λε | H + ddt |∇ u λε | H ≤ C a.e. on (0 , T ) , (3.17)where C := | h | ∞ | Ω | . Similarly, testing Eq. (3.5) by − ∆ u λε we have ddt |∇ u λε | H + | ∆ u λε | H ≤ C a.e. on (0 , T ) . (3.18)Next, testing Eq. (3.6) by v ′ λε we see in view of Lemma 3.1 that12 | v ′ λε | H + ddt (cid:16) κ |∇ v λε | H + ˜ I λε ( u λε ; v λε ) (cid:17) ≤ L | u ′ λε | H + 14 (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + C (3.19)a.e. on (0 , T ), where C := | g | ∞ | Ω | . Then, testing Eq. (3.6) by − κ ∆ v λε yields κ | ∆ v λε | H + ddt (cid:16) κ |∇ v λε | H (cid:17) ≤ D κ ∆ v λε , ∂ ˜ I λε ( u λε ; v λε ) E H + 12 κ | ∆ v λε | H + 4 C (3.20)a.e. on (0 , T ). We evaluate the first term on the right-hand side of this inequalityas follows D ∂ ˜ I λε ( u λε ; v λε ) , κ ∆ v λε E H = (cid:28) κλ h v λε − ˜ f ∗ ε ( u λε ) i + , ∆ (cid:16) v λε − ˜ f ∗ ε ( u λε ) (cid:17)(cid:29) H + (cid:28) κλ h v λε − ˜ f ∗ ε ( u λε ) i + , ∆ ˜ f ∗ ε ( u λε ) (cid:29) H + (cid:28) κλ h ˜ f ∗ ε ( u λε ) − v λε i + , ∆ (cid:16) ˜ f ∗ ε ( u λε ) − v λε (cid:17)(cid:29) H + (cid:28) κλ h ˜ f ∗ ε ( u λε ) − v λε i + , − ∆ ˜ f ∗ ε ( u λε ) (cid:29) H = − κλ (cid:12)(cid:12)(cid:12)(cid:12) ∇ h v λε − ˜ f ∗ ε ( u λε ) i + (cid:12)(cid:12)(cid:12)(cid:12) H + (cid:28) κλ h v λε − ˜ f ∗ ε ( u λε ) i + , ∆ ˜ f ∗ ε ( u λε ) (cid:29) H − κλ (cid:12)(cid:12)(cid:12)(cid:12) ∇ h ˜ f ∗ ε ( u λε ) − v λε i + (cid:12)(cid:12)(cid:12)(cid:12) H + (cid:28) κλ h ˜ f ∗ ε ( u λε ) − v λε i + , − ∆ ˜ f ∗ ε ( u λε ) (cid:29) H ≤ λ ((cid:12)(cid:12)(cid:12)(cid:12)h v λε − ˜ f ∗ ε ( u λε ) i + (cid:12)(cid:12)(cid:12)(cid:12) H + (cid:12)(cid:12)(cid:12)(cid:12)h ˜ f ∗ ε ( u λε ) − v λε i + (cid:12)(cid:12)(cid:12)(cid:12) H ) + 2 κ (cid:18)(cid:12)(cid:12)(cid:12) ∆ ˜ f ∗ ε ( u λε ) (cid:12)(cid:12)(cid:12) H + (cid:12)(cid:12)(cid:12) ∆ ˜ f ∗ ε ( u λε ) (cid:12)(cid:12)(cid:12) H (cid:19) ≤ (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + 2 κ (cid:18)(cid:12)(cid:12)(cid:12) ∆ ˜ f ∗ ε ( u λε ) (cid:12)(cid:12)(cid:12) H + (cid:12)(cid:12)(cid:12) ∆ ˜ f ∗ ε ( u λε ) (cid:12)(cid:12)(cid:12) H (cid:19) . Observing that ∆ f ( u ) = f ′′ ( u ) |∇ u | H + f ′ ( u )∆ u and invoking the Gagliardo-Nirenberginequality from the last inequality we obtain D ∂ ˜ I λε ( u λε ; v λε ) , κ ∆ v λε E H ≤ (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + 2 κ C (cid:0) | u λε | H + | ∆ u λε | H (cid:1) for a constant C = C ( | f ∗′ | ∞ , | f ∗′ | ∞ , | f ∗′′ | ∞ , | f ∗′′ | ∞ , | Ω | ) >
1, so that (3.20) impliesthat κ | ∆ v λε | H + ddt (cid:16) κ |∇ v λε | H (cid:17) ≤ (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + 2 κ C (cid:0) | u λε | H + | ∆ u λε | H (cid:1) (3.21)a.e. on (0 , T ). Similarly, testing Eq. (3.6) by ∂ ˜ I λε ( u λε ; v λε ) and using Lemma 3.1we see that 12 (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + ddt ˜ I λε ( u λε ; v λε ) ≤ L | u ′ λε | H + 2 κ C (cid:0) | u λε | H + | ∆ u λε | H (cid:1) + 4 C (3.22)a.e. on (0 , T ).Calculating 4 L × { (3 .
17) + (3 . } + (3 .
19) + (3 .
21) + (3 .
22) we obtain2 L | u ′ λε | H + 12 | v ′ λε | H + 4( L − κ C ) | ∆ u λε | H + κ | ∆ v λε | H + 18 (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + 2 ddt φ λε ( u λε ; v λε ) ≤ C C P κ L φ λε ( u λε , v λε ) + C (3.23)a.e. on (0 , T ), where C := 8 L C + 5 C . Now, we calculate σ × (3 .
23) + (3 .
16) forsome σ > L (cid:18) σ − γ (cid:19) | u ′ λε | H + (cid:18) σ − γ (cid:19) | v ′ λε | H + 4 σ ( L − κ C ) | ∆ u λε | H + σκ | ∆ v λε | H + 12 (cid:18) σ − L γ (cid:19) (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) H + (cid:18) − σC C P κ L (cid:19) φ λε ( u λε , v λε ) + 2 σ ddt φ λε ( u λε ; v λε ) ≤ C + σC a.e. on (0 , T ). Taking κ ≤ κ := C P C and choosing σ in the interval (cid:16) L C P κ , L C C P κ (cid:17) we see that all the coefficients in the last inequality are positive so that there existtwo positive constants α and β such that ddt φ λε ( u λε , v λε ) + αφ λε ( u λε , v λε ) ≤ β. Therefore, by virtue of Proposition A.1 in [17] one can take a constant R ≥ β/α > φ λε ( u λε ( T ) , v λε ( T )) ≤ R if φ λε ( u , v ) ≤ R. Consequently, we conclude that P λε maps the set B R into itself for κ ∈ (0 , κ ]. (cid:3) eriodic solutions of a phase-field model with hysteresis Now we are in a position to finish the proof of Theorem 3.1.From the definition of the set B R by the Rellich-Kondrachov theorem we seethat the set B R is compact. Since it it obviously convex, combining Propositions4.1 and 4.2 and applying Schauder’s fixed point theorem to P λε : B R → B R we cantake a fixed point ( u ∗ , v ∗ ) ∈ B R such that P λε ( u ∗ , v ∗ ) = ( u ∗ , v ∗ ), which providesthe desired solution of Problem ( P ) λ,ε , λ, ε > ( P ) In this section, first we derive uniform a priori estimates independent of the pa-rameter ε > { u λε , v λε } of the approximate periodic Problem ( P ) λ,ε ,which will allow us to derive the convergence of { u λε , v λε } as ε → { u λ , v λ } of an intermediate approximate problem depending on the parameter λ only. Then, we establish uniform bounds independent of the parameter λ for solu-tions { u λ , v λ } of the latter system and finally pass to the limit as λ → P ).We note that the a priori estimates for solutions of the approximate Cauchyproblem ( C ) λ,ε of the previous section do not depend on the initial values in (3 . P ) λ,ε .We will use this fact throughout this section without further mentioning it. ε → First, we fix λ > .
1) by u λε to obtain12 ddt | u λε | H + |∇ u λε | H ≤ C P | h | ∞ | Ω | + 12 C P | u λε | H a.e. on (0 , T ) , where recall that C P > ddt | u λε | H + |∇ u λε | H ≤ C P | h | ∞ | Ω | a.e. on (0 , T ) . Integrating this inequality from 0 to T and taking account of (3 .
4) we obtain Z T |∇ u λε ( τ ) | H dτ ≤ R , (4.1)where R := C P | h | ∞ | Ω | T . Next, we show that the sequence {∇ u λε } ε> is bounded in L ∞ (0 , T ; H ) . (4.2)Reasoning by contradiction, suppose that there exists a subsequence {∇ u λε n } n ≥ of this sequence such that |∇ u λε n | L ∞ (0 ,T ; H ) → + ∞ as n → ∞ . From (3 .
17) we seethat ddt |∇ u λε n | H ≤ C a.e. on [0 , T ] . (4.3) Integrating this inequality over [0 , t ], t ∈ (0 , T ], we obtain |∇ u λε n ( t ) | H ≤ |∇ u λε n (0) | H + R for all t ∈ [0 , T ] , n ≥ , where R := C T . This inequality implies that |∇ u λε n | L ∞ (0 ,T ; H ) ≤ |∇ u λε n (0) | H + R and, thus, we have |∇ u λε n ( T ) | H = |∇ u λε n (0) | H → + ∞ as n → ∞ (4.4)by our assumption. Integrating (4 .
3) over [ t, T ], t ∈ [0 , T ), gives |∇ u λε n ( T ) | H ≤ |∇ u λε n ( t ) | H + R for all t ∈ [0 , T ] , n ≥ . Hence, by virtue of (4 .
4) we derive that Z T |∇ u λε n ( τ ) | H dτ → + ∞ as n → ∞ , which is in contradiction with (4 . . .
18) from 0 to T and taking account of the periodicitycondition (3 .
4) and (4 .
2) we see that the following uniform with respect to theparameter ε > u λε , v λε ) ofthe approximate periodic problem ( P ) λε : | u ′ λε | L (0 ,T ; H ) + | ∆ u λε | L (0 ,T ; H ) + |∇ u λε | L ∞ (0 ,T ; H ) ≤ R , (4.5)for a positive constant R independent of ε . On account of these uniform estimates,by weak and weak-star compactness results, there exists a null sequence ε n , n ≥ ,
1] and a function u λ such that u λε n → u λ weakly in W , (0 , T ; H ) ∩ L (0 , T ; H (Ω))and weakly-star in L ∞ (0 , T ; V ) . (4.6)In particular, we also have u λε n → u λ in C ([0 , T ]; H ) . (4.7)Invoking the Poincar´e inequality from (4 .
1) we obtain Z T | u λε ( τ ) | H dτ ≤ C P R . (4.8)Now, taking the sum of the inequalities (3 . . . T we see in view of (3 . . .
8) that | v ′ λε | L (0 ,T ; H ) + κ | ∆ v λε | L (0 ,T ; H ) + (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) L (0 ,T ; H ) ≤ R (4.9) eriodic solutions of a phase-field model with hysteresis for a constant R > ε . The last inequality implies that thereexists a function v λ such that v λε n → v λ weakly in W , (0 , T ; H ) (4.10)and κ ∆ v λε n → κ ∆ v λ weakly in L (0 , T ; H ) . (4.11)Below, we show that along with the convergences (4 . . . .
11) we alsohave v λε n → v λ strongly in C ([0 , T ]; H ) . (4.12)To this end, take two arbitrary i, j ≥ i = j and denote u k := u λε k , v k := v λε k , f ∗ k := ˜ f ∗ ε k , f ∗ k := ˜ f ∗ ε k , I k := ˜ I λε k , k = i, j . Then, from (3 .
2) it follows that v ′ j − v ′ i − κ (∆ v j − ∆ v i ) + ∂I j ( u j ; v j ) − ∂I i ( u i ; v i ) = g ( u j , v j ) − g ( u i , v i ) . Testing this equality by v j − v i , using the Lipschitz continuity of g and invokingYoung’s inequality we have12 ddt | v j − v i | H + κ |∇ ( v j − v i ) | H + h ∂I j ( u j ; v j ) − ∂I i ( u i ; v i ) , v j − v i i H ≤ L ( | v j − v i | H + | u j − u i | H ) . (4.13)Setting S εij := h ∂I j ( u j ; v j ) − ∂I i ( u i ; v i ) , v j − v i i H . (4.14)we see from (2 .
2) that S εij = (cid:28) λ [ v j − f ∗ j ( u j )] + − λ [ f ∗ j ( u j ) − v j ] + − λ [ v i − f ∗ i ( u i )] + + 1 λ [ f ∗ i ( u i ) − v i ] + , v j − v i (cid:29) H . We have nine possible cases to estimate the value of S εij from below. First, assumingthat v j ≥ f ∗ j ( u j ), v i ≥ f ∗ i ( u i ) we obtain S εij = (cid:28) λ ( v j − f ∗ j ( u j )) − λ ( v i − f ∗ i ( u i )) , v j − v i (cid:29) H ≥ − λ | v j − v i | H − λ (cid:8) | f ∗ j ( u j ) − f ∗ j ( u i ) | H + | f ∗ j ( u i ) − f ∗ i ( u i ) | H (cid:9) . Second, when v j ≥ f ∗ j ( u j ), f ∗ i ( u i ) ≤ v i < f ∗ i ( u i ) we see that S εij = (cid:28) λ ( v j − f ∗ j ( u j )) , v j − v i (cid:29) H ≥ −| ∂I j ( u j ; v j ) | H (cid:8) | f ∗ j ( u j ) − f ∗ j ( u i ) | H + | f ∗ j ( u i ) − f ∗ i ( u i ) | H (cid:9) . Third, if v j ≥ f ∗ j ( u j ), v i < f ∗ i ( u i ), then S εij = (cid:28) λ ( v j − f ∗ j ( u j )) + 1 λ ( f ∗ i ( u i ) − v i ) , v j − v i (cid:29) H ≥ − ( | ∂I j ( u j ; v j ) | H + | ∂I i ( u i ; v i ) | H ) (cid:8) | f ∗ j ( u j ) − f ∗ j ( u i ) | H + | f ∗ j ( u i ) − f ∗ i ( u i ) | H (cid:9) . The reasoning in the remaining cases: f ∗ j ( u j ) < v j < f ∗ j ( u j ) , v i ≥ f ∗ i ( u i ) f ∗ j ( u j ) < v j < f ∗ j ( u j ) , f ∗ i ( u i ) ≤ v i < f ∗ i ( u i ) f ∗ j ( u j ) < v j < f ∗ j ( u j ) , v i < f ∗ i ( u i ) v j ≤ f ∗ i ( u j ) , v i ≥ f ∗ i ( u i ) v j ≤ f ∗ i ( u j ) , f ∗ i ( u i ) ≤ v i < f ∗ i ( u i ) v j ≤ f ∗ i ( u j ) , v i < f ∗ i ( u i )is fully symmetric and is left to the reader. Consequently, we always have S εij ≥ − λ | v j − v i | H − λ {| f ∗ j ( u j ) − f ∗ j ( u i ) | H + | f ∗ j ( u i ) − f ∗ i ( u i ) | H + | f ∗ j ( u j ) − f ∗ j ( u i ) | H + | f ∗ j ( u i ) − f ∗ i ( u i ) | H }− ( | ∂I j ( u j ; v j ) | H + | ∂I i ( u i ; v i ) | H ) {| f ∗ j ( u j ) − f ∗ j ( u i ) | H + | f ∗ j ( u i ) − f ∗ i ( u i ) | H + | f ∗ j ( u j ) − f ∗ j ( u i ) | H + | f ∗ j ( u i ) − f ∗ i ( u i ) | H } =: − λ | v j − v i | H − δ εij . Therefore, integrating inequality (4 .
13) from 0 to t ∈ [0 , T ] we infer in view of (4 . | v j − v i | H ( t ) ≤ (cid:18) L + 1 λ (cid:19) Z t | v j − v i | H ( τ ) dτ + 4 L Z t | u j − u i | H ( τ ) dτ + 2 Z t δ εij ( τ ) dτ. Applying Gronwall’s inequality to this inequality we conclude in view of the con-vergence (4 . .
9) that v i , i ≥
1, is a Cauchy sequence in thespace C ([0 , T ]; H ). Hence, according to (4 .
10) we obtain the convergence (4 . . . . .
12) and Lemma 2.1 (2), (3)we see that the pair { u λ , v λ } , λ >
0, is a solution of the following system, which wedenote by ( P ) λ : u ′ − ∆ u = h ( u, v ) in H a.e. on [0 , T ] , (4.15) v ′ − κ ∆ v + ∂ ˜ I λ ( u ; v ) = g ( u, v ) in H a.e. on [0 , T ] , (4.16) u = v = 0 on ∂ Ω a.e. on [0 , T ] , (4.17) eriodic solutions of a phase-field model with hysteresis u (0) = u ( T ) , v (0) = v ( T ) in H, (4.18)where ∂ ˜ I λ ( u ; v ) is defined as ∂I λ ( u ; v ) in (2 .
2) with f ∗ and f ∗ replaced by ˜ f ∗ and˜ f ∗ , respectively.A solution to ( P ) λ is a pair of functions { u, v } such that u, v ∈ W , (0 , T ; H ) ∩ L ∞ (0 , T ; V ) ∩ L (0 , T ; H (Ω)) and (4 . .
18) hold.We note that the validity of the periodic condition (4 .
18) follows from (3 . . . λ → We now derive a priori estimates uniform with respect to the parameter λ > u λ , v λ ) of Problem ( P ) λ .To this aim, we note that the constants R i , i = 1 , . . . ,
4, in the uniformestimates of the previous subsection do not depend on λ . Hence, repeating thereasoning in derivation of (4 .
5) and (4 .
9) we obtain | u ′ λε | L (0 ,T ; H ) + | ∆ u λε | L (0 ,T ; H ) + |∇ u λε | L ∞ (0 ,T ; H ) + | v ′ λε | L (0 ,T ; H ) + κ | ∆ v λε | L (0 ,T ; H ) + (cid:12)(cid:12)(cid:12) ∂ ˜ I λε ( u λε ; v λε ) (cid:12)(cid:12)(cid:12) L (0 ,T ; H ) ≤ R for a constant R > λ . In particular, as above we conclude thatthere exists a null sequence λ n , n ≥
1, in (0 ,
1] and functions u, v such that u λ n → u weakly in W , (0 , T ; H ) ∩ L (0 , T ; H (Ω))and weakly-star in L ∞ (0 , T ; V )and, thus, strongly in C ([0 , T ]; H ) , (4.19) v λ n → v weakly in W , (0 , T ; H ) , (4.20) κ ∆ v λ n → κ ∆ v weakly in L (0 , T ; H ) , (4.21) ∂ ˜ I λ n ( u n , v n ) → ξ weakly in L (0 , T ; H ) (4.22)for some function ξ ∈ L (0 , T ; H ).Below, we show that along with the convergences (4 . .
22) we also have v λ n → v strongly in C ([0 , T ]; H ) . (4.23)To this end, take two arbitrary i, j ≥ i = j and denote u i := u λ i , v i := v λ i .Then, from (4 .
16) it follows that v ′ j − v ′ i − κ (∆ v j − ∆ v i ) + ∂ ˜ I λ j ( u j ; v j ) − ∂ ˜ I λ i ( u i ; v i ) = g ( u j , v j ) − g ( u i , v i ) , Testing this equality by v j − v i , using the Lipschitz continuity of g and invokingYoung’s inequality we have12 ddt | v j − v i | H + κ |∇ ( v j − v i ) | H + h ∂ ˜ I λ j ( u j ; v j ) − ∂ ˜ I λ i ( u i ; v i ) , v j − v i i H ≤ L ( | v j − v i | H + | u j − u i | H ) . (4.24) Setting S λij = h ∂ ˜ I λ j ( u j ; v j ) − ∂ ˜ I λ i ( u i ; v i ) , v j − v i i H . (4.25)we see from (2 .
2) that S λij = (cid:28) λ j [ v j − ˜ f ∗ ( u j )] + − λ j [ ˜ f ∗ ( u j ) − v j ] + − λ i [ v i − ˜ f ∗ ( u i )] + + 1 λ i [ ˜ f ∗ ( u i ) − v i ] + , v j − v i (cid:29) H . We have nine possible cases to estimate the value of S λij from below. First, assumingthat v j ≥ ˜ f ∗ ( u j ), v i ≥ ˜ f ∗ ( u i ) we obtain S λij = (cid:28) h λ j ( v j − ˜ f ∗ ( u j )) − λ i ( v i − ˜ f ∗ ( u i )) ,λ j λ j ( v j − ˜ f ∗ ( u j )) − λ i λ i ( v i − ˜ f ∗ ( u i )) + ˜ f ∗ ( u j ) − ˜ f ∗ ( u i ) (cid:29) H ≥ λ j | ∂ ˜ I λ j ( u j ; v j ) | H + λ i | ∂ ˜ I λ i ( u i ; v i ) | H − ( λ j + λ i ) | ∂ ˜ I λ j ( u j ; v j ) | H | ∂ ˜ I λ i ( u i ; v i ) | H − ( | ∂ ˜ I λ j ( u j ; v j ) | H + | ∂ ˜ I λ i ( u i ; v i ) | H ) | ˜ f ∗ ( u j ) − ˜ f ∗ ( u i ) | H . Second, when v j ≥ ˜ f ∗ ( u j ), ˜ f ∗ ( u i ) ≤ v i < ˜ f ∗ ( u i ) we see that S λij = (cid:28) λ j ( v j − ˜ f ∗ ( u j )) , v j − v i (cid:29) H ≥ −| ∂ ˜ I λ j ( u j ; v j ) | H | ˜ f ∗ ( u j ) − ˜ f ∗ ( u i ) | H . Third, if v j ≥ ˜ f ∗ ( u j ), v i < ˜ f ∗ ( u i ), then S λij = (cid:28) λ j ( v j − ˜ f ∗ ( u j )) + 1 λ i ( ˜ f ∗ ( u i ) − v i ) , v j − v i (cid:29) H ≥ − (cid:16) | ∂ ˜ I λ j ( u j ; v j ) | H + | ∂ ˜ I λ i ( u i ; v i ) | H (cid:17) | ˜ f ∗ ( u j ) − ˜ f ∗ ( u i ) | H . The reasoning in the remaining cases:˜ f ∗ ( u j ) < v j < ˜ f ∗ ( u j ) , v i ≥ ˜ f ∗ ( u i )˜ f ∗ ( u j ) < v j < ˜ f ∗ ( u j ) , ˜ f ∗ ( u i ) ≤ v i < ˜ f ∗ ( u i )˜ f ∗ ( u j ) < v j < ˜ f ∗ ( u j ) , v i < ˜ f ∗ ( u i ) v j ≤ ˜ f ∗ ( u j ) , v i ≥ ˜ f ∗ ( u i ) v j ≤ ˜ f ∗ ( u j ) , ˜ f ∗ ( u i ) ≤ v i < ˜ f ∗ ( u i ) v j ≤ ˜ f ∗ ( u j ) , v i < ˜ f ∗ ( u i ) eriodic solutions of a phase-field model with hysteresis is fully symmetric and is left to the reader. Consequently, we always have S λij ≥ − ( λ j + λ i ) | ∂ ˜ I λ j ( u j ; v j ) | H | ∂ ˜ I λ i ( u i ; v i ) | H − ( | ∂ ˜ I λ j ( u j ; v j ) | H + | ∂ ˜ I λ i ( u i ; v i ) | H ) (cid:16) | ˜ f ∗ ( u j ) − ˜ f ∗ ( u i ) | H + | ˜ f ∗ ( u j ) − ˜ f ∗ ( u i ) | H (cid:17) =: δ λij . Therefore, integrating inequality (4 .
24) from 0 to t ∈ [0 , T ] we infer in view of (4 . | v j − v i | H ( t ) ≤ L Z t | v j − v i | H ( τ ) dτ + 4 L Z t | u j − u i | H ( τ ) dτ + 2 Z t δ λij ( τ ) dτ. Applying Gronwall’s inequality to this inequality we conclude in view of the conver-gence (4 .
19) that v i , i ≥
1, is a Cauchy sequence in the space C ([0 , T ]; H ). Hence,according to (4 .
20) we obtain the convergence (4 . . . { u, v } is a solution to Problem ( P ) it remains to show that ξ ∈ ∂I ( u ; v ) a.e. on (0 , T ) . (4.26)To this end, let z be an arbitrary function from L (0 , T ; H ) such that z ∈ [ f ∗ ( u ) , f ∗ ( u )]a.e. on Q . For every n ≥
1, define z n to be the pointwise projection of z onto theset [ ˜ f ∗ ( u n ) , ˜ f ∗ ( u n )] . Then, z n ∈ [ ˜ f ∗ ( u n ) , ˜ f ∗ ( u n )] a.e. on Q , n ≥
1, and z n → z in L (0 , T ; H ) as n → ∞ . Consequently, since the operator ∂ ˜ I λ n ( u n ; · ) is the subdif-ferential of the function ˜ I λ n ( u n ; · ), from the definition of subdifferential we have h ∂ ˜ I λ n ( u n ; v n ) , z n − v n i H ≤ ˜ I λ n ( u n ; z n ) − ˜ I λ n ( u n ; v n ) = 0 , (4.27) n ≥ . On the other hand, from (2 .
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