The Property Of Smooth Solution To The Incompressible Viscoelastic Flow

In this dissertation, we study the incompressible viscoelastic fluids systems of the 01-droyd model in Rn. where u(t,x) is the velocity field, p is the pressure,?is the viscosity coefficient and F is the deformation tensor. The Oldroyd model (0.0.3) describes an incompressible non-Newtonian fluid, which bears the elastic property. For the details on this model see [4].For the smooth initial data, we know the systems (0.0.3) possesses a unique local smooth solution. Whether the local smooth solution is a global one? Lin, Liu and Zhang [4] obtained a continuation criterion ?0T|?u?H22ds<+? Yuan [25] extended the result in L?. In this thesis, we improve these results to the space BMO, which includes the space L?. Using the energy method and the properties of the space BMO and stokes system, we obtain the blow-up criteria of a smooth solution to the incompressible viscoelastic flow in space BMO.(1) Let u0 ? H2(R2) and F0 ? H2(R2) with ?·u0=0, ?·F.k,0=0 for k=1,2. Assume the pair of solution u?L?([0,T]; H2(R2))?L2([0,T]; H3(R2)), F?L?([0,T];H2(R2)) is a smooth solution to the Oldroyd model (0.0.3). Then (u,F) is smooth up to time T provided that(2) Let u0 ? H2(R3) and F0 ? H2(R3) with ?·u0= 0, ?·F.k,0=0 for k= 1,2,3. Assume the pair of solution u?L([0,T]; H2(R3))? L2([0,T]; H3(R3)), F ? L([0, T]; H2(R3)) is a smooth solution to the Oldroyd model (0.0.3). If T* is the maximal time of existence, then In the fourth chapter, we study the well-posedness and continuation criterion of the generalized incompressible viscoelastic flow. where, ?=: (-?)1/2, defined by Fourier transform. ?f(?) =|?|f)?).When ?=1, the systems (0.0.4) reduces to the standard incompressible viscoelas-tic flow. In this paper, we prove that the generalized incompressible viscoelastic flow possesses a unique local smooth solution in Hs by Friedrich' method.(3) Assume (u0, F0)?Hs with s > max{a, 1+n/2}, Then, there exists a local time T=T(?upHs,?F0?Hs), such that (0.0.4) has a unique local smooth solution on [0, T] with u?L? ([0, T]; Hs(Rn))?L2([0, T]; Ha+s(Rn)), F?L?([0, T]; Hs(Rn)).we improve the regularity step by step, by the logarithmic Sobolev inequality one can deduce some continuation criteria in B?,?0.(4) Let n/2 <a and (u0, F0)?Hs(Rn) for s? 3 and n=2,3. Assume the pail" of solution u?L?([0,T];H2(Rn))?L2([0,T];Ha+2(Rn)), F?L?([O,T];H2(Rn)) is a smooth solution to the generalized Oldroyd model (0.0.4). Then the smooth solution (u,F) can be extended in (0, T*)(T* > T) provided that When ?=0, the system (0.0.4) reduces to the ideal viscoelastic flow, which possesses a unique local smooth solution. By virtue of the energy method and logarithmic Sobolev inequality, we obtain an improved continuation criterion in B?,?0.(5) Let u0, F0?Hs(Rn) for s? 3, n=2,3. Assume the pail" of solution u? L? ([0, T]; H2(Rn)), F?L?([0, T]; H2 (Rn)) is a smooth solution to the ideal viscoelastic flow system. If T* is the maximal time of existence, then.