Affiliations: Department of Mathematics, Florida State University, Tallahassee, FL, USA. E-mail: wxm@math.fsu.edu | School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, China. E-mail: haowufd@yahoo.com
Note: [] Corresponding author: Hao Wu, School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China. Tel.: +86 21 65642216; E-mail: haowufd@yahoo.com.
Abstract: We study the Hele–Shaw–Cahn–Hilliard system that models two phase incompressible Darcian flow in porous media with matched density but arbitrary viscosity contrast. In the 3D case, we prove eventual regularity of weak solutions, as well as existence of global classical solutions if either the Péclet number is sufficiently small or the initial datum is close to one local energy minimizer of the free energy. In both 2D and 3D, we demonstrate that the ω-limit set of each trajectory consists of a single steady state. Finally, stability of local minimizers is established.
Keywords: Hele–Shaw–Cahn–Hilliard system, eventual regularity, convergence to equilibrium, stability
