On Energy Inequality for the Problem on the Evolution of Two Fluids of Different Types Without Surface Tension

Abstract

The paper deals with the motion of two immiscible viscous fluids in a container, one of the fluids being compressible while another one being incompressible. The interface between the fluids is an unknown closed surface where surface tension is neglected. We assume the compressible fluid to be barotropic, the pressure being given by an arbitrary smooth increasing function. This problem is considered in anisotropic Sobolev–Slobodetskiǐ spaces. We show that the L2-norms of the velocity and deviation of compressible fluid density from the mean value decay exponentially with respect to time. The proof is based on a local existence theorem (Denisova, Interfaces Free Bound 2:283–312, 2000) and on the idea of constructing a function of generalized energy, proposed by Padula (J Math Fluid Mech 1:62–77, 1999). In addition, we eliminate the restrictions for the viscosities which appeared in Denisova (Interfaces Free Bound 2:283–312, 2000).