Well-posedness of the Hele–Shaw–Cahn–Hilliard system

  • Xiaoming Wang

    Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, United States
  • Zhifei Zhang

    School of Mathematical Science, Peking University, Beijing 100871, China

Abstract

We study the well-posedness of the Hele–Shaw–Cahn–Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in , , the existence and uniqueness of solution in that is global in time in the two dimensional case () and local in time in the three dimensional case () are established. Several blow-up criterions in the three dimensional case are provided as well. One of the tools that we utilized is the Littlewood–Paley theory in order to establish certain key commutator estimates.

Cite this article

Xiaoming Wang, Zhifei Zhang, Well-posedness of the Hele–Shaw–Cahn–Hilliard system. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 3, pp. 367–384

DOI 10.1016/J.ANIHPC.2012.06.003