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Elliptic-Parabolic Variational Inequalities with Time-Dependent Constraints

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Elliptic-Parabolic Variational Inequalities with Time-Dependent Constraints 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 Mathematics 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 <t<T, (b(u)t,u − v)+ a(x, b(u), ∇u) · ∇(u − v)dx ≤ (f,u − v), v ∈ K(t), 0 <t<T, ZΩ b(u(0, ·)) = b0 in Ω. 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 set 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 mathematical analysis 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 convex set 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 inequality 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.
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