Mathematical Theory Of Viscoelastic Fluid Systems Of The Oldroyd Model

In this thesis, we study the well-posedness viscoelastic ?uids systems of the Ol-droyd model. In the space with dimension no less than two, we prove the well-posedness for the cauchy problem of both of the incompressible and the compressibleviscoelastic ?uids systems: the local solutions are proved to exist under the assumptionthat the initial data is in the Besov space of critical regularity. Moreoever, if the theinitial data are small under certain norms, the solutions exist globally in time. Also, inthe space of two or three dimension, the global solution is proved to exsits in Sobolevspace H2 near the equilibrium. As compared to the elastic Dumbbell model and itsclosure model, the elastic stress here has no damp mechanic. New method are usedhere to overcome this diffculty.The well-posedness results in critical space of the homogenous incompressiblemodel system generalize the existing results that the system having small smooth globalsolution. To prove the local well posedness, the approximating solutions are con-structed with their uniform estmates given. The blow up condition for the local smoothsolution is also used. To prove the global existence result, by using the intrisic prop-erties of deformation tensor, we have transformed the system to find that the resultantsystem having the same structure with the lineanerized compressible Navier-Stokessystem. By applying the estimates for the later system, the global well posedness resultcan be proved.For the Cauchy problem or the initial-boundary value problem of the compress-ible model system, the global well posedness result is given for the first time in thisartical. Still, the initial value problem is studied in framework of the critical space.To prove the local well posedness of such a system, we have used the sharp estimatesfor the linear momentum equation with variable coefficient in weighted Besove space,which is obtained recently by Z. Zhang, and also the Hukuhara fixed point theorem;The main diffculty for proving the global well posedness lies in the lack of dampingmachinic for the density and the deformation tensor, which satisfy the transport eqution.To overcome, inspried by Danchin [15], by applying the Littlewood-Paley decomposa- tion, we have deduced a uniform estimate for a linearied hypobolic-parabolic systemwith convection terms by using the intrinsic properties of the viscoelastic ?uids system.Furtherly, to solve the initial boundary value problem, we have used the introduced thevariable, which is physically meaningful. By applying the techanique of estimatingthe tangential derivative and norm derivative of the unkowns seperately, with the aid ofregular estimates for the Stokes equations, the global existence result is finally proved.Here, the intrinsic properties of the model also playes an important role in the proof.In the nonhomogenous incompressible model studied in the last chapter, the pres-sure satisfies a second order elliptic equation. Under the assumption that the initialdendity is small, we proved the local well-posedness of the model in critical Besovspace. In the global well-posedness result, the dendity of the global small solutionshows no dissipation.