On The Energy Conservation Of The Weak Solutions For The Cahn-Hilliard-Navier-Stokes Equations

Navier-Stokes equations describe the dynamic properties of viscous fluid from three aspects:mass conservation,energy conservation and momentum conservation;Cahn-Hilliard equations describe the mixed diffusion properties of two incompatible fluids.There will be a certain micro thin interface lay-er between the two macroscopic incompatible fluids,such as water and oil.The Cahn-Hilliard-Navier-Stokes equations describe the diffusion and convec-tion process of the mixed fluids with hydrodynamic properties.The equations have important application in practical problems,and theoretical research has certain difficulty and challenge,so it has attracted more and more attention recently.Because of the strong nonlinearity and coupling,the regularity theory of the weak solution is a very challenging problem.Therefore,the regularity of the weak solutions,the energy conservation of the weak solutions and other problems are always concerned by mathematicians and physicists.Inspired by the related research of Navier-Stokes equations,this paper considers the incom-pressible and compressible Cahn-Hilliard-Naiver-Stokes equations respectively.The added regularity conditions are based on Berselli and Chiodaroli^[6].The main methods used in this paper are smoothing approximation and commu-tator to get that the energy equation is valid in the sense of distribution.Finally,we can prove that the energy equation is almost always valid by using the Lions-Aubin lemma.