Study Of The Properties Of The Solutions To Some Class Of Nonl Inear Problem Arising From ER Fluids And MHD Equations

In this paper,we mainly discuss the content of two parts,the third chapter and the fourth chapter make up the first part.in this part,which is concerned with nonlinear problems arising from Electrorheological fluids,we study the partial regularity of weak solutions and discussed the Hausdorff dimension of the singular set in time of weak solutions to non-stationary Navier-Stokes equation on the three dimensional.The other part is the fifth chapter,including the compressible viscous magnetohydrodynamic equations,we study the optimal time decay estimate of the solution.In Chapter 3,we study the Holder continuity of weak solution u to an equation arising in the Stationary motion of Electrorheological fluids system.To this end,we first obtain higher integrability of Du in our case by establishing a reverse H?lder inequality.Next,based on the result of locally higher integrability of Du and difference quotient argument,we derive a Nikolskii type inequality;then in view of the fractional Sobolev embedding theorem and a bootstrap argument we obtain the continuity of weak solution u.In Chapter 4,we investigate the Hausdorff dimension of the singular set in time of weak solutions to non-stationary Navier-Stokes equation on the three dimensional under some growth conditions.According to the definition of singular set,we obtain dimension Hausdorff measure of the time singular set is zero.In Chapter 5,in this paper we consider the compressible viscous magnetohydrodynamic equations with Coulomb force.It is assumed that the equation satisfies the Cauchy initial boundary value condition.By spectral analysis and energy method,we obtain the optimal time decay estimate of the solution.We show that the global classical solution converges to its equilibrium state at the same decay rate as the solution of the linearized equations.