DISCRETE AND CONTINUOUS Website: www.aimSciences.org DYNAMICAL SYSTEMS Supplement 2011 pp. 437–446 VARIATIONAL INEQUALITY FOR THE STOKES EQUATIONS WITH CONSTRAINT Takeshi Fukao Department of Mathematics, Kyoto University of Education 1 Fujinomori, Fukakusa, Fushimi-ku Kyoto 612-8522, Japan Abstract. In this paper, the existence problem of the variational inequal- ity for the constrained Stokes equations is considered, in 2- and 3-dimensions with bounded domain. The evolution equation is governed by the subdifferen- tial which formulates the pointwise constraint by the time-dependent obstacle functions. Thanks to the penalty method due to Temam, the hindrance of divergence freeness is avoided. We then characterize the Yosida approximation of the subdifferential and obtain suitable limit conditions. 1. Introduction. The solid-liquid phase transition is an interesting problem in material science. From the view point of the partial differential equations, it is a free boundary problem. For example, we consider the system of equations for the thermodynamics and the fluid dynamics. The domain Ω is occupied by the solid and liquid material, for example, the ice and water. The solid-liquid phase changes at the critical temperature. It is natural to consider the equation for the fluid dynamics in the liquid region. But the liquid region is defined by the unknown, for example, the temperature, because the phase changes at the critical temperature. So the problem of interpreting the liquid region, or more precisely the regularity of the unknown function, is very important. If we use the classical Stefan problem for the thermodynamics, then we get the smoothness of the liquid region as the solution. But if we use the enthalpy formulation of the Stefan problem, the issue of defining the liquid region and obtaining its smoothness becomes very delicate. This issue in using the Stefan problem has been considered, for example by Rodrigues [21] in many of his papers. Now consider the variational formulation of the Navier-Stokes equations in a material region Ω with any test function whose support is included in the unknown liquid region Ω ℓ ⊂ Ω. To consider the variational formulation on the exact liquid region Ω ℓ , we need to get at least continuity of the unknown function, defining the liquid region Ω ℓ . But if the coupled equation for the thermodynamics is nonlinear, then it may be difficult to obtain the continuity of the unknown function. The condition for the dimension and the regularity problem is found in many papers in the references of [21] . For the phase field equations, see the paper of Planas and Boldrini [16]. They use the system of modified Navier-Stokes equations where the penalty for the Navier-Stokes has a physical meaning called the Carman-Kozeny penalty. 2000 Mathematics Subject Classification. Primary: 35K86, 35Q35; Secondary: 76D07. Key words and phrases. Stokes equations, obstacle problem, pointwise constraint. Supported by a Grant-in-Aid for Encouragement of Young Scientists (B) (No.21740130), JSPS. 437