Well-posedness Of Magnetohydrodynamic Equations And Double Diffusion Convection Equation

In this paper,we research well-pocedness of Cauchy problem for the full compressible magnetohydrodynamic equations in R~3 and double-diffusive convection equations in R~2.Where,magnetohydrodynamic describes the physical motions for plasma,liquid metals and electrolytes in a magnetic field.And double-diffusive convection means the convective phenomena cased by varying density which is influenced by some components of different molecular diffusion rates.On the one hand,firstly,we define new modified effective viscous flux and vorticity of the full compressible magnetohydrodynamic equations in R~3,and obtain energy esti-mates of the coupling terms and magnetic field terms;Secondly,we get the global a priori estimates of solution to the 3-D magnetohydrodynamic equations.Meanwhile,we use the div-curl decomposition technique to calculate estimates of L~3 norm for gradient of velocity and magnetic fields;Finally,we obtain global existence and uniqueness for the strong so-lution with the initial energy being suitably small and the initial data belonging to a class of space with lower regularity than H~2(R~3).On the other hand,we use the interpolation inequality to cope with the convection term in velocity field equation for double-diffusive convection equations in R~2.When the initial data dose not have any smallness condition,with the help of contraction mapping principle and energy method,we obtain global unique solution for the 2-D double-diffusive convection equations inH~i(R~2)(i≥1).