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-?-Oldroyd-B type models,can be shown as?0.0.4?Here?t,x?? R+× R2,v? 0 is the viscosity coefficient,the parameter ?? 0 is the reciprocal of relaxation time,?>0 and a>0 are defined by the dynamic viscosity of fluid,retardation time and the parameter ?.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,and?=??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???is called the deformation tensor and is the symmetric part of the velocity gradient.W?u?=1?2??-???u???is the vorticity tensor of the fluid,which is the symmetric part of the velocity gradient.u=?u1,u2?is the"filtered" velocity,?>0 is the length scale parameter that represents the width of the filters.v0?x?and ?0?x?are the given initial data satisfying ?·u0=0.By replacing u·?v in?0.0.4?into v·?u,we obtain another Leray-?-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 ?=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 ?>0 and?v0,?0??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??,??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 ?1,?2 the the nonnegative diffusion indices,the bilinear term Q??v,??=W?v??-?W?v?+b?Dv?+?Dv?,the constant b ?[-1,1].If ?1=1,?2=0,b=0,we call thesystem?0.0.6?as the classical Oldroyd-B type models in the corotational case.Zygmund operator?-??^?is defined through the Fourier transform:?-??^?f???= |?|^2? ??????.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 ? > 0,? > 0,b = 0,for T > 0 and?v₀,?₀?? H^s?Rⁿ??s ? 1 +n/2?,if ?₁?1/2+n/4,?₂> 0,?₁+?₂? 1 +n/2,then there exists a unique global strong solution?v,???L??[0,T];H^s?Rⁿ??.