On The Energy Equality For Distributional Solutions To Navier-Stokes Equations Of 3-Dimensions

In this paper,we study the distributional solutions to the Cauchy problem of the Navier-Stokes equations in three-dimensions,under certain conditions,the solutions satisfy the energy equality.Namely,if the distributional solution v ver-ifies v ? L?(0,T;L_?~2(R~3))? L~q(0,T;L~p(R~3)),where1/p+1/q=1/2,p?4,then v satisfies the energy equality.The paper consists of two chapters.The first chapter mainly introduces the research history and status of Navier-Stokes equations,and gives the conclusion drawn in this paper as well as the sym-bols and inequalities.In the second chapter,we mainly apply the classical Galerkin method to ex-plore the existence of the weak solution to the dual equation of the original Cauchy problem,in addition,the solution satisfies the energy estimation.If the distribu-tional solution v of the original Cauchy problem meets v ?L?(0,T;L_?~2(R~3))?L~q(0,T;L~p(R~3)),we can deduce v? L?(0,T;L_?~2(R~3))?L~2(0,T;H~1(R~3)),so the energy equality holds.