DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 19, Number 2, October 2007 pp. 335–359

ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES WITH TIME-DEPENDENT CONSTRAINTS

Masahiro Kubo

Department of Nagoya Institute of Technology Gokiso-cho, Showa-ku, Nagoya-city Aichi, 466-8555, Japan Noriaki Yamazaki

Department of Mathematical Science Common Subject Division Muroran Institute of Technology 27-1 Mizumoto-ch¯o, Muroran Hokkaido, 050-8585, Japan

Abstract. We study variational inequalities for quasilinear elliptic-parabolic equations with time-dependent constraints. Introducing a general condition for the time-dependence of convex sets defining the constraints, we establish theo- rems concerning existence, uniqueness as well as an order property of solutions. Some applications of the general results are given.

1. Introduction. This paper aims to establish a general theory of variational in- equalities with time-dependent constraints for elliptic-parabolic equations:

b(u)t − ∇ · a(x, b(u), ∇u)= f(t, x) in (0,T ) × Ω. (1.1) Here Ω is a bounded domain in RN (N ≥ 1), b : R → R is a given bounded, nondecreasing and Lipschitz continuous function, the term a(x,s,p) is a quasi-linear elliptic vector field satisfying some structure condition, in particular we assume N a(x,s,p)= ∂pA(x,s,p) for a potential function A :Ω × R × R → R, and f(t, x) is a given function on (0,T ) × Ω. The equation (1.1) is called an elliptic-parabolic equation, since it is elliptic in the region {b′(u)=0} and parabolic in {b′(u) > 0}, respectively. A time-dependent constraint is given by a family of time-dependent convex sets. Under an appropriate condition on the convex sets, we prove the existence, uniqueness as well as an order property of solutions with strong time- derivatives of b(u) in L2(Ω) to the following: Problem (P) : u(t) ∈ K(t), 0

(b(u)t,u − v)+ a(x, b(u), ∇u) · ∇(u − v)dx ≤ (f,u − v), v ∈ K(t), 0

2000 Mathematics Subject Classification. Primary: 35K85, 35K65; Secondary 47J35, 76S05. Key words and phrases. Elliptic-parabolic variational inequalities, flows in porous media, non- linear evolution equations.

335 336 M. KUBO AND N. YAMAZAKI

1 Here K(t) is a time-dependent convex in H (Ω) and b0 is a given initial value. There have been many mathematical studies concerning elliptic-parabolic equa- tions motivated by both mathematical and physical interests. In of elliptic-parabolic equations, two kinds of solutions have been studied: the weak solutions and the strong solutions. The strong solutions admit strong time derivatives of b(u) in L2(Ω) (or L1(Ω)), while the weak solutions do not. The notion of weak solutions was introduced by H. W. Alt and S. Luckhaus. In a fundamental paper [1], they studied a general system of elliptic-parabolic equations with Dirichlet-Neumann boundary conditions, and problems with convex constraints. They introduced a class of weak solutions, without any strong time- derivative of b(u) in L1(Ω), based on an energy integral defined by the Legendre transform of the primitive of b. Later, Otto [28] proved the uniqueness of the weak solution of [1] by an elegant technique: the doubling of variables. Other kinds of weak solutions of elliptic-parabolic equations without a strong time-derivative in L1(Ω): mild solutions and renormalized solutions, and their relations to the weak solutions of Alt-Luckhaus [1] have also been studied. We refer, for instance, to Carrillo-Wittbold [10]. We note that in these works of weak, mild or renormalized solutions, it is assumed only that b is nondecreasing and continuous, and the elliptic vector field a does not need to have a potential A. The strong solutions, which admit strong time-derivatives of b(u) in L2(Ω), have been studied when the function b is assumed to be Lipschitz continuous and the vector field a admits a potential: a = ∂pA. In this case, the equation (1.1) (or the simpler one (1.3) below) has been studied with various constraints posed on the boundary or in the interior of the domain. We refer, for instance, to Alt-Luckhaus [1, Theorem 2.3], Bertsch-Hulshoff [5], Hornung [13], Hulshoff [15], Kenmochi-Pawlow [19, 20] and Rodrigues [31]. In physical applications, the elliptic-parabolic equation (1.1) and variational in- equalities for it appear in models of flows in partially saturated porous media (cf. van Duyn-Pelelier [11] and Fasano-Primicerio [12]). In these models, the spatial domain Ω is occupied with a porous medium, the unknown function u refers to the pressure, and s = b(u) is the saturation depending on the pressure u via the function b. A typical form of the equations is as follows:

b(u)t − ∇ · a(x)[∇u + k(b(u))] = f(t, x) in (0,T ) × Ω. (1.2)

The positive definite matrix a = (ai,j ) describes the permeability of the fluid. The term k(b(u)) refers to the mobility effect caused by the gravitational force, where k : R → RN is a given function modeling the dependence on the saturation. Finally, f refers to a given source of the fluid. For (1.2), we put a(x,s,p) = a(x)[p + k(s)] in (1.1). Most of the above cited works were motivated by this model of partially sat- urated flows. In particular, Alt-Luckhaus-Visintin [2] studied the equation (1.2) with time-dependent boundary constraints referring to time-changing water levels of reservoirs. They studied the problem in the framework of weak solutions of Alt-Luckhaus [1], and proved the existence of a weak solution. Its uniqueness was proved later by Otto [29]. Weak solutions for other obstacle problems were also studied by Ivanov-Rodrigues [16]. By applying their abstract theory in [19], Kenmochi-Pawlow [20] gave a general framework of strong solutions to variational inequalities for the following equation ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 337

(1.3): b(u)t − ∆u = f(t, x) in (0,T ) × Ω. (1.3) Although their result can be applied to problems with various time-dependent con- straints posed on the boundary or in the interior of the domain, such as considered in the problems of [1, 2, 28, 29], it is not applicable to equations of the form (1.1) or (1.2), which is important in studying partially saturated flows taking account of the effect of gravitational force. Now, the aim of the present paper is to establish a framework of strong solutions of the problem (P) with general time-dependent constraints implied by the K(t). Hence we extend the framework of elliptic-parabolic variational inequal- ities of Kenmochi-Pawlow [20], so that we can treat equations of the form (1.1) or (1.2), and can treat, as applications, models of partially saturated flows with the gravitational force. Our solutions have strong time-derivatives of b(u) in L2(Ω), and our results can be applied, for instance, to the problems with constraints as studied in [1, 2, 20, 28, 29]. We assume that b is bounded and Lipschitz continuous, and that the elliptic vector field a admits a potential (cf. assumptions (A) and (B) below), which is natural in applications to partially saturated flows. Our result can be regarded as the regularity property of the weak solutions in [1, 2, 28, 29]. In fact, every solution in our sense is a weak solution in their sense (cf. [1, Section 1.4, Lemma 1.5], [29, p.541 Remark]), and the uniqueness of weak solution holds by the result of Otto [28, 29]. For example, the regularity theorem of weak solutions to the Dirichlet-Neumann problem in [1, Theorem 2.3] can be reproduced by our result as a very special case (cf. Section 5.1). Since our solutions have strong time-derivatives in L2(Ω), the order property (and hence uniqueness) of solutions can be proved by a usual test function technique, which is much simpler than the uniqueness proof of weak solutions as given by [28, 29]. There seems, however, no result so far that is directly applicable to our problem (P) in general. Hence, we give a proof of it (cf. Remark 2.3). Moreover, working with strong solutions has an advantage in studying the large time behavior of solutions. In fact, we prove in [25, 26] that when the data f(t) and K(t) are periodic in time, there exists a unique and asymptotically stable periodic solution. We further plan to study various asymptotic behavior in our frame work of strong solutions. The main part of this paper is the proof of existence of a solution to the problem (P). For that, we generalize and refine the idea of Kenmochi-Kubo [18], which is a fixed point argument combined with an energy introduced by Kenmochi- Pawlow [19, 20]. We note that the method of the energy inequality in [19, 20] originates in the theory of time-dependent subdifferential evolution equations, for which we refer to [4, 6, 14, 17, 23, 24, 27, 33, 34] and their references (cf. Remark 2.2). The plan of this paper is as follows. In the next Section 2, the main results concerning existence-uniqueness (Theorem 2.1) and an order property (Theorem 2.2) of solutions are stated after assumptions (K1)–(K4) on K(t) and a definition of a solution are given. Theorem 2.2 is proved at the end of Section 2. The existence of a solution is proved in Sections 3 and 4, In the final Section 5, we give some applications of the main results to some models of partially saturated flows. Notation and basic assumptions. Throughout this paper, we put H := L2(Ω) with usual real Hilbert space structure. The inner product and norm in H are 338 M. KUBO AND N. YAMAZAKI

1 denoted by (·, ·) and by | · |H , respectively. We also put V := H (Ω) with the usual 1 2 2 2 norm |u|V := |u|H + |∇u|H . Various L∞-norms, e.g., norms in L∞(Ω),L∞(R), etc., are all denoted by the  same symbol | · |∞. In the proof of existence result (Sections 3 and 4), we use some techniques of proper (i.e., not identically equal to infinity), l.s.c. (lower semi-continuous) and convex functions and their subdifferentials, which are useful in the systematic study of variational inequalities. Let us here prepare some notations and definitions. For a proper, l.s.c., convex function ϕ : H → R ∪{+∞}, the effective domain D(ϕ) is defined by D(ϕ) := {z ∈ H; ϕ(z) < +∞} and the subdifferential of ϕ is a possibly multi-valued operator in H and is defined by z∗ ∈ ∂ϕ(z) if and only if z ∈ D(ϕ) and (z∗,y − z) ≤ ϕ(y) − ϕ(z) for all y ∈ H. For various properties and related notions of proper, l.s.c., convex functions and their subdifferentials, we refer to a monograph by Br´ezis [8]. Also, for the sake of order relation, we use the following notation: u ∨ v := max{u, v}, u ∧ v := min{u, v}. Also, we denote the positive part of u by [u]+, namely, [u]+ := u ∨ 0. Let us now give some assumptions on data. Throughout this paper, the elliptic vector field a and the nondecreasing function b are assumed to satisfy the following conditions (A) and (B), respectively.

(A) a(x,s,p)= ∂pA(x,s,p) for some potential function A(x,s,p). a(x, ·, ·) : R × RN → RN is continuous for a.e. x ∈ Ω, a(·,s,p):Ω → RN is measurable for any s ∈ R, p ∈ RN , A(x, ·, ·) : R × RN → R is continuous for a.e. x ∈ Ω, A(·,s,p):Ω → R is measurable for any s ∈ R, p ∈ RN .

Moreover, there exist constants µ > 0, C1 = C1(a) > 0 and C2 = C2(a) > 0 such that [a(x,s,p) − a(x, s, pˆ)] · (p − pˆ) ≥ µ|p − pˆ|2, 2 2 2 2 |a(x,s,p)| + |A(x,s,p)| + |∂sA(x,s,p)| ≤ C1(1 + |s| + |p| ),

|a(x,s,p) − a(x, s,pˆ )|≤ C2(1 + |p|)|s − sˆ| for all x ∈ Ω, s, sˆ ∈ R, p, pˆ ∈ RN . (B) b : R → R is bounded, nondecreasing and Lipschitz continuous with a Lips- chitz constant Cb > 0. Note that the function A(x,s,p) is convex in p ∈ RN by condition (A) and [30, Theorem 42B]. As for the data {b0,f,K(t)}, the following condition (C) is always assumed. 2 (C) b0 ∈ H,f ∈ L (0,T ; H) and K(t) is a non-empty, closed and convex set in V for all t ∈ [0,T ].

Finally, throughout this paper, Ci = Ci(·),i = 1, 2, 3, · · · , denotes positive (or nonnegative) constants, depending on its argument(s). ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 339

2. Main results.

2.1. Theorems. We begin by defining the notion of a solution.

Definition 2.1. A function u : [0,T ] → V is a solution of (P), or (P;b0,f,K(t)) when the data are specified, if the following items (a)–(d) are satisfied. (a) u ∈ L∞(0,T ; V ) and there exists u∗ ∈ W 1,2(0,T ; H) such that b(u)= u∗ for a.e. t ∈ (0,T ) (cf. Remark 2.1). (b) u ∈ K(t) for a.e. t ∈ (0,T ). (c) For a.e. t ∈ (0,T ) the following inequality holds:

∗ (ut ,u − v)+ a(x, b(u), ∇u) · ∇(u − v)dx ≤ (f,u − v) ZΩ for all v ∈ K(t). ∗ (d) u (0) = b0 in H.

Remark 2.1. In what follows, we identify the function u∗ with b(u) in the condition (a) of Definition 2.1, and always write b(u) for u∗.

Next, we list the assumptions (K1)–(K4) for the convex sets K(t), 0 ≤ t ≤ T . (K1) There is a function α ∈ W 1,2(0,T ) satisfying the following property: for any 0 ≤ s

|z˜ − z|H ≤ |α(t) − α(s)|(1 + |z|V ),

A(x, w(x), ∇z˜(x))dx − A(x, w(x), ∇z(x))dx ZΩ ZΩ 2 ≤ |α(t) − α(s)|(1 + |z|V + |w|V |z|V ).

(K2) There is a constant C3 = C3(K) > 0 such that

|z|V ≤ C3(1 + |∇z|H ) for all z ∈ K(t) and t ∈ [0,T ]. (K3) For any z, z ∈ K(t) and w, w ∈ V with w ≤ z, z ≤ w, we have w ∨ z, z ∧ w ∈ K(t). (K4) If z, z ∈ K(t) and ∇[z − z]+ ≡ 0, then z ≤ z.

Remark 2.2. Conditions (K1) and (K2) are used to prove the existence of a solu- tion, while conditions (K3) and (K4) are related to an order property of the solutions (cf. Theorem 2.2). Note that the condition (K1) represents smooth dependence of the convex set K(t) upon the time t. If the convex set is time-independent, i.e., K(t) ≡ K, the condition (K1) is trivially satisfied by α ≡ 0 andz ˜ = z. When the potential function A(x,s,p) does not depend on s, the condition (K1) is largely the same as the one in Kenmochi-Pawlow [19]. As for time-dependence conditions on constraints in parabolic variational inequalities, we refer to Hu-Papageorgiou [14], Kenmochi [17] and their references. 340 M. KUBO AND N. YAMAZAKI

We can now state the theorem concerning the existence and uniqueness of a solution. Theorem 2.1 (existence and uniqueness of a solution). Assume (K1)–(K4) are 1,2 satisfied and let f ∈ W (0,T ; H) and b0 = b(u0) for some u0 ∈ K(0). Then, there is a unique solution of (P;b0,f,K(t)). In order to state the second theorem, we define an order relation between convex sets. Definition 2.2 (cf. [7, (1.36)]). For two convex sets K, K ⊂ H, we write K ≤∗ K, if the following holds: z ∧ z ∈ K and z ∨ z ∈ K for all z ∈ K, z ∈ K. The second theorem is concerned with an order property of solutions with respect to given data b0,f and K(t).

Theorem 2.2 (comparison of solutions). Let u and u be solutions to (P;b0,f,K(t)) and (P;b0, f, K(t)), respectively for two sets of data {b0,f,K(t)} and {b0, f, K(t)}. Suppose that both K(t) and K(t) satisfy condition (K3). Assume K(t) ≤∗ K(t) for all t ∈ [0,T ]. Then, we have the following:

[b(u(t, x)) − b(u(t, x))]+dx ZΩ t + + ≤ [b0(x) − b0(x)] dx + dτ [f(τ, x) − f(τ, x)] dx (2.1) ZΩ Z0 ZΩ for all t ∈ [0,T ]. If furthermore b0 ≤ b0,f ≤ f and the following property (2.2) (cf. (K4)) holds: if z ∈ K(t), z ∈ K(t) and ∇[z − z]+ ≡ 0, then z ≤ z, (2.2) then, we have u ≤ u. The uniqueness part of Theorem 2.1 can be obtained by applying Theorem 2.2 with {b0,f,K(t)} = {b0, f, K(t)}. The existence part of Theorem 2.1 will be proved in Sections 3 and 4. We give the proof of Theorem 2.2 below.

2.2. Proof of Theorem 2.2. We employ the techniques used in the proofs of [1, Theorem 2.2], [20, Lemma 5.4] and [18, Proposition 2.1] with some necessary modifications (cf. Remark 2.3). First we see that

u − εσε(u − u) = (u − ε) ∨ (u ∧ u), u + εσε(u − u) = (u ∨ u) ∧ (u + ε), (2.3) where σε : R → R,ε> 0, is defined by 0 (r ≤ 0), σ (r) := ε−1r (0

Therefore, we conclude u − εσε(u − u) ∈ K(t) (cf. (2.3)). Similarly, we have u + εσε(u − u) ∈ K(t). Now, we take v = u−εσε(u−u) in (c) of Definition 2.1 for (P;b0,f,K(t)). Then, dividing by ε> 0, we obtain

(b(u)t, σε(u − u)) + a(x, b(u), ∇u) · ∇(σε(u − u))dx ≤ (f, σε(u − u)). (2.4) ZΩ Also, we can take v = u + εσε(u − u) in (c) of Definition 2.1 for (P;b0, f, K(t)). Then, dividing by ε> 0, we obtain

−(b(u)t, σε(u − u)) − a(x, b(u), ∇u) · ∇(σε(u − u))dx ≤−(f, σε(u − u)). (2.5) ZΩ Adding (2.5) to (2.4), we have

(b(u)t − b(u)t, σε(u − u))

+ {a(x, b(u), ∇u) − a(x, b(u), ∇u)} · ∇(σε(u − u))dx ZΩ ≤ (f − f, σε(u − u)). (2.6) Here we observe by assumption (A) for the elliptic vector field a

{a(x, b(u), ∇u) − a(x, b(u), ∇u)} · ∇(σε(u − u))dx ZΩ ′ + = {a(x, b(u), ∇u) − a(x, b(u), ∇u)} · (σε(u − u)∇[u − u] )dx ZΩ ′ + + {a(x, b(u), ∇u) − a(x, b(u), ∇u)} · (σε(u − u)∇[u − u] )dx ZΩ C (a)|b′| ≥− 2 ∞ (1 + |∇u|)|u − u||∇[u − u]+|dx ε Z{00} ZΩ + ≤ (f − f,χ{u−u>0}) ≤ [f − f] dx. ZΩ Hence, we have the desired inequality by integration. Here, χ{u−u>0} is the char- acteristic function of the set {x ∈ Ω; u − u > 0}, i.e. χ{u−u>0} = 1 on the set {x ∈ Ω; u − u> 0} and χ{u−u>0} =0 on the set Ω \{x ∈ Ω; u − u> 0}. Next, assume b0 ≤ b0,f ≤ f, then by (2.1) we have b(u) ≤ b(u). Therefore, we have the following relations

{x ∈ Ω; σε(u − u) > 0} = {x ∈ Ω; u − u> 0}⊂{x ∈ Ω; b(u)= b(u)}, which implies that the first term of the left hand side in (2.6) is equal to 0, i.e.,

(b(u)t − b(u)t, σε(u − u)) = 0 for a.e. t ∈ (0,T ) and any ε> 0. 342 M. KUBO AND N. YAMAZAKI

Hence by (2.6) and assumption (A) we have µ |∇(u − u)|2dx ≤ 0 for a.e. t ∈ (0,T ) and any ε> 0. ε Z{0 0 since the set {x ∈ Ω;0

Remark 2.3. The fundamental idea behind the above proof of the order property is a standard test function technique. However, so far no results seem directly applicable to our problem (P). In fact, we need to introduce some technicalities: the expression (2.3) of the test functions and the condition (K3) on the convex set K(t).

3. Auxiliary lemmas. In this and the next sections, we suppose that all the assumptions of Theorem 2.1 hold and prove the existence part of it. First, we rewrite the problem (P) in a form of an evolution equation in H. Define, for each t ∈ [0,T ], a function ϕt : H × H → R ∪{+∞} by

A(x, w(x), ∇z(x))dx + C , if z ∈ K(t), ϕt(w; z) := 4 (3.1)  ZΩ  +∞, otherwise, C C ϕ where 4 = 4( ) is a constant specified in the following lemma. Lemma 3.1. For all t ∈ [0,T ] and w ∈ H, ϕt(w; ·) : H → R ∪{+∞} is a proper, l.s.c. and convex function with effective domain D(ϕt(w; ·)) = K(t). Moreover, by choosing C4 in (3.1) in an appropriate manner we can find a con- stant C5 = C5(ϕ) > 0 such that t 2 ϕ (w; z) ≥ C5|z|V +1 (3.2) for all t ∈ [0,T ],z ∈ K(t) and w ∈ H with |w| ≤ |b|∞ a.e. in Ω. Proof. By assumptions (A) and (C), we see that ϕt(w; ·) is proper and convex. In order to prove (3.2), we notice by (A) 1 d 1 A(x,w,p) − A(x, w, 0) = A(x,w,ξp)dξ = a(x,w,ξp) · pdξ dξ Z0 Z0 µ|p|2 1 1 ≥ a(x, w, 0) · p + ≥− C (1 + |w|2)+ µ|p|2. 2 µ 1 4 Therefore, for w ∈ H with |w| ≤ |b|∞ 1 1 A(x,w,p) ≥−(1 + )C (1 + |b|2 )+ µ|p|2, µ 1 ∞ 4 and hence for z ∈ K(t) we have by (K2) 1 A(x, w(x), ∇z(x))dx ≥−C(Ω,a,b)+ µ|∇z|2 4 H ZΩ 2 ≥−C4(ϕ)+ C5(ϕ)|z|V +1, ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 343 where 1 C(Ω,a,b)= |Ω|(1 + )C (1 + |b|2 ), µ 1 ∞ µ C (ϕ)= C(Ω,a,b)+1+ , 4 4 1 C (ϕ)= µC (K)−2. 5 8 3 Thus we have (3.2). Finally, we shall prove the lower-semicontinuity. First note that, by a standard result (cf. [32, Theorem C.2], [22, Theorem I.2.1]), the function z 7→ ϕt(w; z) is continuous on K(t) with respect to the topology of V . Therefore the set {z ∈ V ; ϕt(w; z) ≤ r} is a closed and convex set in V , and hence weakly closed in V . t Now if zn ∈ {z ∈ V ; ϕ (w; z) ≤ r} and zn → z in H, then, by (3.2) we have that zn → z weakly in V and hence by the weak closedness we have that z ∈{z ∈ V ; ϕt(w; z) ≤ r}. Therefore, we conclude that the set {z ∈ V ; ϕt(w; z) ≤ r} is closed in H, which means that the function z 7→ ϕt(w; z) is lower-semicontinuous. Thus we have proved the lemma.

By the above lemma, for each w ∈ H, the subdifferential ∂ϕt(w; ·) of ϕt(w; ·) is defined by z∗ ∈ ∂ϕt(w; z) if and only if z ∈ K(t)(= D(ϕt(w; ·))) and (z∗, v − z) ≤ ϕt(w; v) − ϕt(w; z) (3.3) for all v ∈ H. The next lemma characterizes ∂ϕt(w; ·). Lemma 3.2. Let t ∈ [0,T ] and w ∈ H. Then, for z,z∗ ∈ H, z∗ ∈ ∂ϕt(w; z) if and only if z ∈ K(t) and

(−z∗,z − v)+ a(x, w, ∇z) · ∇(z − v)dx ≤ 0 (3.4) ZΩ for all v ∈ K(t).

Proof. Let z∗ ∈ ∂ϕt(w; z). Replace v by ξv + (1 − ξ)z in (3.3). Then, we have

ξ(z∗, v − z) ≤ {A(x, w, ∇(ξv + (1 − ξ)z)) − A(x, w, ∇z)}dx. ZΩ Dividing by ξ > 0 and letting ξ ↓ 0, we have (3.4). Conversely, (3.4) implies (3.3) by the following inequality (cf. [30, Theorem 42A]) (a consequence of the convexity of A(x,s,p) in p): a(x,s,p) · (ˆp − p) ≤ A(x, s, pˆ) − A(x,s,p) for all x ∈ Ω,s ∈ R and p, pˆ ∈ RN .

By Lemma 3.2, we have the following reformulation of problem (P). Proposition 3.1. A function u : [0,T ] → H is a solution of problem (P) if and only if the following (a), (b) and (c) are satisfied. (a) u ∈ L∞(0,T ; V ) and b(u) ∈ W 1,2(0,T ; H). (b) b(u)′(t)+ ∂ϕt(b(u)(t); u(t)) ∋ f(t) in H for a.e. t ∈ (0,T ). (c) b(u)(0) = b0 in H. 344 M. KUBO AND N. YAMAZAKI

Here b(u) := b(u(t, ·)) and b(u)′ is its derivative as an H-valued function. Definition 3.1. A function u : [0,T ] → H is called a solution of (CP) if the items (a)–(c) in Proposition 3.1 are satisfied. Now the problem (P) is transformed to (CP). The idea for showing the existence of a solution of (CP) is as follows (cf. [18]). Given w : [0,T ] → H, we first solve the following problem (CP;w): b(u)′(t)+ ∂ϕt(w(t); u(t)) ∋ f(t), 0

4. Proof of existence. 4.1. Local solution. Put for T > 0 1,2 ∞ E(T ) := w ∈ W (0,T ; H) ∩ L (0,T ; V ); |w| ≤ |b|∞ a.e. in (0,T ) × Ω .  Lemma 4.1. Let w ∈ E(T ) and put t t Φw(z) := ϕ (w(t); z) for z ∈ H. s Then, we have the following: for any 0 ≤ s 0 is a constant. Proof. By (K1) and noting (3.2) we have

|z˜ − z|H ≤ |α(t) − α(s)|(1 + |z|V ) −1/2 s 1/2 ≤ |α(t) − α(s)|(1 + C5 (Φw(z)) ) −1/2 s 1/2 ≤ (1 + C5 )|α(t) − α(s)|(1 + (Φw(z)) ). Next note that by (A) and |w| ≤ |b|∞

A(x, w(t), ∇z˜)dx − A(x, w(s), ∇z˜)dx ZΩ ZΩ w(t,x) ∂A = (x, σ, ∇z˜(x))dσdx ∂σ ZΩ Zw(s,x) w(t,x) 1/2 ≤ C1 (1 + |σ| + |∇z˜(x)|)dσ dx Ω w(s,x) Z Z

ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 345

1/2 ≤ C1 |w(t, x) − w(s, x)|(1 + |b|∞ + |∇z˜(x)|)dx ZΩ 1/2 1/2 ≤ C1 (1 + |b|∞)|w(t) − w(s)|H (|Ω| + |∇z˜|H ). (4.1) s s 1/2 s Therefore, by (K1) and noting that Φw(z) ≥ (Φw(z)) since Φw(z) ≥ 1 by (3.2), we have t s Φw(˜z) − Φw(z) = A(x, w(t), ∇z˜)dx − A(x, w(s), ∇z)dx ZΩ ZΩ = A(x, w(t), ∇z˜)dx − A(x, w(s), ∇z˜)dx ZΩ ZΩ + A(x, w(s), ∇z˜)dx − A(x, w(s), ∇z)dx ZΩ ZΩ 1/2 1/2 ≤ C1 (1 + |b|∞)|w(t) − w(s)|H (|Ω| + |∇z˜|H ) 2 +|α(t) − α(s)|(1 + |z|V + |w(s)|V |z|V ) s t 1/2 ≤ C7 |α(t) − α(s)| (1+Φw(z)) + |w(t) − w(s)|H Φw(˜z) n s 1/2  +|α(t) − α(s)||w(s)|V (Φw(z)) , (4.2) where the constant C7 > 0 depends only on a, b, ϕ, Ω. o t 1/2 Let us now estimate the term (Φw(˜z)) in the right hand side of (4.2) (cf. (4.3) below). Using the inequality (4.2) and noting that |w| ≤ |b|∞, we obtain t s Φw(˜z) − Φw(z) s s 1/2 ≤ C8 (1+Φw(z)) + |α(t) − α(s)||w(s)|V (Φw(z))

n t 1/2 o +C9 Φw(˜z) s s 1/2 ≤ C8 (1+Φw(z)) + |α(t) − α(s)||w(s)|V (Φw(z)) n 1 1 o + Φt (˜z) + C2 2 w 2 9 for some constants C8(C7, α) > 0 and C9(C8, b, Ω) > 0. Therefore, we have by the s s 1/2 Schwarz inequality and noting Φw(z) ≥ (Φw(z)) ≥ 1 (cf. (3.2)) t Φw(˜z) s s s 1/2 2 ≤ 2Φw(z)+2C8 (1+Φw(z)) + |α(t) − α(s)||w(s)|V (Φw(z)) + C9 n2 s s o 1/2 ≤ (2+2C8 + C9 )(1+Φw(z))+2C8|α(t) − α(s)||w(s)|V (Φw(z)) 2 2 s 2 2 s ≤ (2+2C8 + C9 + C8 )(1+Φw(z)) + |α(t) − α(s)| |w(s)|V Φw(z) s 2 2 s ≤ C10 Φw(z)+ |α(t) − α(s)| |w(s)|V Φw(z) , where C10 depends only on C8 and C9. Hence we obtain  t 1/2 1/2 s 1/2 s 1/2 Φw(˜z) ≤ C10 (Φw(z)) + |α(t) − α(s)||w(s)|V (Φw(z)) , n o and furthermore  we have since |w| ≤ |b|∞ t 1/2 |w(t) − w(s)|H Φw(˜z) 1/2 s 1/2 ≤ |w(t) − w(s)|H C10 (Φw(z)) 1/2 1/2 s 1/2 +2|b|∞|Ω| C10 |α(t) − α(s)||w(s)|V (Φw(z)) . (4.3) 346 M. KUBO AND N. YAMAZAKI

Using (4.3) in the right hand side of (4.2), we have the Lemma for an appropriate constant C6 > 0. The notion of a solution of (CP;w) is defined in a way similar to that in Defini- tion 3.1 by replacing the term ∂ϕt(b(u)(t); u(t)) in (b) by ∂ϕt(w(t); u(t)). For its solvability we have the following. Proposition 4.1. For each w ∈ E(T ), there exists a unique solution of (CP;w). Proof. Note from the Lipschitz continuity of b (cf. (B)) that

(b(z1) − b(z2),z1 − z2) = |b(z1)(x) − b(z2)(x)||z1(x) − z2(x)|dx Ω Z −1 2 ≥ Cb |b(z1) − b(z2)|H , ∀z1,z2 ∈ H. By this inequality and Lemma 4.1, we can apply Kenmochi-Pawlow [19, Theorem 1.1] to show the existence of a solution. The uniqueness can be proved in the same way as Theorem 2.2, or by applying [19, Theorem 1.2]. By virtue of Proposition 4.1, we can define a mapping Q : E(T ) → E(T ) by Qw := b(u), where u is the solution of (CP;w). Now we show that the mapping Q has a fixed point in the set E(T ), if we choose the time T > 0 to be sufficiently small. To this end, let us consider the following approximate problem (CP;w)ε,λ (0 < ε, λ ≤ 1) of (CP;w): b (u )′(t)+ ∂ϕt (w(t); u (t)) = f(t), 0

t 1 2 t ϕλ(w(t); z) := inf |z − y|H + ϕ (w(t); y) , z ∈ H. y∈H 2λ   We refer to [8] for the basic property of Yosida-approximations of proper, l.s.c. con- vex functions. We notice here that the following inequality holds as a consequence of (3.2) (cf. [19, Lemma 2.2]): t ′ 2 ϕλ(w; z) ≥ C5|z|H +1, (4.4) ′ where C5 > 0 is independent of t,w,z and 0 < λ ≤ 1. Employing the arguments in [19], we can show that the problem (CP;w)ε,λ admits an unique solution uε,λ which converges to the solution of (CP;w) in an appropriate sense. Moreover, with a slight modification of [19, Lemma 2.3], we can conclude that the function [0,T ] ∋ t → Φε,λ(t) defined by t Φε,λ(t) := ϕλ(w(t); uε,λ(t)) is a function of bounded variation and satisfies t ′ ′ Φε,λ(t) − Φε,λ(s)+ (bε(uε,λ) (τ) − f(τ),uε,λ(τ))dτ Zs ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 347

t ′ ′ 1/2 ≤ C6 |α (τ)||bε(uε,λ) (τ) − f(τ)|H {1+(Φε,λ(τ)) } s Z h ′ +|α (τ)|(1+Φε,λ(τ)) ′ ′ 1/2 +(|w (τ)|H + |α (τ)||w(τ)|V )(Φε,λ(τ)) dτ (4.5) for 0 ≤ s ≤ t ≤ T . i Here we notice the following relations: ′ ′ ′ ′ ′ (bε(uε,λ) ,bε(uε,λ) ) = (bε(uε,λ) ,bε(uε,λ)uε,λ) ′ ′ ′ ≤ (|b |∞ + ε)(bε(uε,λ) ,uε,λ), (4.6) t t ′ ′ (f,uε,λ)dτ = − (f ,uε,λ)dτ + (f(t),uε,λ(t)) − (f(s),uε,λ(s)) s s Z t Z ′ 2 ≤ {|f |H + C11Φε,λ}dτ + (f(t),uε,λ(t)) − (f(s),uε,λ(s)), (4.7) Zs ′ ′ 1/2 |α ||bε(uε,λ) − f|H {1+(Φε,λ) } ′ 2 −1 ′ 2 ≤ δ|bε(uε,λ) − f|H + δ |α | (1+Φε,λ) ′ 2 2 −1 ′ 2 ≤ 2δ|bε(uε,λ) |H +2δ|f|H + δ |α | (1+Φε,λ), (4.8) 1/2 (Φε,λ) ≤ (1+Φε,λ), (4.9) ′ where the constant C11 in (4.7) is related to the constant C5 in (4.4) and the constant δ > 0 in (4.8) will be specified below. Using (4.6)–(4.9) in (4.5), we obtain t ′ 2 Fε,λ(t) − Fε,λ(s)+ C12 |bε(uε,λ) (τ)|H dτ Zs t 1/2 ≤ C13 G(τ)(1 + Φε,λ(τ)) + W (τ)(Φε,λ(τ)) dτ (4.10) Zs for 0 ≤ s ≤ t ≤ T and 0

Fε,λ(t):=Φε,λ(t) − (f(t),uε,λ(t)), 2 ′ 2 ′ 2 G(t) := |f(t)|H + |f (t)|H + |α (t)| +1, ′ ′ W (t) := |w (t)|H + |α (t)||w(t)|V , ′ −1 C12 := (|b |∞ + 1) − 2δC6, where the constant δ > 0 (cf. (4.8)) is chosen so that C12 > 0, and the constant ′ C13 > 0 is determined only by C6, C11 and |b |∞. Here we note that by (4.4)

Fε,λ =Φε,λ − (f,uε,λ) 2 −1 2 ≥ Φε,λ − δ1|uε,λ|H − δ1 |f|H −1 2 ≥ C14Φε,λ − δ1 |f|H , (4.11) for some constant C14 > 0 by choosing δ1 > 0 to be sufficiently small. By (4.10) and (4.11), we obtain t ′ 2 Fε,λ(t) − Fε,λ(s)+ C12 |bε(uε,λ) (τ)|H dτ Zs t ≤ C15 {G(τ)+ W (τ)} (1 + Fε,λ(τ))dτ (4.12) Zs 348 M. KUBO AND N. YAMAZAKI for 0 ≤ s ≤ t ≤ T and 0 < ε,λ ≤ 1, where C15 > 0 depends on C14,δ1 and |f|L∞(0,T ;H). Applying a Gronwall-type inequality (e.g., [17, Proposition 0.4.1]) to (4.12), we obtain with the aid of (4.11) T ′ 2 sup Φε,λ(t)+ |bε(uε,λ) (t)|H dt 0≤t≤T Z0

−1 −1 −1 2 C15(|G|L1(0,T )+|W |L1(0,T )) ≤ (C14 + C12 ) δ1 |f|L∞(0,T ;H) + e h ×{Φε,λ(0) + |f(0)|H |u0|H + C15(|G|L1(0,T ) + |W |L1(0,T ))} . (4.13) Here, by taking account of (4.4), let λ → 0 in (4.13). Then, we have T t ′ 2 sup ϕ (w(t); uε(t)) + |bε(uε) (t)|H dt 0≤t≤T Z0

C15(|G|L1(0,T )+|W |L1(0,T )) ≤ C16e 1+ C15(|G|L1(0,T ) + |W |L1(0,T )) , (4.14) where C16 depends on the given data. By the assumption (B) of b, we note that |b(z)|V ≤ C17 (|z|V + 1) for all z ∈ V, (4.15) where C17 > 0 depends only on b and Ω. From (K2), (3.2), (4.14) and (4.15), it follows that T 2 2 ′ 2 sup |uε(t)|V + sup |bε(uε)(t)|V + |bε(uε) (t)|H dt 0≤t≤T 0≤t≤T Z0

C15(|G|L1(0,T )+|W |L1(0,T )) ≤ C18e 1+ C15(|G|L1(0,T ) + |W |L1(0,T )) , (4.16) where C18 depends on C5, C16 and C17. Next, passing to the limit in (4.16) as ε → 0, we get T 2 2 ′ 2 ess sup |u(t)|V + sup |b(u)(t)|V + |b(u) (t)|H dt 0≤t≤T 0≤t≤T Z0

C15(|G|L1(0,T )+|W |L1(0,T )) ≤ C18e 1+ C15(|G|L1(0,T ) + |W |L1(0,T )) . (4.17) The above arguments are quite standard, so we omit the detailed proof. For in- stance, we refer to [19, Sections 3,4]. Now put for M > 0

E(T,M) := {w ∈ E(T ); |w|E ≤ M}, where T 2 2 ′ 2 |w|E := sup |w(t)|V + |w (t)|H dt 0≤t≤T Z0 and we show that the mapping Q maps the set E(T,M) into itself for appropriately chosen T > 0 and M > 0. Note first that for w ∈ E(T,M) we have T T ′ ′ |W |L1(0,T ) = |w (t)|H dt + |α (t)||w(t)|V dt Z0 Z0 1/2 1/2 ′ ≤ T M + T |α |L2(0,T )M. Take M > 0 so large that

2C15 2 C18e × (1+2C15) ≤ M ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 349 and then choose T > 0 so small that 1/2 1/2 ′ |G|L1(0,T ) ≤ 1,T M + T |α |L2(0,T )M ≤ 1. Then, the estimate (4.17) implies that Qw(= b(u)) belongs to the set E(T,M) for w ∈ E(T,M). Thus the mapping Q maps the set E(T,M) into itself for T,M chosen as above. It is easy to see that the set E(T,M) is convex and compact in C([0,T ]; H). Also, the continuity of Q : E(T,M) → E(T,M) with respect to the topology of C([0,T ]; H) can be proved by a standard technique of evolution equations generated by subdifferentials (cf. [19, Section 6]). In fact, let {wn} ⊂ E(T,M) and w ∈ E(T,M). Then, by the similar calculation in (4.1) we observe that t t |ϕ (wn(t); z) − ϕ (w(t); z)|

= A(x, wn(t), ∇z)dx − A(x, w(t), ∇z)dx ZΩ ZΩ 1/2 ≤ C (1 + |b| )|w − w| (|Ω|1/2 + |∇z | ) 1 ∞ n C([0,T ];H) H −→ 0 if wn → w in C([0,T ]; H) as n → +∞ for any t ∈ [0,T ] and z ∈ K(t). Therefore, we can apply the abstract convergence theory of evolution equations generated by subdifferentials (cf. [19, Section 6]). Thus, we get the continuity of Q : E(T,M) → E(T,M) with respect to the topology of C([0,T ]; H). Now, we apply Schauder’s fixed point theorem to the mapping Q. Then, we conclude the existence of a fixed point of Q, and hence of a local solution of (CP). 4.2. Global solution. We start with the inequality (4.10). Applying the Schwarz 1/2 inequality to the term W (τ)(Φε,λ(τ)) and using (4.11), we obtain t ′ 2 Fε,λ(t) − Fε,λ(s)+ C12 |bε(uε,λ) (τ)|H dτ Zs t t ′ 2 2 ≤ C19 G(τ)(1 + Fε,λ(τ))dτ + γ (|w (τ)|H + |w(τ)|V )dτ (4.18) Zs Zs for 0 ≤ s ≤ t ≤ T and 0 < ε,λ ≤ 1, where γ > 0 is a small constant specified later (cf. (4.19) and (4.20) below) and C19 > 0 depends on C13, C14,δ1, |f|L∞(0,T ;H) and γ. Applying a Gronwall-type inequality (e.g., [17, Proposition 0.4.1]) to (4.18), we obtain t t C19 τ G(s)ds ′ 2 Fε,λ(t)+ e C12|bε(uε,λ) (τ)|H dτ Z0 R T T C19 G(s)ds ≤ e 0 Fε,λ(0) + C19 G(s)ds R ( Z0 ) t t C19 τ G(s)ds ′ 2 2 +γ e (|w (τ)|H + |w(τ)|V )dτ. Z0 R Hence t t C19 τ G(s)ds ′ 2 ′ 2 2 Fε,λ(t)+ e C12|bε(uε,λ) (τ)|H − γ(|w (τ)|H + |w(τ)|V ) dτ Z0 R  ≤ C20, (4.19) where the constant C20 > 0 depends only on the given data. 350 M. KUBO AND N. YAMAZAKI

We let ε, λ → 0 in (4.19), noting (K2), (3.2), (4.4), (4.11) and (4.15). Then, we obtain (cf. (4.17)) 2 2 |u|L∞(0,t;V ) + |b(u)|L∞(0,t;V ) t t C19 τ G(s)ds ′ 2 ′ 2 2 + e {C12|b(u) (τ)|H − γ(|w (τ)|H + |w(τ)|V )}dτ ≤ C21 (4.20) Z0 R for 0 ≤ t ≤ T , where the constant C21 > 0 depends only on the given data. Notice that the inequality (4.20) holds for any w ∈ E(T ). By the local existence result in Section 4.1, we can take w = b(u) ∈ E(T0),u as being the solution of (CP) on a small time interval [0,T0] with 0

T0 2 2 ′ 2 u ∞ b u ∞ b u t dt C , | |L (0,T0;V ) + | ( )|L (0,T0;V ) + | ( ) ( )|H ≤ 23 (4.22) Z0 where C23 depends only on the given data and is independent of T0 ≤ T . Now we shall prove the existence of a global solution by employing the estimate (4.22). Put first ∗ T := sup{T0; (CP) has a solution on [0,T0]}. Then, from the local existence result in Section 4.1, we have T ∗ > 0. By the definition of T ∗ there exists a function u : [0,T ∗) → V such that u is a solution of ∗ (CP) on [0,T0] for any T0

∗ Therefore, we can conclude that the limit b := limt↑T ∗ b(u)(t) exists strongly in H and weakly in V . Moreover, by virtue of the following Lemma 4.2, we have that b∗ = b(u∗) for some u∗ ∈ K(T ∗). Hence, by using the local existence result and taking b∗ as the initial value at t = T ∗, we can prolong the solution u beyond the time interval [0,T ∗]. Therefore, the solution must exist for the whole time interval [0,T ]. Thus the proof of Theorem 2.1 will be completed when we show the following lemma. ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 351

∗ Lemma 4.2. Assume that un ∈ K(tn) for tn ↑ T and that {un} is bounded in ∗ ∗ V . Then, there exists u ∈ K(T ) such that for some subsequence {unk } we have ∗ unk → u weakly in V and strongly in H. ∗ ∗ Proof. By (K1), (A) and Lemma 3.1, there exists un ∈ K(T ) such that ∗ ∗ |un − un|H ≤ |α(T ) − α(tn)|(1 + |un|V ) ∗ and {un} is bounded in V . ∗ Therefore we have for some u ∈ V and for some subsequence {unk } ∗ ∗ unk → u weakly in V and strongly in H and ∗ unk → u weakly in V and strongly in H. Since K(T ∗) is closed and convex in V , it is weakly closed in V , and hence we have u∗ ∈ K(T ∗). Remark 4.1. By carefully reconstructing the argument in Sections 3 and 4, we could reformulate our result for an abstract Cauchy problem like (CP) in Definition 3.1 in an abstract Hilbert space H. However, we would need very many assumptions, if we formulate the problem in such an abstract from. Thus, in this paper, we have chosen to formulate the problem in L2(Ω) with given functions a and b, and to let only the convex sets K(t) be unspecified.

5. Applications. The main results stated in the Section 2 (Theorems 2.1 and 2.2) can be applied to problems with various constraints by choosing the convex set K(t) in an appropriate manner for each problem. In this section, we exemplify the application of Theorems 2.1 and 2.2 by using some problems in flows in partially sat- urated porous media. What we have to do is to define K(t) satisfying assumptions (K1)–(K4). 5.1. Mixed boundary condition. Let us consider the initial boundary value problem with a Signorini-Dirichlet-Neumann type mixed boundary condition. This kind of problem arises in the model of flows in partially saturated porous media. In fact, let us assume that the boundary Γ := ∂Ω of the domain Ω is smooth and admits a mutually disjoint decomposition such as

Γ=ΓS ∪ ΓD ∪ ΓN , where ΓS, ΓD and ΓN are measurable of Γ, and ΓD has positive surface measure. In the model of partially saturated porous media, ΓS, ΓD and ΓN refer to the parts of the boundary in contact with the atmosphere, reservoirs and impervious layer, respectively. In a special case where the Signorini part is empty (ΓS = ∅), the problem is the one studied by Alt-Luckhaus [1] and Otto [28]. Hence, in this case, our solution coincides with theirs by the uniqueness result of [28], and our result deduces the regularity theorem of [1, Theorem 2.3]. The problem is given as follows. Problem 1 :

b(u)t − ∇ · a(x, b(u), ∇u)= f(t, x) in (0,T ) × Ω, u ≤ p(t), ν · a(x, b(u), ∇u) ≤ 0

and (u − p(t))ν · a(x, b(u), ∇u)=0 onΓS, 352 M. KUBO AND N. YAMAZAKI

u = p(t) onΓD,

ν · a(x, b(u), ∇u)=0 onΓN ,

b(u(0, ·)) = b0 in Ω, where ν is the outward normal vector on the boundary, and the function p is given so that the following condition is satisfied: p ∈ W 1,2(0,T ; V ).

The function p refers to the pressure in the reservoirs on ΓD, and to the atmospheric pressure on ΓS. For each t ∈ [0,T ], define a convex set K1(t) by

K1(t) := {z ∈ V ; z ≤ p(t) on ΓS and z = p(t) on ΓD}.

Then, one can see that the problem (P;b0,f,K1(t)) is the weak (variational) formu- lation of the above Problem 1 (cf. [2], [18], [20], [28]). We shall verify assumptions (K1)–(K4). In order to show (K1), we put z˜ := z − p(s)+ p(t).

Then, we havez ˜ ∈ K1(t) if z ∈ K1(s). Moreover we have t ′ |z˜ − z|H = |p(t) − p(s)|H ≤ |p (τ)|H dτ. Zs And by (A) we have for w ∈ V with |w(x)| ≤ |b|∞ a.e. in Ω

A(x, w, ∇z˜)dx − A(x, w, ∇z)dx ZΩ ZΩ 1 d = A(x, w, ∇(z + ξ(p(t) − p(s))))dξdx dξ ZΩ Z0 1 = a(x, w, ∇(z + ξ(p(t) − p(s)))) · ∇(p(t) − p(s))dξdx ZΩ Z0 1/2 1/2 1/2 ≤ C1(a) (|Ω| + |b|∞|Ω| + |∇z|H +2|p|L∞(0,T ;V ))|p(t) − p(s)|V . From these inequalities, we see that (K1) holds with t 1/2 1/2 1/2 ′ α(t)=(1+ C1(a) )(1 + |Ω| + |b|∞|Ω| +2|p|L∞(0,T ;V )) |p (τ)|V dτ. Z0 Condition (K3) is easy to verify. Conditions (K2) and (K4) are verified by noting that ΓD has positive surface measure and by using the Poincar´einequality. Therefore, we can apply Theorem 2.1 to the problem (P;b0,f,K1(t)). Next, define another convex set by

K1(t) := {z ∈ V ; z ≤ p(t) on ΓS and z = p(t) on ΓD}, where we have another decomposition of the boundary

Γ= ΓS ∪ ΓD ∪ ΓN , and p ∈ W 1,2(0,T ; V ). Assume that

ΓD ⊂ ΓD, ΓN = ΓN and p ≤ p on (0,T ) × Γ \ ΓN . ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 353

∗ Then, we can easily see that K1(t) ≤ K1(t) and that the condition (2.2) in Theorem 2.2 is satisfied. Hence comparison of solutions to problems (P;b0,f,K1(t)) and (P;b0, f, K1(t)) holds.

5.2. Time-dependent water levels of reservoirs. We study here the problem proposed by Alt-Luckhaus-Visintin [2] and Otto [29], and show that our Theorems 2.1 and 2.2 can be applied to it. That is, we consider the time-dependent boundary decomposition (cf. the condition (a) below), which refers to the time-dependent change of water levels of reservoirs. Our assumption on the regularity of data and the obtained solutions are stronger than those in [2, 29]. The solution in our sense is a weak solution of [2, 29] (cf. [29, p.541 Remark]), and hence they coincide with each other by the uniqueness result of [29]. Therefore, our result asserts a regularity property of the weak solution of [2, 29]. We assume that the function b is nondecreasing, bounded and Lipschitz continu- ous, while in [2, 29] it is assumed only to be nondecreasing and continuous. Further, we assume more regularity for the water level change (cf. (b)). One can see that our regularity of b and the water level change are physically natural assumptions. Next, the assumptions regarding the boundary data p in [2, 29], which are also weaker than ours (cf. (c)), correspond to those of the so-called ’Dam Problem’ (cf. Carrillo [9]). The ’Dam Problem’ is a free boundary problem with a weak formulation that is problematic like ours, with the function b replaced by the Heaviside function. As shown in Alt-Luckhaus-Visintin [2], the elliptic-parabolic problem can be regarded as its regularization. From this view-point, our stronger assumptions regarding the boundary data are also justified as a regularization of the limit problem and cause no drawback in physical applications. Suppose now that the water levels of reservoirs change over time and the equation is of the form (1.2), then the problem is given as follows. Problem 2 :

b(u)t − ∇ · a(x)[∇u + k(b(u))] = f(t, x) in (0,T ) × Ω,

u ≤ p(t), ν · a(x)[∇u + k(b(u))] ≤ 0

and (u − p(t))ν · a(x)[∇u + k(b(u))]=0 onΓS(t),

u = p(t) onΓD(t),

ν · a(x)[∇u + k(b(u))]=0 onΓN ,

b(u(0, ·)) = b0 in Ω, where the time-dependence of the water levels is expressed by that of the boundary decomposition. We assume that the following condition (A)’ is satisfied. 1 (A)’ a(x) = (ai,j (x)) is a symmetric and positive definite matrix with ai,j ∈ C (Ω), and k : R → RN is Lipschitz continuous. There exists a constant µ> 0 such that a(x)[p − pˆ] · (p − pˆ) ≥ µ|p − pˆ|2 for all x ∈ Ω and p, pˆ ∈ RN . 354 M. KUBO AND N. YAMAZAKI

Define 1 A(x,s,p) := a(x)[ p + k(s)] · p 2 for (x,s,p) ∈ Ω × R × RN . Then, we have

∂pA(x,s,p)= a(x)[p + k(s)]. Note that (A)’ implies (A) for a(x,s,p)= a(x)[p+k(s)]. Without loss of generality, we may assume that k is bounded, since only its value on the range of b appears in the problem. Next, we assume according to [20] that the following (a), (b) and (c) are satisfied. (a) For each t ∈ [0,T ], the boundary Γ of the domain Ω admits a mutually disjoint decomposition such as

Γ=ΓS(t) ∪ ΓD(t) ∪ ΓN ,

where ΓS(t), ΓD(t)andΓN are measurable subsets of Γ, and ΓD(t) has positive surface measure. (b) There is a family Θ = (θ1, ..., θN ) = {Θ(t); t ∈ [0,T ]} of C1-diffeomorphisms i i on Ω such that Θ(0, ·)= IdΩ, Θ(t, Γj(0)) = Γj (t)(j = S,D), and ∂tθ , ∂xj θ , i ∂t∂xj θ ∈ C([0,T ] × Ω)(i, j =1, ..., N). (c) p ∈ W 1,2(0,T ; V ) ∩ L∞(0,T ; H2(Ω)). −1 −1 We write Θt(x) =Θ(t, x) and Θt,s =Θs ◦ Θt , where Θt is the inverse of Θt. 1 N 1 Then, Θt,s = (θt,s, ..., θt,s) is a C -diffeomorphism on Ω such that −1 Θt,s(Γj (t))=Γj (s), j = S,D and Θt,s =Θs,t . (5.1)

Furthermore, by elementary calculations we can show that there is a constant R0 > 0 such that |∂tΘt,s(x)|≤ R0 (x ∈ Ω), (5.2a) i i |∂xj θt,s(x) − δj |≤ R0|t − s| (i, j =1, · · · ,N,x ∈ Ω), (5.2b)

||JΘt,s |− 1|≤ R0|t − s| (x ∈ Ω) (5.2c) for all s,t ∈ [0,T ], where |JΘt,s | is the Jacobian of Θt,s. Now, for each t ∈ [0,T ], define a convex set K2(t) by

K2(t) := {z ∈ V ; z ≤ p(t) on ΓS(t) and z = p(t) on ΓD(t)}.

Then, the problem (P;b0,f,K2(t)) is the weak formulation of Problem 2 (cf. [2], [18], [20], [28]). We verify the assumptions (K1)–(K4). As before, (K2)–(K4) are easy to be verified. We verify (K1) by employing the technique of [20] and [18], where the case a = δi,j , k ≡ 0 and the case a = δi,j are treated, respectively. Here, we give a rather detailed calculation to clarify how to handle the terms a and k. Notice that (cf. [20, Lemma 5.1])

|z ◦ Θt,s − z|H ≤ R1|t − s||∇z|H (z ∈ V, t,s ∈ [0,T ]), (5.3) where R1 > 0 is a constant depending only on the family Θ. In order to verify (K1), we take (cf. [20, Lemma 6.1])

z˜ = z ◦ Θt,s − p(s) ◦ Θt,s + p(t) for 0 ≤ s ≤ t ≤ T and z ∈ K2(s). Then, by (5.1) we seez ˜ ∈ K2(t). We have by (5.3)

|z˜ − z|H ≤ |z ◦ Θt,s − z|H + |p(s) ◦ Θt,s − p(s)|H + |p(t) − p(s)|H ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 355

t ′ ≤ R1|t − s|(|∇z|H + |∇p(s)|H )+ |p (τ)|H dτ Zs ≤ |α1(t) − α1(s)|(1 + |z|V ), (5.4) where

t ′ α1(t)= R1(1 + |p|L∞(0,T ;V ))t + |p (τ)|H dτ. Z0 Next we have for w ∈ V , writing ∂i := ∂xi , k := (k1, ..., kN ) and using summation convention, A(x, w(x), ∇z˜(x))dx − A(x, w(x), ∇z(x))dx ZΩ ZΩ 1 = a (∂ z∂˜ z˜ − ∂ z∂ z)dx + a k (w)(∂ z˜ − ∂ z)dx 2 i,j i j i j i,j i j j ZΩ ZΩ =: I+II. (5.5) The integrals I and II will be estimated in what follows. First we see the term I in (5.5): 1 I= a (∂ (z ◦ Θ )∂ (z ◦ Θ ) − ∂ z∂ z) 2 i,j i t,s j t,s i j Z 1 + a ∂ (z ◦ Θ )∂ (−p(s) ◦ Θ + p(t)) 2 i,j i t,s j t,s Z 1 + a ∂ (−p(s) ◦ Θ + p(t))∂ (z ◦ Θ − p(s) ◦ Θ + p(t)) 2 i,j i t,s j t,s t,s Z =: I1 +I2 +I3.

By using (5.2) and (5.3), the terms I1, I2, I3 are estimated as below. 1 I = a (∂ z) ◦ Θ (∂ θl − δl)(∂ z) ◦ Θ ∂ θm 1 2 i,j l t,s i t,s i m t,s j t,s Z 1 + a (∂ z) ◦ Θ (∂ z) ◦ Θ (∂ θm − δm) 2 i,j i t,s m t,s j t,s j Z 1 1 + a (∂ z) ◦ Θ (∂ z) ◦ Θ − a ∂ z∂ z 2 i,j i t,s j t,s 2 i,j i j Z Z 1 ≤ |a| R |t − s||∂ Θ| |J | |∇z|2 2 ∞ 0 x ∞ Θt,s ∞ H 1 + |a| R |t − s||J | |∇z|2 2 ∞ 0 Θt,s ∞ H 1 + (a ◦ Θ−1 − a )∂ z∂ z 2 i,j t,s i,j i j Z 1 ≤ C(a, Θ)|t − s||∇z|2 + |∂ a| R |t − s||∇z|2 H 2 x ∞ 0 H 2 ≤ C(a, Θ)|t − s||∇z|H . (5.6) 1 I = a ∂ (z ◦ Θ )(−(∂ p(s)) ◦ Θ (∂ θl − δl )) 2 2 i,j i t,s l t,s j t,s j Z 1 + a ∂ (z ◦ Θ )(−(∂ p(s)) ◦ Θ + ∂ p(s)) 2 i,j i t,s j t,s j Z 1 + a ∂ (z ◦ Θ )(−∂ p(s)+ ∂ p(t)) 2 i,j i t,s j j Z ≤ C(a, Θ, |p|L∞(0,T ;V ))R0|t − s||∇z|H 356 M. KUBO AND N. YAMAZAKI

+C(a, Θ)R1|t − s||p(s)|H2(Ω)|∇z|H

+C(a, Θ)|p(t) − p(s)|V |∇z|H

≤ C(a, Θ, |p|L∞(0,T ;H2(Ω)))(|t − s| + |p(t) − p(s)|V )|∇z|H . (5.7) 1 I = a ∂ (−p(s) ◦ Θ + p(s))∂ (z ◦ Θ − p(s) ◦ Θ + p(t)) 3 2 i,j i t,s j t,s t,s Z 1 + a ∂ (−p(s)+ p(t))∂ (z ◦ Θ − p(s) ◦ Θ + p(t)) 2 i,j i j t,s t,s Z 1 = a (−(∂ p(s)) ◦ Θ (∂ θl − δl))∂ (z ◦ Θ − p(s) ◦ Θ + p(t)) 2 i,j l t,s i t,s i j t,s t,s Z 1 + a (−(∂ p(s)) ◦ Θ + ∂ p(s))∂ (z ◦ Θ − p(s) ◦ Θ + p(t)) 2 i,j i t,s i j t,s t,s Z 1 + a ∂ (−p(s)+ p(t))∂ (z ◦ Θ − p(s) ◦ Θ + p(t)) 2 i,j i j t,s t,s Z ≤ C(a, Θ, |p|L∞(0,T ;V ))R0|t − s|(1 + |∇z|H )

+C(a, Θ, |p|L∞(0,T ;V ))R1|t − s||p(s)|H2 (Ω)(1 + |∇z|H )

+C(a, Θ, |p|L∞(0,T ;V ))|p(t) − p(s)|V (1 + |∇z|H )

≤ C(a, Θ, |p|L∞(0,T ;H2(Ω)))(|t − s| + |p(t) − p(s)|V )(1 + |∇z|H ). (5.8) Next, we see the term II in (5.5):

II = ai,j ki(w)(∂j (z ◦ Θt,s) − ∂j z) Z

+ ai,j ki(w)∂j (−p(s) ◦ Θt,s + p(t)) Z l l = ai,j ki(w)(∂lz) ◦ Θt,s(∂j θt,s − δj ) Z

+ ai,j ki(w)((∂j z) ◦ Θt,s − ∂j z) Z

+ ai,j ki(w)∂j (−p(s) ◦ Θt,s + p(t)) Z =: II1 +II2 +II3, where

II1 ≤ |a|∞|k|∞|JΘt,s |∞R0|t − s||∇z|H , (5.9) and −1 −1 II2 = (ai,j ◦ Θt,s ki(w ◦ Θt,s ) − ai,j ki(w))∂j z Z −1 −1 −1 = {(ai,j ◦ Θt,s − ai,j )ki(w ◦ Θt,s )+ ai,j (ki(w ◦ Θt,s ) − ki(w))}∂j z Z 1/2 ≤ |∂xa|∞R0|t − s||k|∞|Ω| |∇z|H ′ −1 +|a|∞|k |∞|w ◦ Θt,s − w|H |∇z|H

≤ C(a,k, Θ, |Ω|)|t − s||∇z|H + C(a, k)R1|t − s||∇w|H |∇z|H

≤ C(a,k, Θ, |Ω|)|t − s|(|∇z|H + |∇w|H |∇z|H ), (5.10) also as in (5.7) we have

II3 ≤ C(a, Θ, |p|L∞(0,T ;H2(Ω)), k, |Ω|)(|t − s| + |p(t) − p(s)|V ). (5.11) ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES 357

Thus, by (5.5)–(5.11) we have

A(x, w(x), ∇z˜(x))dx − A(x, w(x), ∇z(x))dx ZΩ ZΩ ≤ C(a,k, Θ, |p|L∞(0,T ;H2(Ω)), |Ω|) 2 × (|t − s| + |p(t) − p(s)|V )(1 + |∇z|H )+ |t − s||∇w|H |∇z|H . (5.12) By (5.4) and (5.12) we have (K1). Thus, we can apply Theorem 2.1 to the problem (P;b0,f,K2(t)). Next define

K2(t) := {z ∈ V ; z ≤ p(t) on ΓS(t) and z = p(t) on ΓD(t)}, where the boundary decomposition ΓS(t) ∪ ΓD(t) ∪ ΓN of Γ and the function p(t) are assumed to satisfy (a), (b) and (c). Also suppose that for all t ∈ [0,T ]

ΓD(t) ⊂ ΓD(t), ΓN = ΓN and p(t) ≤ p(t) onΓ \ ΓN . ∗ Then, it is not difficult to see that K2(t) ≤ K2(t) and the condition (2.2) of Theorem 2.2 is satisfied. Therefore, the comparison of solutions holds for problems (P;b0,f,K2(t)) and (P;b0, f, K2(t)). 5.3. Constraints on a compact set. Let z ∈ V (= H1(Ω)) and E ⊂ Ω. Following [21, p. 35], we say that z ≥ 0 (resp. z = 0) on E in the sense of V , if there exists 0,1 a sequence zn ∈ C (Ω) such that zn ≥ 0 (resp. zn = 0) on E and zn → z in V . Other relations, e.g., z ≥ g and z = g are naturally defined by z − g ≥ 0 and z − g = 0, respectively. Now let 1,2 1 g ∈ W (0,T ; H0 (Ω)) be given, and define 1 K3(t) := {z ∈ H0 (Ω); z ≥ g(t) on E in the sense of V }, 1 K4(t) := {z ∈ H0 (Ω); z = g(t) on E in the sense of V }, K5(t) := K4(t) ∩{z ∈ V ; z ≥ g(t) a.e. on Ω}. Then, for i =3, 4, 5, consider the problems defined below:

u(t) ∈ Ki(t), 0

(b(u)t,u − v)+ a(x, b(u), ∇u) · ∇(u − v)dx ≤ (f,u − v), v ∈ Ki(t), 0

In the same way as in Section 5.1, we can see that the convex sets Ki(t),i =3, 4, 5, satisfy the conditions (K1)–(K4) (putz ˜ = z − g(s)+ g(t) and so on). Therefore we can apply Theorems 2.1 and 2.2 to the corresponding problems. Next, we assume that the set E changes over time. That is, given a time- dependent set E(t)(⊂ Ω), 0 ≤ t ≤ T , we define 1 K6(t) := {z ∈ H0 (Ω); z ≥ g(t) on E(t) in the sense of V }, 1 K7(t) := {z ∈ H0 (Ω); z = g(t) on E(t) in the sense of V }, K8(t) := K7(t) ∩{z ∈ V ; z ≥ g(t) a.e. on Ω}. Assume that the set E(t) depends on t smoothly in the sense as in the condition (b) in Section 5.2, that the equation is of the form (1.2) with condition (A)’ in Section 5.2, and that 1,2 1 ∞ 2 g ∈ W (0,T ; H0 (Ω)) ∩ L (0,T ; H (Ω)). 358 M. KUBO AND N. YAMAZAKI

Then, we can verify in the same way as in Section 5.2 that the convex sets Ki(t),i = 6, 7, 8, satisfy the conditions (K1)–(K4). Therefore, our Theorems 2.1 and 2.2 can be applied to the problems defined by these convex sets. For parabolic variational inequalities, constraints of these kinds have been studied by Hu-Papageorgiou [14] and Kenmochi [17]. We notice that our Theorems 2.1 and 2.2 are also applicable to other kinds of constraints considered by Kenmochi-Pawlow [20] and by Hu-Papageorgiou [14] and Kenmochi [17], under appropriate additional assumptions like (A)’, if necessary.

Acknowledgements. The authors wish to thank an anonymous referee for review- ing the original manuscript and for many valuable comments that helped clarify and refine this paper.

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