Stability Of Steady Navier-Stokes Flows In 3D Exterior Domains And The Derived Operator Semigroup

The dissertation is devoted to study the stability problem of physically reasonable solutions to Navier-Stokes equations with rotating and translating effect in the three dimensional exterior domains.More precisely,we investigate when the class of physi-cally reasonable solutions can be seen as corresponding nonstationary solutions as time tends to infinity.Recently,this stability problem has attracted the attention and research of many famous mathematicians.The Navier-Stokes equation with rotating and trans-lating effect in 3D exterior domains,describe the motion of an incompressible viscous fluid filling the whole space exterior to a moving with rotation and translation obsta-cle.From a physical point of view,the study of this stability problem can shows when the incompressible viscous fluid pasting a moving with rotation and translation obsta-cle gradually tends to be stable over time.From a mathematical point of view,if the rotation is nonzero,the stability problem becomes complicated because there are some interesting feature including hyperbolic aspect caused by the presence of spin.So the stability of the steady-state Navier-Stokes flow with rotation and translation effects in the exterior domains has important research significance.In the study process of stability of the stationary flows,the key tool is the LP-Lq estimates of the operator semigroups generated by different forms of linear operators,such as the LP-Lq estimates of the operator semigroups generated by the Stokes operator with rotating and translating effects in 3D exterior domains.From the perspective of theoretical research,to study Lp-Lq estimates of these semigroups is also an independent research topic.In this dissertation,we focus on the case that translation is present and the direction of translation is perpendicular to the axis of rotation(if the rotation exists).Unlike the usual method,that is to establish the Lp-Lq estimates of the semigroup generated by the Stokes operator with rotating and translating effects in 3D exterior domains and view the perturbation of the steady Navier-Stokes flow as the Duhamel term form this semi-group,we will prove the Stokes operator with rotating and translating effects around this steady flow in 3D exterior domains generates a C0-semigroup in Lp space and then develop a series of LP-Lq estimates of such semigroup.In this study process,we first reduce the LP-Lq estimates of this C0-semigroup to estimates of the solution operators of corresponding resolvent problem in exterior domains by means of the Bogovskii op-erator,the rotation transform in whole space on account of the rotation and the Laplace transform of C0 semigroups.Next,we split the resolvent problem in exterior domain into the resolvent problem in whole space and the resolvent problem in bounded do-main.For the resolvent problem in bounded domain,we deal with it by regarding it as a perturbation of the resolvent problem in bounded domain of the classical Stokes operator;For the resolvent problem in whole space,we view the linear term invoking the steady Navier-Stokes flow as a perturbation of the Stokes operator with rotating and translating effects to formally make the Neumann extension for its resolvent operator,and then investigate further by making use of the Fourier multiplier representation and the pointwise estimates of the kernel function associated with the semigroup generated by the Stokes operator with rotating and translating effects,Fourier multiplier theorem,the domain decomposition,self-similar iteration and other tools.Finally,as an application of LP-Lq estimates of this C0-semigroup as established above,we give out the stability of the Navier-Stokes flow with rotation and translation effects in the exterior domains in the framework of L3 if the initial disturbance in L3 and the steady flow are sufficiently small when the translation is present and the direction of translation is perpendicular to the axis of rotation(if the rotation exists).