Global Well-posedness Of The Generalized Incompressible Oldroyd-B Type Models

In this dissertation,we study the global well-posedness of the generalized incom-pressible Oldroyd-B type models in the corotational case.First,we consider the following two classes of the regularized Oldroyd-B type models in R2:The first class of the regularized Oldroyd-B type models,which is also called the Leray-α-Oldroyd-B type models,can be shown as（0.0.4）Here（t,x）∈ R+× R2,v≥ 0 is the viscosity coefficient,the parameter β≥ 0 is the reciprocal of relaxation time,κ>0 and a>0 are defined by the dynamic viscosity of fluid,retardation time and the parameter β.The vector v=v（t,x）∈ R2 denotes the velocity of the fluid,the scalar p=p（t,x）∈ R denotes the pressure of the fluid,andτ=τ（t,x）is the non-Newtonian part of the stress tensor,which can be seen as a 2 × 2 symmetric matrix.Du=1/2（▽u+（▽u）τ）is called the deformation tensor and is the symmetric part of the velocity gradient.W（u）=1、2（▽-。（▽u）τ）is the vorticity tensor of the fluid,which is the symmetric part of the velocity gradient.u=（u1,u2）is the"filtered" velocity,α>0 is the length scale parameter that represents the width of the filters.v0（x）and τ0（x）are the given initial data satisfying ▽·u0=0.By replacing u·▽v in（0.0.4）into v·▽u,we obtain another Leray-α-Oldroyd-B model,which is recorded as（0.0.4）*.Next,we introduce the second class of the regularized Oldroyd-B type models as（0.0.5）Similarly,by replacing u·▽v in（0.0.5）into v·▽u.we have another regularized Oldroyd-B model,which is denoted by（0.0.5）*.When α=0,the systems（0.0.4）-（0.0.4）*and（0.0.5）-（0.0.5）*reduce to the classical Oldroy-B type models in the corotational case.Thanks to the classical Oldroy-B type models only contain the velocity Laplacian dissipation,it remains unknown whether or not smooth solution of the classical Oldroy-B type models can develop finite-time singularities,even in the two-dimensional corotational case.In the third chapter,we study this difficulty by regularizing the velocity field,that is,we are interested in the global regularity of the above four regularized Oldroyd-B type models.By employing the standard energy methods,together with two logarithmic Sobolev inequalities,we obtain that,for α>0 and（v0,τ0）∈Hs（R2）（s>2）,the systems（0.0.4）-（0.0.4）*and（0.0.5）-（0.0.5）*have a unique global regular solution such that for any given T>0,v∈L∞（[0,T];H2（R2））∩L2（[0,T];H2+1（R2））,τ∈L∞（[0,T];Hs（R2））.Besides,we consider the following n-D generalized incompressible Oldroyd-B type models with fractional Laplacian dissipation:（0.0.6）where η1,η2 the the nonnegative diffusion indices,the bilinear term Q（▽v,τ）=W（v）τ-τW（v）+b（Dvτ+τDv）,the constant b ∈[-1,1].If η1=1,η2=0,b=0,we call thesystem（0.0.6）as the classical Oldroyd-B type models in the corotational case.Zygmund operator（-?）^ηis defined through the Fourier transform:（-?）^ηf（ξ）= |ξ|^2η （?）（ξ）.In the fourth chapter,we study the global well-posedness problem of the system（0.0.6）.By using useful tools such as the Littlewood-Paley decomposition,the Bony decomposition,a special kind of commutator estimates and Bernstein’s inequality,combining with energy methods,we obtain that when ν > 0,μ > 0,b = 0,for T > 0 and（v₀,τ₀）∈ H^s（Rⁿ）（s ≥ 1 +n/2）,if η₁≥1/2+n/4,η₂> 0,η₁+η₂≥ 1 +n/2,then there exists a unique global strong solution（v,τ）∈L∞（[0,T];H^s（Rⁿ））.