Uniform regularity of the weak solution to higher-order Navier-Stokes-Cahn-Hilliard systems

Abstract Higher-order Navier-Stokes-Cahn-Hilliard system is a noted interface system that describes the evolution of two immiscible incompressible fluids. Regularity of the solutions to this system is a primary issue from a mathematical point of view. The purpose of this paper is to improve the known results regarding the regularity with respect to time of weak solutions to two-dimensional higher-order Navier-Stokes-Cahn-Hilliard system. Namely, ( H 1 , H k ) -regularity of the first and second derivatives of the weak solutions to two-dimensional higher-order Navier-Stokes-Cahn-Hilliard system is obtained with the methods of energy estimates.