On The Local Well-posedness And Regularity Of Solutions For The Generalized Boussinesq Equations

In this thesis, we study the local existence and uniqueness of strong solutions for the generalized Boussinesq equations with fractional dissipation and the problem of regularity criteria for the 2D generalized Boussinesq equations. The thesis contains three chapters.In chapter one, we briefly introduce the background and significance, meanwhile, we introduce the related definitions, notations, known results as well as some lemmas which will play an important role in following parts.In chapter two, we consider the the local existence and uniqueness of strong solu-tions for the generalized Boussinesq equations in Hs(s>max{n/2+1-α,1}).In chapter three, we establish various regularity criteria for the 2D generalized Boussinesq equations to guarantee smoothness of solutions.