The Unsteady Thermally Coupled Stokes Problem

Several typical kinds of PDE models called coupled nonlinear equations have been considered in this thesis and the existence , uniqueness and blowup results of the respective elliptic equations are obtained.The model in this thesis is:This thesis is mainly composed of three parts of contents:In the first part, under the condition of the variational formulation, existence of the unsteady thermally coupled Stokes problem which describing the unsteady flow of a quasi-Newtonian fluid with temperature-dependent viscosity and with a viscous heating is proved under Faedo-Galerkin method.In the second part, by Meyers estimate, Schauder fixed point theorem together we study the boundary value problem for the thermally coupled Stokes problem and the coexistence of existence is obtained.The third part studies the estimate of the weak solution which depends on the initial and boundary conditions is established to prove the uniqueness. Results on blowup of the weak solution is also studied.Differential equations are transformed to variational formulation all the above parts. By the existence of the fixed point for the operaor equation, we obtain the existence for the differential equations, in which the Meyers estimate, Faedo-Galerkin method and Schauder fixed point theorem play an important role.