A vanishing diffusion limit in a nonstandard system of phase field equations
Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels
aa r X i v : . [ m a t h . A P ] D ec A vanishing diffusion limit in a nonstandardsystem of phase field equations
Pierluigi Colli (1) e-mail: [email protected]
Gianni Gilardi (1) e-mail: [email protected]
Pavel Krejˇc´ı (2) e-mail: [email protected]
J¨urgen Sprekels (3) e-mail: [email protected] (1)
Dipartimento di Matematica “F. Casorati”, Universit`a di Paviavia Ferrata 1, 27100 Pavia, Italy (2)
Institute of Mathematics, Czech Academy of SciencesˇZitn´a 25, CZ-11567 Praha 1, Czech Republic (3)
Weierstraß-Institut f¨ur Angewandte Analysis und StochastikMohrenstraße 39, 10117 Berlin, Germany
Abstract
We are concerned with a nonstandard phase field model of Cahn-Hilliard type.The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describestwo-species phase segregation and consists of a system of two highly nonlinearlycoupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli,and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treatthe very general case in which the diffusivity coefficient of the parabolic PDE isallowed to depend nonlinearly on both variables. In the same framework, this pa-per investigates the asymptotic limit of the solutions to the initial-boundary valueproblems as the diffusion coefficient σ in the equation governing the evolution ofthe order parameter tends to zero. We prove that such a limit actually exists andsolves the limit problem, which couples a nonlinear PDE of parabolic type with anODE accounting for the phase dynamics. In the case of a constant diffusivity, weare able to show uniqueness and to improve the regularity of the solution. Key words: nonstandard phase field system, nonlinear partial differential equa-tions, asympotic limit, convergence of solutions
AMS (MOS) Subject Classification: Vanishing diffusion limit in nonstandard phase field systems
In this paper, we consider the following system (cid:0) g ( ρ ) (cid:1) ∂ t µ + µ g ′ ( ρ ) ∂ t ρ − div (cid:0) κ ( µ, ρ ) ∇ µ (cid:1) = 0 (1.1) ∂ t ρ − σ ∆ ρ + f ′ ( ρ ) = µ g ′ ( ρ ) (1.2) (cid:0) κ ( µ, ρ ) ∇ µ (cid:1) · ν | Γ = 0 and ∂ ν ρ | Γ = 0 (1.3) µ (0) = µ and ρ (0) = ρ , (1.4)in the unknown fields µ and ρ , where the partial differential equations (1.1)–(1.2) aremeant to hold in a three-dimensional bounded domain Ω, endowed with a smooth bound-ary Γ, and in some time interval (0 , T ). Relations (1.4) specify the initial conditions for µ and ρ , while (1.3) are nothing but homogeneous boundary conditions of Neumann type,involving precisely those boundary operators that match the elliptic differential operatorsin (1.1)–(1.2).This system has been recently addressed in the paper [6]: the existence of solutionshas been proved, thus complementing and extending the results of the papers [3, 4, 5]concerned with simpler or reduced versions of the problem.Here, we are interested to investigate the asymptotic behavior of the above initial-boundary value problem (1.1)–(1.4) as the positive diffusion coefficient σ appearing in(1.2) tends to 0.Let us briefly explain the modelling background for (1.1)–(1.4). Such a system comesfrom a generalization of the phase-field model of viscous Cahn-Hilliard type originallyproposed in [14], and it aims to describe the phase segregation of two species (atomsand vacancies, say) on a lattice in presence of diffusion. The state variables are the order parameter ρ , interpreted as the volume density of one of the two species, and the chemical potential µ . For physical reasons, µ is required to be nonnegative, while thephase parameter ρ must of course take values in the domain of f ′ .We also recall the features of [3] and what has been generalized in [5, 6]. Firstly, thenonlinearity f considered in [3] is a double-well potential defined in (0 , f ′ diverges at the endpoints ρ = 0 and ρ = 1: e.g., for f = f + f with f smooth, onecan take f ( ρ ) = c ( ρ log( ρ ) + (1 − ρ ) log(1 − ρ )) , (1.5)with c a positive constant. In this paper, we let f : R → [0 , + ∞ ] be a convex, properand lower semicontinuous function so that its subdifferential (and not the derivative) isa maximal monotone graph from R to R . Then, we rewrite equation (1.2) as a differ-ential inclusion, in which the derivative of the convex part f of f is replaced by thesubdifferential β := ∂f , i.e., ∂ t ρ − σ ∆ ρ + ξ + f ′ ( ρ ) = µg ′ ( ρ ) with ξ ∈ β ( ρ ) . (1.6)Note that f need not be differentiable in its domain, so that its possibly nonsmooth andmultivalued subdifferential β := ∂f appears in (1.2) in place of f ′ . In general, β is onlya graph, not necessarily a function, and it may include vertical (and horizontal) lines, asfor example when f ( ρ ) = I [0 , ( ρ ) = (cid:26) ≤ ρ ≤ ∞ elsewhere (1.7) olli — Gilardi — Krejˇc´ı — Sprekels β = ∂I [0 , is specified by ξ ∈ β ( ρ ) if and only if ξ ≤ ρ = 0= 0 if 0 < ρ < ≥ ρ = 1 . (1.8)Secondly, while in [3] g was simply taken as the identity map g ( ρ ) = ρ , in [5, 6] g isallowed be any nonnegative smooth function, defined (at least) in the domain where f and its subdifferential live. The presence of such a function g allows for a more generalbehavior of (the related term in) the free energy, which reads ψ ( ρ, ∇ ρ, µ ) = − µ − µ g ( ρ ) + f ( ρ ) + σ |∇ ρ | . (1.9)Indeed, in particular g ( ρ ) is not obliged, as it was instead for g ( ρ ) = ρ , to take its minimumvalue at ρ = 0, be increasing and with maximum value at ρ = 1 (when D ( f ) = [0 , g around theextremal points of the domain of f . Here, we have to impose an additional restrictionon g , which however looks reasonable from the modelling point of view: we postulate that g is a (smooth) concave function, which in turn implies convexity with respect to ρ ofthe term − µ g ( ρ ) in the free energy (1.9). However, let us recall that f may stand fora multi-well potential in which the nonconvex perturbations are incorporated into f , sothat ψ in its entirety needs not be convex with respect to ρ .An important generalization that is considered in this paper concerns the diffusivity κ .In [3], κ was just assumed to be a constant function, but it can be a positive-valued,continuous, bounded, and nonlinear function of µ (and this was the setting of [5]), orof µ and ρ as it is postulated in [6]. For simplicity, we confine ourselves to study ofthe convergence properties of the solution under an assumption that guarantees uniformparabolicity, i.e., κ ≥ κ ∗ >
0. We point out that [5] treats the situation of κ dependingonly on µ and possibly degenerating somewhere.Therefore, the system (cid:0) g ( ρ ) (cid:1) ∂ t µ + µ g ′ ( ρ ) ∂ t ρ − div (cid:0) κ ( µ, ρ ) ∇ µ (cid:1) = 0 (1.10) ∂ t ρ − σ ∆ ρ + ξ + f ′ ( ρ ) = µg ′ ( ρ ) with ξ ∈ β ( ρ ) , (1.11) (cid:0) κ ( µ, ρ ) ∇ µ (cid:1) · ν | Γ = 0 and ∂ ν ρ | Γ = 0 (1.12) µ (0) = µ and ρ (0) = ρ , (1.13)turns out the initial and boundary value problem for a nonstandard and highly nonlinearphase field system in which however the role usually played by the temperature is hereconducted by the chemical potential µ . In the study of phase field systems, it has beenalways considered rather important to analyze the behavior of the problem as the coef-ficient σ of the diffusion term in the phase parameter equation tends to 0. The limitingcase σ = 0 corresponds indeed to a pointwise ordinary differential equation (or inclusion) ∂ t ρ + ξ + f ′ ( ρ ) = µg ′ ( ρ ) , ξ ∈ β ( ρ ) , (1.14)in place of (1.11), and to an expression for the free energy (1.9) in which the last quadraticterm accounting for nonlocal interactions is removed. Vanishing diffusion limit in nonstandard phase field systems
In fact, especially for the choice (1.7)–(1.8), the limiting problem can be formulatedin terms of hysteresis operators: in particular, the so-called stop and play operators areinvolved; the interested reader can find some discussion and various results on this classof problems in [7, 8, 9, 10, 11, 12, 13].By collecting a number of estimates independent of σ for the solution ( µ σ , ρ σ ) to theproblem (1.10)–(1.13), by weak and weak star compactness we prove that any limit in asuitable topology of a convergent subsequence of { ( µ σ , ρ σ ) } yields a solution to the limitingproblem in which (1.11) is replaced by (1.14). Furthermore, under natural compatibilityconditions on the nonlinearities and the initial data, we show boundedness for all thecomponents of any solution to the limit problem. Finally, in the special case of a constantmobility κ in (1.10), we prove that the solution is unique and more regular than required.The paper is organized as follows. In the next section, we state precise assumptionsalong with our results. The basic a priori estimates independent of σ are proved in Sec-tion 3 and they allow us to pass to the limit by compactness and monotonicity techniques.Finally, the last section is devoted to the study of the limit problem and our boundedness,uniqueness, and further regularity properties are proved. The aim of this section is to introduce precise assumptions on the data for the mathe-matical problem under investigation, and establish our main result. We assume Ω to be abounded connected open set in R with smooth boundary Γ (treating lower-dimensionalcases would require only minor changes) and let T ∈ (0 , + ∞ ) stand for a final time. Weintroduce the spaces V := H (Ω) , H := L (Ω) , W := { v ∈ H (Ω) : ∂ ν v = 0 on Γ } (2.1)and endow them with their standard norms, for which we use a self-explanatory notationlike k · k V . For powers of these spaces, norms are denoted by the same symbols. We remarkthat the embeddings W ⊂ V ⊂ H are compact, because Ω is bounded and smooth. Thesymbol h · , · i denotes the duality product between V ∗ , the dual space of V , and V itself.Moreover, for p ∈ [1 , + ∞ ], we write k · k p for the usual norm in L p (Ω); as no confusioncan arise, the symbol k · k p is used for the norm in L p ( Q ) as well, where Q := Ω × (0 , T ).Now, we present the structural assumptions we make. It is useful to fix an upperbound for σ , that is, 0 < σ ≤ . (2.2)Then, for the diffusivity coefficient κ we assume that κ : ( m, r ) κ ( m, r ) is continuous from [0 , + ∞ ) × R to R , (2.3)the partial derivatives ∂ r κ and ∂ r κ exist and are continuous , (2.4) κ ∗ , κ ∗ ∈ (0 , + ∞ ) , (2.5) κ ∗ ≤ κ ( m, r ) ≤ κ ∗ , | ∂ r κ ( m, r ) | ≤ κ ∗ , | ∂ r κ ( m, r ) | ≤ κ ∗ for m ≥ r ∈ R , (2.6) olli — Gilardi — Krejˇc´ı — Sprekels f = f + f , f : R → [0 , + ∞ ] , f : R → R , (2.7) f is convex, proper, l.s.c. and f is a C function , (2.8) g ∈ C ( R ), g ( r ) ≥ g ′′ ( r ) ≤ r ∈ R , (2.9) f ′ , g , and g ′ are Lipschitz continuous . (2.10)It is convenient to introduce the notations κ ′ := ∂ r κ, κ ′′ := ∂ r κ, β := ∂f , and π := f ′ (2.11) K ( m, r ) := Z m κ ( s, r ) ds, K ( m, r ) := Z m κ ′ ( s, r ) ds, K ( m, r ) := Z m κ ′′ ( s, r ) ds for m ≥ r ∈ R . (2.12)We denote by D ( f ) and D ( β ) the effective domains of f and β , respectively. Thanks to(2.6), it is clear thatmax {| K ( m, r ) | , | K ( m, r ) | , | K ( m, r ) |} ≤ κ ∗ m for every m ≥ r ∈ R . (2.13)We also note that the structural assumptions of [5] are fulfilled if κ only depends on m ,and that, due to the presence of β ( ρ ), a strong singularity in equation (1.11) is allowed.On the other hand, equation (1.10) is uniformly parabolic, since g is nonnegative and κ is bounded away from zero. Remark 2.1.
Let us recall that any convex, proper, l.s.c. function is bounded from belowby an affine function (cf., e.g., [1, Prop. 2.1, p. 51]), whence the assumption f ≥ f . Moreover, we pointout that the sign conditions g ≥ g ′′ ≤ D ( β ), for g canbe extended outside of D ( β ) accordingly.Concerning the initial data, we require that µ ∈ V, µ ≥ , (2.14) ρ ∈ V, ρ ∈ D ( f ) a.e. in Ω , f ( ρ ) ∈ L (Ω) (2.15)and point out that the above assumptions regard the initial data for the limiting problem,i.e., the one with (1.14) in place of (1.11). On the other hand, let us consider a family ofinitial data µ σ , ρ σ with µ σ ∈ V ∩ L ∞ (Ω) , µ σ ≥ , (2.16) ρ σ ∈ W, there is ξ σ ∈ H such that ρ σ ∈ D ( β ) , ξ σ ∈ β ( ρ σ ) a.e. in Ω , (2.17)that approximate µ , ρ in the sense that µ σ → µ and ρ σ → ρ weakly in V, (2.18) k f ( ρ σ ) k is bounded independently of σ. (2.19)For the reader’s convenience, we show that such a family { µ σ , ρ σ } actually exists. Ofcourse, if µ L ∞ (Ω) we can take as µ σ some truncation of µ , e.g., µ σ = min { µ , /σ } .Concerning ρ σ , one possible choice is letting ρ σ ∈ W denote the solution to ρ σ − σ ∆ ρ σ + σξ σ = ρ , with ξ σ ∈ β ( ρ σ ) , a.e. in Ω . (2.20) Vanishing diffusion limit in nonstandard phase field systems
Indeed, the elliptic problem (2.20) has a unique solution for all σ >
0, since − ∆ + β is amaximal monotone graph in H × H with effective domain { v ∈ W : ∃ η ∈ H such that v ∈ D ( β ) , η ∈ β ( v ) a.e. in Ω } . Thus, ρ σ is nothing but the outcome of the application of the resolvent of − ∆ + β to ρ (let us refer to [1] and [2] for basic definitions and properties of maximal monotone oper-ators). A formal test of the equality in (2.20) by ξ σ and the definition of subdifferentiallead us to the estimate Z Ω f ( ρ σ ) + σ k ξ σ k H ≤ Z Ω f ( ρ ) , (2.21)which ensures (2.17) and (2.19), thanks to the nonnegativity of f . A rigorous way ofproving the existence of ρ σ and estimate (2.21) passes through the use of the Yosidaapproximation β σ (see, e.g., [2, p. 28]) in place of β .Now, we recall the result proved in [6] that allows us to specify a solution to theproblem (1.10)–(1.12), with σ >
0, which fulfills the appropriate initial conditions.
Proposition 2.2.
Assume that both (2.3) – (2.12) and (2.16) – (2.17) hold. Then, thereexists at least one triplet ( µ σ , ρ σ , ξ σ ) satisfying ρ σ ∈ W , ∞ (0 , T ; H ) ∩ H (0 , T ; V ) ∩ L ∞ (0 , T ; W ) , (2.22) ξ σ ∈ L ∞ (0 , T ; H ) , (2.23) µ σ ∈ H (0 , T ; H ) ∩ L ∞ (0 , T ; V ) ∩ L ∞ ( Q ) , µ σ ≥ a.e. in Q, (2.24)div (cid:0) κ ( µ σ , ρ σ ) ∇ µ σ (cid:1) ∈ L ( Q ) and (cid:0) κ ( µ σ , ρ σ ) ∇ µ (cid:1) · ν = 0 a.e. on Σ , (2.25) and solving the system of equations and conditions in the following strong form (cid:0) g ( ρ σ ) (cid:1) ∂ t µ σ + µ σ g ′ ( ρ σ ) ∂ t ρ σ − div (cid:0) κ ( µ σ , ρ σ ) ∇ µ σ (cid:1) = 0 a.e. in Q, (2.26) ∂ t ρ σ − σ ∆ ρ σ + ξ σ + π ( ρ σ ) = µ σ g ′ ( ρ σ ) and ξ σ ∈ β ( ρ σ ) a.e. in Q, (2.27) µ σ (0) = µ σ and ρ σ (0) = ρ σ a.e. in Ω . (2.28)Let us point out that equation (2.26) can be rewritten as ∂ t u σ − div (cid:0) κ ( µ σ , ρ σ ) ∇ µ σ (cid:1) = µ σ g ′ ( ρ σ ) ∂ t ρ σ , where u σ = (1 + 2 g ( ρ σ )) µ σ , a.e. in Q, (2.29)and the auxiliary variable u σ has been added. Now, we take advantage of a variationalformulation of (2.29) which also accounts for the boundary condition in (2.25), that is, h ∂ t u σ ( t ) , v i + Z Ω (cid:0) κ ( µ σ , ρ σ ) ∇ µ σ (cid:1) ( t ) · ∇ v = Z Ω µ σ g ′ ( ρ σ ) ∂ t ρ σ v for all v ∈ V and a.a. t ∈ (0 , T ) . (2.30)The main result of this paper reads as follows. olli — Gilardi — Krejˇc´ı — Sprekels Theorem 2.3.
Assume that (2.3) – (2.12) and (2.14) – (2.19) hold. For any σ ∈ (0 , let ( µ σ , ρ σ , ξ σ ) be the triplet defined by Proposition 2.2 and let u σ := (1 + 2 g ( ρ σ )) µ σ . Then,there exists a subsequence, still labelled by the parameter σ , and a quadruplet ( µ, ρ, ξ, u ) such that µ σ → µ weakly star in L ∞ (0 , T ; H ) ∩ L (0 , T ; V ) , (2.31) ρ σ → ρ weakly star in H (0 , T ; H ) ∩ L ∞ (0 , T ; V ) , (2.32) ξ σ → ξ weakly in L ( Q ) , (2.33) u σ → u weakly in W , / (0 , T ; V ∗ ) ∩ L (0 , T ; W , / (Ω)) (2.34) as σ ց . Moreover, any quadruplet ( µ, ρ, ξ, u ) that is found as limit of converging subse-quences yields a solution to the following limit problem h ∂ t u ( t ) , v i + R Ω κ ( µ, ρ ) ∇ µ ( t ) · ∇ v = R Ω µ g ′ ( ρ ) ∂ t ρ v for all v ∈ V and a.a. t ∈ (0 , T ) , (2.35) u = (1 + 2 g ( ρ )) µ a.e. in Q, (2.36) ∂ t ρ + ξ + π ( ρ ) = µ g ′ ( ρ ) and ξ ∈ β ( ρ ) a.e. in Q, (2.37) µ (0) = µ and ρ (0) = ρ a.e. in Ω . (2.38) Remark 2.4.
The nonnegativity property µ ≥ Q plainly follows from (2.24)and (2.31). Remark 2.5.
One standard situation for the limit problem (2.35)–(2.38) is obtained for β = ∂I [0 , (cf. (1.7)–(1.8)). In this case (2.37) becomes − π ( ρ ) + µ g ′ ( ρ ) − ∂ t ρ ∈ ∂I [0 , ( ρ ) a.e. in Q. (2.39)Then, if one introduces the generalized “freezing index” w ( x, t ) := Z t ( − π ( ρ ) + µ g ′ ( ρ ))( x, s ) ds, ( x, t ) ∈ Q, we thus have ∂ t w − ∂ t ρ ∈ ∂I [0 , ( ρ ), or equivalently, ρ = S K [ w ], where S K is the stophysteresis operator associated with the closed convex set K = [0 ,
1] (see, e.g., [10, 11, 12]).Hence, we may rewrite (2.39) as ∂ t w = − π ( S K [ w ]) + µ g ′ ( S K [ w ]) a.e. in Q. In addition to the convergence result stated in Theorem 2.3, one can derive bound-edness for both the components ρ and ξ of any solution to the limit problem, providedthat special additional requirements are satisfied, namely, by assuming that there existreal constants ρ ∗ , ρ ∗ , ξ ∗ , ξ ∗ such that ρ ∗ , ρ ∗ ∈ D ( β ) , ξ ∗ ∈ β ( ρ ∗ ) , ξ ∗ ∈ β ( ρ ∗ ) , (2.40) ξ ∗ + π ( ρ ∗ ) ≤ , ξ ∗ + π ( ρ ∗ ) ≥ , (2.41) g ′ ( ρ ∗ ) ≥ , g ′ ( ρ ∗ ) ≤ . (2.42) Vanishing diffusion limit in nonstandard phase field systems
Theorem 2.6.
In addition to the assumptions of Theorem 2.3, suppose that (2.40) – (2.42) and ρ ∗ ≤ ρ ≤ ρ ∗ a.e. in Ω (2.43) hold. Then, the components ρ and ξ of any solution ( µ, ρ, ξ, u ) to problem (2.35) – (2.38) satisfy ρ ∗ ≤ ρ ≤ ρ ∗ and ξ ∗ ≤ ξ ≤ ξ ∗ a.e. in Q. (2.44) If moreover µ ∈ L ∞ (Ω) (2.45) and κ = κ is constant, then the solution of Problem (2.35) – (2.38) is unique and µ ∈ H (0 , T ; H ) ∩ L ∞ (0 , T ; V ) ∩ L (0 , T ; W ) . (2.46) Remark 2.7.
We observe that the above result is very general. Indeed, assumptions(2.40)–(2.42) are fulfilled with suitable constants for any graph β with bounded domainthat generalizes the examples (1.5) or (1.7). Of course, the decreasing function g ′ (cf. (2.9))should not assume a definite sign on D ( β ).Now, we list a number of tools and notations we owe to throughout the paper. Werepeatedly use the elementary Young inequalities a b ≤ γa + 14 γ b and a b ≤ ϑa ϑ + (1 − ϑ ) b − ϑ for every a, b ≥ γ >
0, and ϑ ∈ (0 ,
1) (2.47)as well as the H¨older and Sobolev inequalities. The precise form of the latter we use isthe following W ,p (Ω) ⊂ L q (Ω) and k v k q ≤ C p,q k v k W ,p (Ω) for every v ∈ W ,p (Ω),provided that 1 ≤ p < ≤ q ≤ p ∗ := 3 p − p (2.48)with a constant C p,q in (2.48) depending only on Ω, p , and q , since Ω ⊂ R . Moreoverthe embedding W ,p (Ω) ⊂ L q (Ω) is compact if 1 ≤ q < p ∗ . (2.49)The particular case p = 2 of (2.48) becomes V ⊂ L q (Ω) and k v k q ≤ C k v k V for every v ∈ V and q ∈ [1 ,
6] (2.50)where C depends only on Ω. Moreover, the compactness inequality k v k q ≤ ε k∇ v k + C q,ε k v k for every v ∈ V , q ∈ [1 , ε > C q,ε depending on Ω, q , and ε , only. We also recall the interpo-lation inequalities, which hold for any ϑ ∈ [0 , k v k r ≤ k v k ϑp k v k − ϑq ∀ v ∈ L p (Ω) ∩ L q (Ω) , where p, q, r ∈ [1 , + ∞ ] and 1 r = ϑp + 1 − ϑq . (2.52) k v k L r (0 ,T ; L r (Ω)) ≤ k v k ϑL p (0 ,T ; L p (Ω)) k v k − ϑL q (0 ,T ; L q (Ω)) ∀ v ∈ L p (0 , T ; L p (Ω)) ∩ L q (0 , T ; L q (Ω)) , where p i , q i , r i ∈ [1 , + ∞ ] and 1 r i = ϑp i + 1 − ϑq i for i = 1 , . (2.53) olli — Gilardi — Krejˇc´ı — Sprekels k v k r ≤ ϑ k v k p + (1 − ϑ ) k v k q for every v ∈ L p (Ω) ∩ L q (Ω)thanks to the Young inequality, and a similar remark holds for (2.53). Thus, we have thecontinuous embeddings L p (Ω) ∩ L q (Ω) ⊂ L r (Ω) and L p (0 , T ; L p (Ω)) ∩ L q (0 , T ; L q (Ω)) ⊂ L r (0 , T ; L r (Ω)) . We stress the important case of the space L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; L (Ω)), which occursseveral times in the sequel and corresponds to p = ∞ , p = 2, q = 2, and q = 6. Inparticular, the choices ϑ = 2 / ϑ = 1 / v of the abovespace) and the continuous embeddings k v k L / ( Q ) ≤ k v k / X k v k / Y and X ∩ Y ⊂ L / ( Q ) (2.54) k v k L / (0 ,T ; L / (Ω)) ≤ k v k / X k v k / Y and X ∩ Y ⊂ L / (0 , T ; L / (Ω)) (2.55)where X := L ∞ (0 , T ; L (Ω)) and Y := L (0 , T ; L (Ω)) . Notice that we can take v ∈ L ∞ (0 , T ; H ) ∩ L (0 , T ; V ) in (2.54)–(2.55), since V ⊂ L (Ω).Finally, we set Q t := Ω × (0 , t ) for t ∈ [0 , T ] , (2.56)and, again throughout the paper, we use a small-case italic c for different constants, thatmay only depend on Ω, the final time T , the shape of the nonlinearities f and g , andthe properties of the data involved in the statements at hand; a notation like c ε signalsa constant that depends also on the parameter ε . The reader should keep in mind thatthe meaning of c and c ε might change from line to line and even in the same chain ofinequalities, whereas those constants we need to refer to are always denoted by capitalletters, just like C in (2.50). In this section, we prove Theorem 2.3, which ensures the existence of a solution to prob-lem (2.35)–(2.38) along with the convergence properties stated in (2.31)–(2.34).Then, for any σ ∈ (0 ,
1] we let ( µ σ , ρ σ , ξ σ ) denote the triplet defined by Proposition 2.2and set u σ := (1+2 g ( ρ σ )) µ σ . The existence of ( µ σ , ρ σ , ξ σ ) has been proved in [6]: we followin parts the arguments developed there in order to recover useful estimates independentof σ . Before that, let us remark that the property µ σ ≥ − µ − σ , the negative part of µ σ , and integrate over Q t . Inprinciple, in this computation one has to define κ everywhere, e.g., by taking an evenextension ¯ κ with respect to the first variable. We observe that (cid:2)(cid:0) g ( ρ σ ( t )) (cid:1) ∂ t µ σ + µ σ g ( ρ σ ) ∂ t ρ σ (cid:3) ( − µ − σ ) = 12 ∂ t (cid:0) (1 + 2 g ( ρ σ ( t ))) | µ − σ | (cid:1) . Hence, by using µ σ ≥ Z Ω (1 + 2 g ( ρ σ ( t ))) | µ − σ ( t ) | + Z Q t ¯ κ ( µ σ , ρ σ ) |∇ µ − σ | = 0 for a.a. t ∈ (0 , T ) . Vanishing diffusion limit in nonstandard phase field systems
As both g and ¯ κ are nonnegative, this implies µ − σ = 0, that is, µ σ ≥ Q . First a priori estimate.
We test (2.26) by µ σ and point out that (cid:2)(cid:0) g ( ρ σ ) (cid:1) ∂ t µ σ + µ σ g ′ ( ρ σ ) ∂ t ρ σ (cid:3) µ σ = 12 ∂ t (cid:2) (1 + 2 g ( ρ σ ) µ σ (cid:3) . (3.1)Thus, by integrating over (0 , t ), where t ∈ [0 , T ] is arbitrary, we obtain Z Ω (cid:0) g ( ρ σ ( t )) (cid:1) | µ σ ( t ) | + 2 Z Q t κ ( µ σ ( s ) , ρ σ ( s )) |∇ µ σ | = Z Ω (1 + 2 g ( ρ σ )) µ σ . We recall that g is nonnegative and Lipschitz continuous (cf. (2.9)–(2.10)). Moreover, ρ σ , µ σ are both uniformly bounded in V by (2.18), whence Z Ω (1 + 2 g ( ρ σ )) µ σ ≤ c (cid:0) k µ σ k + k ρ σ k k µ σ k (cid:1) ≤ c owing to the H¨older and Sobolev inequalities (see (2.50)). Then, in view of g ≥ κ ≥ κ ∗ >
0, from (3.1) it follows that k µ σ k L ∞ (0 ,T ; H ) + k µ σ k L (0 ,T ; V ) ≤ c. (3.2) Second a priori estimate.
We add ρ σ to both sides of (2.27) and test by ∂ t ρ σ . Onaccount of (2.7)–(2.8) and (2.11), we obtain Z Q t | ∂ t ρ σ | + 12 k ρ σ ( t ) k H + σ k∇ ρ σ ( t ) k H + Z Ω f ( ρ σ ( t ))= σ Z Ω |∇ ρ σ | + Z Ω f ( ρ σ ) + 12 Z Ω (cid:0) ρ σ ( t ) − f ( ρ σ ( t )) (cid:1) + Z Q t µ σ g ′ ( ρ σ ) ∂ t ρ σ for every t ∈ [0 , T ]. Then, thanks to the Lipschitz continuity of f ′ and g , and owing tothe bounds entailed by (2.18)–(2.19), we find out that Z Q t | ∂ t ρ σ | + 12 k ρ σ ( t ) k H + σ k∇ ρ σ ( t ) k H + Z Ω f ( ρ σ ( t )) ≤ c + c Z Ω | ρ σ ( t ) | + 14 Z Q t | ∂ t ρ σ | + c k µ σ k L ∞ (0 ,T ; H ) . On the other hand, by the chain rule and the Young inequality (2.47) we have that c Z Ω | ρ σ ( t ) | ≤ c Z Ω | ρ σ | + 14 Z Q t | ∂ t ρ σ | + c Z t k ρ σ ( s ) k H ds. Then, as f is nonnegative, by accounting for (3.2), with the help of the Gronwall lemmawe infer that Z Q t | ∂ t ρ σ | + k ρ σ ( t ) k H + σ k∇ ρ σ ( t ) k H ≤ c for all t ∈ [0 , T ] . olli — Gilardi — Krejˇc´ı — Sprekels k ρ σ k H (0 ,T ; H ) + σ / k ρ σ k L ∞ (0 ,T ; V ) ≤ c. (3.3) Third a priori estimate.
We proceed formally and test (2.27) by − ∆ ρ σ . Hence,integrating by parts and with respect to time, we deduce that12 k∇ ρ σ ( t ) k H + σ Z Q t | ∆ ρ σ | + Z Q t β ′ ( ρ σ ) |∇ ρ σ | ≤ Z Ω |∇ ρ σ | − Z Q t π ′ ( ρ σ ) |∇ ρ σ | + Z Q t g ′ ( ρ σ ) ∇ µ σ · ∇ ρ σ + Z Q t g ′′ ( ρ σ ) µ σ |∇ ρ σ | , (3.4)where the equality ξ σ = β ( ρ σ ) has been used along with the smoothness of β , according toour formal procedure. In fact, what is important is that the related term on the left-handside is nonnegative, i.e., Z Q t β ′ ( ρ σ ) |∇ ρ σ | ≥ . Concerning the right-hand side of (3.4), we have that 12 Z Ω |∇ ρ σ | ≤ c due to (2.18),and the estimate − Z Q t π ′ ( ρ σ ) |∇ ρ σ | + Z Q t g ′ ( ρ σ ) ∇ µ σ · ∇ ρ σ ≤ c Z t k∇ ρ σ ( s ) k H ds + c k µ σ k L (0 ,T ; V ) owing to the boundedness of π ′ and g ′ (see (2.10)–(2.11)). About the last term, (2.9) and(2.24) imply Z Q t g ′′ ( ρ σ ) µ σ |∇ ρ σ | ≤ , so that the sign properties of g ′′ and µ σ become crucial to control this term. Then, inview of (3.2), from (3.4) it follows that12 k∇ ρ σ ( t ) k H + σ Z Q t | ∆ ρ σ | ≤ c + c Z t k∇ ρ σ ( s ) k H ds for all t ∈ [0 , T ] , and the Gronwall lemma and (3.3) allow us to deduce that k ρ σ k L ∞ (0 ,T ; V ) + σ / k ρ σ k L (0 ,T ; W ) ≤ c. (3.5)Note that here we have used the regularity theory for elliptic equations, owing to thebound on σ k ∆ ρ σ k and to the homogeneous Neumann boundary condition satisfied by ρ σ (cf. (2.22)). Finally, an easy consequence of (3.3) and (3.5) comes out from a comparisonof terms in (2.27), which yields k ξ σ k L (0 ,T ; H ) ≤ c. (3.6) Fourth a priori estimate. As u σ = (1 + 2 g ( ρ σ )) µ σ , by (2.10) we have that | u σ | ≤ c (1 + | ρ σ | ) | µ σ | , |∇ u σ | = | g ′ ( ρ σ ) µ σ ∇ ρ σ + (1 + 2 g ( ρ σ )) ∇ µ σ | ≤ c | µ σ | |∇ ρ σ | + c (1 + | ρ σ | ) |∇ µ σ | . Vanishing diffusion limit in nonstandard phase field systems
Now, taking (3.2) into account, we see that |∇ µ σ | is bounded in L (0 , T ; L (Ω)), while | µ σ | is bounded in L (0 , T ; L (Ω)) thanks to the Sobolev inequality (2.50). On the otherhand, (3.5) provides a bound for |∇ ρ σ | in L ∞ (0 , T ; L (Ω)) and for | ρ σ | in L ∞ (0 , T ; L (Ω)).Hence, using H¨older inequality, it is not difficult to check that the products | µ σ | |∇ ρ σ | and | ρ σ | |∇ µ σ | are bounded in L (0 , T ; L / (Ω)), whereas | ρ σ | | µ σ | is even bounded in L (0 , T ; L (Ω)). Therefore, we conclude that k u σ k L (0 ,T ; W , / (Ω)) ≤ c . (3.7) Fifth a priori estimate.
Let us recall that (3.2) and (2.50) imply the boundednessof { µ σ } in the space L ∞ (0 , T ; L (Ω)) ∩ L (0 , T ; L (Ω)). Then, we can apply (2.52) with p = 2, q = 6, ϑ = 1 / r = 3 to see that k µ σ ( t ) k ≤ k µ σ ( t ) k k µ σ ( t ) k for a.a. t ∈ (0 , T ) , whence squaring and integrating with respect to t lead to k µ σ k L (0 ,T ; L (Ω)) ≤ k µ σ k L ∞ (0 ,T ; L (Ω)) k µ σ k L (0 ,T ; L (Ω)) ≤ c. (3.8)Consider now (2.30) which turns out to be a variational formulation of (2.26). As wewant to prove that k ∂ t u σ k L / (0 ,T ; V ∗ ) ≤ c , (3.9)we use (2.30) and let v vary in L (0 , T ; V ). By integrating with respect to time andinvoking (2.6), the boundedness of g ′ and H¨older’s inequality, we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z T h ∂ t u σ ( t ) , v ( t ) i dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ κ ∗ k∇ µ σ k L (0 ,T ; H ) k∇ v k L (0 ,T ; H ) + c Z T k µ σ ( t ) k k ∂ t ρ σ ( t ) k k v ( t ) k dt. Hence, in view of (3.2), by applying the H¨older and Sobolev inequalities (see (2.50)) inthe time integral, we infer that (cid:12)(cid:12)(cid:12)(cid:12)Z T h ∂ t u σ ( t ) , v ( t ) i dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ c k v k L (0 ,T ; V ) + c k µ σ k L (0 ,T ; L (Ω)) k ∂ t ρ σ k L (0 ,T ; H ) k v k L (0 ,T ; V ) . Now, the continuous embedding L (0 , T ; V ) ⊂ L (0 , T ; V ), (3.8) and (3.3) allow us toconclude that (cid:12)(cid:12)(cid:12)(cid:12)Z T h ∂ t u σ ( t ) , v ( t ) i dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ c k v k L (0 ,T ; V ) , whence (3.9) follows. Passage to the limit.
By the above estimates, there are a quadruplet ( µ, ρ, ξ, u ), with µ ≥ Q , and a function k such that (2.31)–(2.34) are satisfied as long as κ ( µ σ , ρ σ ) → k weakly star in L ∞ ( Q ) (3.10) olli — Gilardi — Krejˇc´ı — Sprekels τ = τ i ց
0. By the weak convergence of time derivatives, theCauchy conditions (2.28) hold for the limit pair ( ρ, u ). By (2.32), (2.34), and the compactembedding (2.49), we can apply well-known strong compactness results (see, e.g., [15,Sect. 8, Cor. 4]) and infer that (possibly taking another subsequence) ρ σ → ρ strongly in C ([0 , T ]; L p (Ω)) for p < Q (3.11) u σ → u strongly in L (0 , T ; L p (Ω)) for p < Q. (3.12)The weak convergence (2.33), together with (3.11) with p = 2, implies that ξ ∈ β ( ρ ) a.e.in Q (see, e.g., [2, Prop. 2.5, p. 27]), due to the maximal monotonicity of the operatorinduced by β on L ( Q ). Now, we deal with the other nonlinear terms and the products.We first observe that (3.11) also entails that φ ( ρ σ ) → φ ( ρ ) strongly in C ([0 , T ]; L p (Ω)) for p < Q (3.13)for φ = g, g ′ , π, / (1 + 2 g ), thanks to the Lipschitz continuity of such functions. Thisis sufficient to establish equation (2.37). Indeed, by accounting for (2.31), we see thatthe product µ σ g ( ρ σ ) converges to µg ( ρ ) weakly (e.g.) in L ( Q ). On the other hand,(3.5) implies that σ ∆ ρ σ converges to zero strongly in L ( Q ). Now, we prove equations(2.35)–(2.36), which involve the whole triplet ( µ, ρ, u ). The first step is showing strongconvergence for µ σ and relation (2.36). By combining (3.13) with (3.12), we see that µ σ = u σ g ( ρ σ ) → u g ( ρ ) a.e. in Q. (3.14)This and (2.31) imply µ = u/ (1 + 2 g ( ρ )) and (2.36) is proved. Moreover, as { µ σ } isbounded in L / ( Q ) by (3.2), the Sobolev embedding V ⊂ L (Ω), and (2.54), we can alsodeduce a strong convergence. We summarize as follows: µ σ → µ strongly in L p ( Q ) for every p < / Q. (3.15)From this, we immediately infer that κ ( µ σ , ρ σ ) converges to κ ( µ, ρ ) a.e. in Q , just bycontinuity. Then, (3.10) implies k = κ ( µ, ρ ) and κ ( µ σ , ρ σ ) → κ ( µ, ρ ) strongly in L p ( Q ) for every p < + ∞ . (3.16)Therefore, κ ( µ σ , ρ σ ) ∇ µ σ converges to κ ( µ, ρ ) ∇ µ weakly in L p ( Q ) for every p <
2, thanksto (2.31), and the choice p = 3 / Z Q κ ( µ σ , ρ σ ) ∇ µ σ · ∇ v → Z Q κ ( µ, ρ ) ∇ µ · ∇ v for every v ∈ L (0 , T ; W , (Ω)) . On the other hand, µ σ g ′ ( ρ σ ) ∂ t ρ σ converges to µg ′ ( ρ ) ∂ t ρ weakly at least in L ( Q ), as onecan easily see by combining (2.32), (3.13), and (3.15). It follows that Z Q µ σ g ′ ( ρ σ ) ∂ t ρ σ v → Z Q µg ′ ( ρ ) ∂ t ρ v for every v ∈ L ∞ ( Q ) . Moreover, (2.34) holds. Hence, we can conclude that Z T h ∂ t u ( t ) , v ( t ) i dt + Z Q κ ( µ, ρ ) ∇ µ · ∇ v = Z Q µg ′ ( ρ ) ∂ t ρ v for every v ∈ L (0 , T ; W , (Ω)) ∩ L ∞ ( Q ) . (3.17)4 Vanishing diffusion limit in nonstandard phase field systems
Now, we observe that ∂ t u ∈ L / (0 , T ; V ∗ ) by (2.34) and that κ ( µ, ρ ) ∇ v ∈ L (0 , T ; H )by (2.31) and the boundedness of κ . Finally, µg ′ ( ρ ) ∂ t ρ ∈ L / (0 , T ; L / (Ω)), since g ′ isbounded, ∂ t ρ ∈ L (0 , T ; H ), and µ ∈ L (0 , T ; L (Ω)) as a consequence of (2.31), V ⊂ L (Ω), and (3.8)). Therefore, we can improve (3.17) by a density argument and see thatthe variational equation still holds for any v ∈ L (0 , T ; V ). What we obtain is equivalentto (2.35), and the proof is complete. In this section, we prove Theorem 2.6. In the whole section, it is understood that theassumptions of Theorem 2.6 are satisfied, and sometimes we do not remind the readerabout that. As far as the first part of Theorem 2.6 is concerned, the true result regardsordinary variational inequalities and we present it in the form of a lemma. For convenience,we use the same notation ρ , etc., even though it is clear that everything is independentof x : the dot over the variable ρ denotes the (time) derivative, here. Lemma 4.1.
Let (2.40) – (2.42) hold and ρ ∗ ≤ ρ ≤ ρ ∗ . Then for every nonnegativefunction µ ∈ L (0 , T ) , the differential inclusion ˙ ρ ( t ) + β ( ρ ( t )) + π ( ρ ( t )) − µ ( t ) g ′ ( ρ ( t )) ∋ for a.a. t ∈ (0 , T ) and ρ (0) = ρ (4.1) has a unique solution ρ ∈ W , (0 , T ) such that ρ ∗ ≤ ρ ( t ) ≤ ρ ∗ and ξ ∗ ≤ ξ ( t ) ≤ ξ ∗ for a.a. t ∈ (0 , T ) , (4.2) where ξ ( t ) := − (cid:0) ˙ ρ ( t ) + π ( ρ ( t )) − µ ( t ) g ′ ( ρ ( t )) (cid:1) ∈ β ( ρ ( t )) . Moreover, there exists a constant
C > such that if µ , µ ∈ L (0 , T ) and ρ , ρ are twoinputs and ρ , ρ are the corresponding solutions of (4.1) , then for every t ∈ [0 , T ] wehave | ρ − ρ | ( t ) + Z t | ˙ ρ − ˙ ρ | ( τ ) dτ ≤ C (cid:18) | ρ − ρ | + Z t (cid:0) (1 + µ ) | ρ − ρ | + | µ − µ | (cid:1) ( τ ) dτ (cid:19) . (4.3) Proof.
The existence of a unique solution can easily be proved, e.g., by the iterated BanachContraction Principle, due to the monotonicity of β and to the Lipschitz continuity ofthe other nonlinearities. In (4.2), we only prove the upper inequalities since the proof ofthe lower ones is quite similar. It suffices to prove the desired inequalities for the solution( ρ, ξ ) of the cut-off problem˙ ρ ( t ) + ξ ( t ) + π ∗ ( ρ ( t )) − µ ( t ) g ∗ ( ρ ( t )) = 0 , ξ ( t ) ∈ β ( ρ ( t )) for a.a. t ∈ (0 , T ) , (4.4) ρ (0) = ρ , (4.5)where π ∗ and g ∗ are defined by π ∗ ( r ) := π (min { r, ρ ∗ } ) and g ∗ ( r ) := g ′ (min { r, ρ ∗ } ) . olli — Gilardi — Krejˇc´ı — Sprekels ρ − ρ ∗ ) + and integrate. Recalling (2.40)–(2.42) and noting that ξ ≥ ξ ∗ and g ∗ ( ρ ) = g ′ ( ρ ∗ ) where ρ > ρ ∗ , we obtain12 | ( ρ ( t ) − ρ ∗ ) + | ≤ − Z t (cid:0) ξ − ξ ∗ (cid:1) ( ρ − ρ ∗ ) + − Z t (cid:0) ξ ∗ + π ∗ ( ρ ∗ ) (cid:1) ( ρ − ρ ∗ ) + + Z t (cid:0) π ( ρ ∗ ) − π ( ρ ) (cid:1) ( ρ − ρ ∗ ) + + Z t µ g ∗ ( ρ )( ρ − ρ ∗ ) + ≤ Z t (cid:0) π ( ρ ∗ ) − π ( ρ ) (cid:1) ( ρ − ρ ∗ ) + ≤ c Z t | ( ρ − ρ ∗ ) + | and the assertion is obtained by the Gronwall argument. The second inequality followsfrom the monotonicity of β . Moreover, the lower bounds can be checked in a similar way.To prove (4.3), we set w i ( t ) = µ i ( t ) g ′ ( ρ i ( t )) − π ( ρ i ( t )), ξ i ( t ) = w i ( t ) − ˙ ρ i ( t ), i = 1 , ξ − ξ )( ρ − ρ ) ≥ ξ − ξ ) (withsign(0) = 0) is bounded and measurable, and so is sign( ρ − ρ ). We now claim that bytesting the identity ( ξ − ξ ) + ( ˙ ρ − ˙ ρ ) = w − w (4.6)by sign( ξ − ξ ), we infer that | ξ − ξ | + ddt | ρ − ρ | ≤ | w − w | a.e. in (0 , T ) . (4.7)Indeed, this is obvious for all t such that sign( ξ − ξ )( t ) = sign( ρ − ρ )( t ) or such that ξ ( t ) = ξ ( t ). The remaining case is sign( ξ − ξ )( t ) = 0, sign( ρ − ρ )( t ) = 0. For almostall t with this property, we have ˙ ρ ( t ) = ˙ ρ ( t ), ddt | ρ − ρ | ( t ) = 0, and (4.7) follows. Usingthe Lipschitz continuity properties in (2.10) and integrating (4.7) over (0 , t ), we obtainfor t ∈ (0 , T ) Z t | ξ − ξ | ( s ) ds + | ρ − ρ | ( t ) ≤ c (cid:18) | ρ − ρ | + Z t (cid:0) (1 + µ ) | ρ − ρ | + | µ − µ | (cid:1) ( τ ) dτ (cid:19) . On the other hand, (4.6) yields Z t | ˙ ρ − ˙ ρ | ( s ) ds ≤ Z t (cid:0) | w − w | + | ξ − ξ | (cid:1) ( s ) ds and (4.3) follows from the sum of the last two inequalities.Next, if ( µ, ρ, ξ, u ) is a solution to problem (2.35)–(2.38), it is clear that, for almost all x ∈ Ω, the functions µ ( x, · ) and ρ ( x, · ), and the constant ρ ( x ) satisfy the assumptions ofLemma 4.1. Thus, the first part of Theorem 2.6 concerning bounds (2.44) is proved. Wederive an interesting consequence. Corollary 4.2.
Under the assumptions of Theorem 2.6, let ( µ, ρ, ξ, u ) be a solution toproblem (2.35) – (2.38) satisfying the regularity conditions specified in Theorem 2.3. Then µ ∈ L ∞ ( Q ) and ∂ t ρ ∈ L ∞ ( Q ) . (4.8)6 Vanishing diffusion limit in nonstandard phase field systems
Proof.
We already know that both ξ and π ( ρ ) are bounded. Moreover, µg ′ ( ρ ) belongsto L ∞ (0 , T ; H ) ∩ L (0 , T ; L (Ω)) since µ does so and g ′ ( ρ ) is bounded. We see thatalso ∂ t ρ belongs to such a space, just by comparison in (2.37). It follows that ∂ t ρ ∈ L / (0 , T ; L / (Ω)) by (2.55). From this and assumption (2.45), we derive the bounded-ness of µ . Indeed, we can reproduce the proof carried out in [6, Fifth a priori estimate],since that proof acts only on the equation for µ and works provided that an estimate of ∂ t ρ in L / (0 , T ; L / (Ω)) is known. At this point, by comparing in (2.37) once more, weconclude that ∂ t ρ is bounded as well. Remark 4.3.
The analogous estimate ρ ∗ ≤ ρ σ ≤ ρ ∗ a.e. in Q (4.9)for the solution to problem (2.26)–(2.28) also holds provided that ρ ∗ ≤ ρ σ ≤ ρ ∗ a.e. in Ω . (4.10)We prove one of the inequalities (4.9), the other one being similar. We proceed as in theproof of Lemma 4.1, testing (2.27) by ( ρ σ − ρ ∗ ) + and integrating. By accounting for thesecond inequality (4.10), we easily obtain12 Z Ω | ( ρ σ − ρ ∗ ) + ( t ) | + σ Z Q t |∇ ( ρ σ − ρ ∗ ) + | + Z Q t (cid:0) ξ σ − ξ ∗ (cid:1) ( ρ σ − ρ ∗ ) + + Z Q t (cid:0) ξ ∗ + π ( ρ ∗ ) (cid:1) ( ρ σ − ρ ∗ ) + ≤ Z Q t (cid:0) π ( ρ ∗ ) − π ( ρ σ ) (cid:1) ( ρ σ − ρ ∗ ) + + Z Q t µ σ g ′ ( ρ σ )( ρ σ − ρ ∗ ) + . Now, we observe that all the terms on the left-hand side are nonnegative, the third onethanks to (2.40) and the monotonicity of β (as before, the integrand vanishes whenever ρ σ ≤ ρ ∗ ), the last one due to (2.41). Concerning the right-hand side, we show that the lastintegrand is nonpositive. Indeed, g ′ is decreasing (see (2.9)), whence g ′ ( ρ σ ) ≤ g ′ ( ρ ∗ ) ≤ ρ σ > ρ ∗ , and µ σ ≥
0. By taking all this into account and owing to the Lipschitz continuityof π (cf. (2.11)), we can apply the Gronwall lemma and conclude that ( ρ σ − ρ ∗ ) + = 0, i.e., ρ ≤ ρ ∗ a.e. in Q . Remark 4.4.
A sufficient condition for (4.10) to hold at least for small σ is that ρ σ isgiven by (2.20) and the hypotheses of Theorem 2.6 are reinforced by also assuming thateither inf ess ρ > ρ ∗ and sup ess ρ < ρ ∗ or ξ ∗ ≤ ≤ ξ ∗ . (4.11)The proof is rather simple and we show just one of the desired inequalities since the otherone is quite similar. We test (2.20) by ( ρ σ − ρ ∗ ) + . We easily obtain Z Ω | ( ρ σ − ρ ∗ ) + | + σ Z Ω |∇ ( ρ σ − ρ ∗ ) + | + σ Z Ω ( ξ σ − ξ ∗ )( ρ σ − ρ ∗ ) + = Z Ω ( ρ − ρ ∗ − σξ ∗ )( ρ σ − ρ ∗ ) + . (4.12) olli — Gilardi — Krejˇc´ı — Sprekels δ := ρ ∗ − sup ess ρ and take σ ∗ > σ ∗ | ξ ∗ | ≤ δ .Then, for σ ≤ σ ∗ , we have ρ − ρ ∗ − σξ ∗ ≤ − δ + σ ∗ | ξ ∗ | ≤ β is monotone, and the third integrand vanishes whenever ρ σ ≤ ρ ∗ ), we conclude that( ρ σ − ρ ∗ ) + = 0, whence ρ σ ≤ ρ ∗ . Proof of the second part of Theorem 2.6.
Assume thus that κ ( µ, ρ ) = κ and setfor simplicity κ = 1. The system now reads h ∂ t u ( t ) , v i + Z Ω ∇ µ ( t ) · ∇ v = Z Ω µ g ′ ( ρ ) ∂ t ρ v for all v ∈ V and a.a. t ∈ (0 , T ) , (4.13) u = (1 + 2 g ( ρ )) µ a.e. in Q, (4.14) ∂ t ρ + ξ + π ( ρ ) = µ g ′ ( ρ ) and ξ ∈ β ( ρ ) a.e. in Q, (4.15) µ (0) = µ and ρ (0) = ρ a.e. in Ω . (4.16)Let ( µ i , ρ i , ξ i , u i ), i = 1 , t and subtract the equation with index 2 from the one with index 1. We testthe result by v = ( µ − µ )( t ) and obtain, by virtue of Corollary 4.2, that Z Ω ( u − u )( µ − µ )( t ) + 12 ddt Z Ω (cid:12)(cid:12)(cid:12)(cid:12)Z t ∇ ( µ − µ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ c Z Ω (cid:18) | µ − µ | ( t ) Z t ( | µ − µ | + | ρ − ρ | + | ∂ t ρ − ∂ t ρ | ) ( τ ) dτ (cid:19) . (4.17)In addition, from Lemma 4.1 (see, in particular, (4.3)) and H¨older’s inequality it followsthat Z Ω (cid:18)Z t | ∂ t ρ − ∂ t ρ | ( τ ) dτ (cid:19) ≤ c Z Ω (cid:18)Z t ( | ρ − ρ | + | µ − µ | )( τ ) dτ (cid:19) , (4.18) Z Ω | ρ − ρ | ( s ) ≤ D Z s Z Ω (cid:0) | ρ − ρ | + | µ − µ | (cid:1) ( τ ) dτ (4.19)for every t, s ∈ [0 , T ], thanks to the boundedness for µ ensured by Corollary 4.2. Notethat the constant D in (4.19) is marked for later reference.Now, we observe that the inequalities( u − u )( µ − µ ) ≥ | µ − µ | − µ (cid:0) g ( ρ ) − g ( ρ ) (cid:1) ( µ − µ ) ≥ | µ − µ | − c | ρ − ρ | hold a.e. in Q . Thus, by integrating (4.17) from 0 to s , s ∈ (0 , T ), and ignoring a positiveterm on the left-hand side, we obtain Z s Z Ω | µ − µ | ( t ) dt ≤ c Z s Z Ω | ρ − ρ | ( t ) dt + c (cid:18)Z s Z Ω | µ − µ | ( t ) dt (cid:19) / × Z s Z Ω (cid:18)Z t ( | µ − µ | + | ρ − ρ | + | ∂ t ρ − ∂ t ρ | ) ( τ ) dτ (cid:19) dt ! / . (4.20)8 Vanishing diffusion limit in nonstandard phase field systems
Hence, using Young’s inequality and (4.18), we have that Z s Z Ω | µ − µ | ( t ) dt ≤ c Z s Z Ω | ρ − ρ | ( t ) dt + c Z s Z Ω (cid:18)Z t ( | µ − µ | + | ρ − ρ | ) ( τ ) dτ (cid:19) dt. (4.21)We now multiply (4.21) by 2 D and add it to (4.19). Thus, we obtain an inequality of theform Φ( s ) ≤ c R s Φ( t ) dt , withΦ( s ) = Z Ω | ρ − ρ | ( s ) + Z s Z Ω | µ − µ | ( t ) dt. From the Gronwall argument, it is straightforward to deduce that Φ( s ) = 0 for all s ,hence, µ = µ , ρ = ρ , which implies uniqueness.The L bound for ∂ t µ can be established in the following way. Assume first that µ ∈ W . We extend µ by µ and ρ by ρ for t <
0. Then, equation (4.13) then can bewritten as h ∂ t u ( t ) , v i + Z Ω ∇ µ ( t ) · ∇ v = Z Ω ψ ( t ) v for all v ∈ V and a.a. t ∈ (0 , T ), (4.22)where ψ is defined by ψ ( t ) = (cid:0) µg ′ ( ρ ) ∂ t ρ (cid:1) ( t ) for t > ψ ( t ) = − ∆ µ for t <
0. Weobserve that ψ ∈ L ∞ ( − T, T ; H ) thanks to Corollary 4.2 and to our assumption on µ .Next, we integrate (4.22) in time from ( t − h ) to t for any fixed t ∈ (0 , T ) and a small h > h tend to zero, and test the resulting equality by µ ( t ) − µ ( t − h ).We obtain Z Ω (cid:0) u ( t ) − u ( t − h ) (cid:1)(cid:0) µ ( t ) − µ ( t − h ) (cid:1) + 12 Z Ω ddt (cid:12)(cid:12)(cid:12)(cid:12)Z tt − h ∇ µ ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) = Z Ω (cid:18)Z tt − h ψ ( τ ) dτ (cid:19) (cid:0) µ ( t ) − µ ( t − h ) (cid:1) ≤ Z Ω | µ ( t ) − µ ( t − h ) | + (cid:13)(cid:13)(cid:13)(cid:13)Z tt − h ψ ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) H ≤ Z Ω | µ ( t ) − µ ( t − h ) | + c h (4.23)Now, we recall that (4.14) holds, that g is nonnegative and Lipschitz continuous, and that µ and ∂ t ρ are bounded by Corollary 4.2. Hence, we easily derive that (cid:0) u ( t ) − u ( t − h ) (cid:1)(cid:0) µ ( t ) − µ ( t − h ) (cid:1) ≥ | µ ( t ) − µ ( t − h ) | − µ ( t ) | g ( ρ ( t )) − g ( ρ ( t − h )) | | µ ( t ) − µ ( t − h ) |≥ | µ ( t ) − µ ( t − h ) | − c h | µ ( t ) − µ ( t − h ) | ≥ | µ ( t ) − µ ( t − h ) | − c h . Therefore, by integrating (4.23) from 0 to T , forgetting the nonnegative term that in-volves ∇ µ , and rearranging, we obtain Z T Z Ω | µ ( t ) − µ ( t − h ) | dt ≤ c h + c Z Ω (cid:12)(cid:12)(cid:12)(cid:12)Z − h ∇ µ dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ c h . olli — Gilardi — Krejˇc´ı — Sprekels h > ∂ t µ ∈ L ( Q ). At this point, we are allowedto use the Leibniz rule for the time derivative ∂ t u ; then, from (4.13)–(4.14) we infer thatthe equation (cid:0) g ( ρ ) (cid:1) ∂ t µ + µg ′ ( ρ ) ∂ t ρ − ∆ µ = 0 (4.24)holds at least in the sense of distributions. By comparison, we deduce that ∆ µ ∈ L ( Q ),whence µ ∈ L (0 , T ; W ). Using the identity − Z Ω ∂ t µ ∆ µ = 12 ddt Z Ω |∇ µ | a.e. in (0 , T ) , we see that ∇ µ ∈ L ∞ (0 , T ; L (Ω)). Thus, the regularity (2.46) is established if µ ∈ W .Let now µ ∈ V ∩ L ∞ (Ω) be arbitrary, and consider a sequence { µ k } ⊂ W boundedin L ∞ (Ω) and converging to µ in V as k → ∞ . Let ( µ k , ρ k , ξ k , u k ) be the correspondingsolutions to (4.13)–(4.16). Then, we can use equation (4.24) written with the index k andtest it by ∂ t µ k . We obtain Z Ω | ∂ t µ k ( t ) | + 12 ddt Z Ω |∇ µ k ( t ) | ≤ Z Ω | ψ k ( t ) | | ∂ t µ k ( t ) | , (4.25)with an obvious choice of ψ k ∈ L ( Q ) bounded in this space (even better) independentlyof k . By time integration, it is straightforward to obtain a bound for k ∂ t µ k k L ( Q ) and for k∇ µ k k L ∞ (0 ,T ; H ) independent of k . Then, by weak star compactness we infer that µ k → ˜ µ weakly star in H (0 , T ; H ) ∩ L ∞ (0 , T ; V )at least for a subsequence, which implies (see, e.g., [15, Cor. 4, p. 85]) strong convergencein C ([0 , T ]; H ). In particular, ˜ µ (0) = µ . On the other hand, ( µ k , ρ k , ξ k , u k ) satisfies theestimates stated in Lemma 4.1 and the boundedness properties for µ k and ∂ t ρ k given byCorollary 4.2, which are uniform with respect to k . This yields weak or weak star limits˜ ρ and ˜ ξ . Moreover, strong convergence in L ( Q ) for { ρ k } and { ∂ t ρ k } is ensured via aCauchy sequence argument based on (4.3), integration over Ω, and Gronwall’s lemma.Hence, { µ k } , { ρ k } , { ∂ t ρ k } converge strongly in L p ( Q ) for every p ∈ [1 , ∞ ). At thispoint, it is not difficult to verify that (˜ µ, ˜ ρ, ˜ ξ, ˜ u ), with the corresponding ˜ u , actually solvesproblem (2.35)–(2.38) and thus coincides with the unique solution ( µ, ρ, ξ, u ). Therefore,the proof is complete. Acknowledgments
The authors gratefully acknowledge the warm hospitality of the IMATI of CNR in Pavia,the Institute of Mathematics of the Czech Academy of Sciences in Prague, and theWIAS in Berlin. The present paper benefits from the GA ˇCR Grant P201/10/2315 andRVO: 67985840 for PK, the MIUR-PRIN Grant 2010A2TFX2 “Calculus of variations” forPC and GG, and the FP7-IDEAS-ERC-StG Grant
Matheon in Berlin.0
Vanishing diffusion limit in nonstandard phase field systems
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