Cauchy Problem For Two Kinds Of Hydrodynamic Equations

In this paper,the Cauchy problem of three-dimensional magnetohydrodynamic equations with damping is studied.The existence of global weak solutions is proved by Galerkin method.When the regularity of initial values increases,the existence of unique local strong solutions of the equations is obtained.Secondly,the Cauchy problem of Navier-Stokes equations with damping in three dimensions is studied.The global existence and uniqueness of strong solutions are obtained on the premise that the initial value has a small H2-norm.The main contents of this paper are as follows:In the first chapter,some methods,the relevant research background and the main conclusions of this paper are given.In the second chapter,the definitions of weak solutions and strong solutions for incompressible MHD equations are given and the related lemmas to prove the existence of weak and strong solutions are given.Finally,the existence of global weak solutions and the existence and uniqueness of local strong solutions are obtained by these lemmas and the classical Galerkin method.In the third chapter,the definition of strong solutions for incompressible N-S equation is given.With the help of Gagliardo-Nirenberg inequality,Holder inequality and Young inequality,the existence and uniqueness of global strong solutions are obtained by the Galerkin method.