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,μis 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, Λ=: (-Δ)1/2, defined by Fourier transform. Λf(ξ) =|ξ|f)ξ).When α=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 μ=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.