Research Of Well-posedness And Analytical Solutions To The Fluid Mechanics Equations With Density-dependent

The research of this dissertation includes two aspects. In the first place, under theassumption that the initial density is bounded away from zero we establish the local well-posedness in some critical Besov spaces for the compressible magneto-hydrodynamic equa-tions with density-dependent viscosities in RN(N≥2) by constructing a sequence of smoothsolutions, and using a compactness argument the convergence of the sequence is proved.The compressible magneto-hydrodynamic equations is:In the second place, we use the separation method to construct a family of analyticalsolutions to the N-dimensional isothermal Euler equations, and a family of analytical solu-tions to the N-dimensional the pressureless isothermal Euler equations with frictional damp-ing. We also prove that the analytical solutions of Euler equations satisfy the correspondingNavier-Stokes equations. In particular, the solutions may blow up in a finite time T.