DISCRETE AND CONTINUOUS Website: www.aimSciences.org DYNAMICAL SYSTEMS Supplement 2009 pp. 240–249 TIME-DEPENDENT OBSTACLE PROBLEM IN THERMOHYDRAULICS Takeshi Fukao † Department of Mathematics Kyoto University of Education Fujinomori 1, Fukakusa Fushimi-ku Kyoto 612-8522, Japan Masahiro Kubo ‡ Department of Mathematics Nagoya Institute of Technology Gokiso-cho, Showa-ku Nagoya 466-8555, Japan Abstract. Obstacle problems, mathematical models of some nonlinear phe- nomena accompanying a free boundary, have been well studied. In this pa- per, the existence and uniqueness of a system between the obstacle problem and the Navier-Stokes equations is considered. The abstract theory for evo- lution equations governed by a subdifferential of the indicator functional on a time-dependent, closed, and convex set is applied to show the main theorem. L ∞ -estimate is an important lemma to prove the existence theorem. 1. Introduction. Let 0 <T< +∞,Ω ⊂ R 2 be a bounded domain with a smooth boundary: Γ := ∂ Ω. We consider the following time-dependent single obstacle problem (P):= {(1)–(7)}, for a prescribed obstacle function ψ := ψ(t,x). θ ≥ ψ in Q := (0,T ) × Ω, (1) ∂θ ∂t + v ·∇θ − Δθ = f in Q(θ) := {(t,x) ∈ Q; θ>ψ}, (2) ∂θ ∂t + v ·∇θ − Δθ ≥ f in Q 1 (θ) := {(t,x) ∈ Q; θ = ψ}, (3) ∂ v ∂t +(v ·∇)v − Δv = g(θ) −∇p in Q, (4) divv =0 in Q, (5) θ = h, v =0 on Σ := (0,T ) × Γ, (6) θ(0) = θ 0 , v(0) = v 0 in Ω, (7) where θ = θ(t,x) is the temperature, v := (v 1 (t,x),v 2 (t,x)) is the velocity, and p := p(t,x) is the pressure; f : Q → R, g := (g 1 ,g 2 ): R → R 2 ,h :Σ → R,θ 0 :Ω → R, v 0 :Ω → R 2 are given functions. From a physical viewpoint, g(θ) represents the Boussinesq approximation of the buoyancy force in Navier-Stokes equations. The solvability of the Boussinesq system between the linear heat equation and the Navier-Stokes equations has been treated in many papers. For example, Morimoto 2000 Mathematics Subject Classification. Primary: 35K65, 76D05; Secondary: 35G30. Key words and phrases. Variational inequality, Obstacle problem, Navier-Stokes equations. † Supported by a Grant-in-Aid for Encouragement of Young Scientists (B) (No.18740095), JSPS. ‡ Supported by a Grant-in-Aid for Scientific Research (C) (No.17540166), JSPS. 240