The Regularity Of The Solution To Navier-Stokes-Cahn-Hilliard System

Navier-Stokes-Cahn-Hilliard system is a mathematical model describing the evolution of an incompressible isothermal mixture of binary fluids.From the viewpoint of mathematics,it's significant to study the existence and regularity of the solution to this system.In this thesis,the regularity of the solution to Navier-Stokes-Cahn-Hilliard system is investigated and the research contents mainly contain the following two parts.In the first part,the existence of the global solution to Navier-Stokes-Cahn-Hilliard system in a 2D bounded domain is mainly investigated in higher order Sobolev spaces.Utilizing energy methods,Sobolev embedding theorem,embedding theorem of fractional order spaces and the properties of sectorial operator,we obtain the estimates for the nonlinear terms of the system in higher order Sobolev spaces.And then,the regularity of the global solution to Navier-Stokes-Cahn-Hilliard system is obtained by successive iteration procedure.In the second part,the regularity of the time derivatives of the weak solution to higher-order Navier-Stokes-Cahn-Hilliard system in a 2D bounded domain is mainly considered.Based on basic inequalities,we give the key estimates for the nonlinear terms and higher order terms.Finally,the regularity of the time derivatives of the weak solution to the system is obtained through uniform Gronwall lemma.