Existence And Asymptotic Property Of Statistical Solutions Of Several 3D Incompressible Hydrodynamic Equations

Some important physical quantities of turbulence,such as velocity and pressure,exhibit chaotic and sharp behaviors with changes in space and time,while we can observe some laws by taking the overall mean quantities with respect to time or space.Statistical solutions and invariant measures are strict mathematical concepts that effectively describe motion characteristics of turbulence with the aid of probability distribution functions.Statistical solutions are of great significance for the study of turbulence.In this thesis,we focus on several 3D incompressible hydrodynamic equations closely related to the classical 3D Navier-Stokes equations,and study the existence and asymptotic property of statistical solutions of each equations respectively.For the 3D incompressible B(?)nard equations,we prove an existence theorem of a Vishik-Fursikov measure by Galerkin projection and Krein-Milman theorem.We introduce the regularized 3D B(?)nard-α equations and establish the asymptotic relationship between their weak solutions and those of the B(?)nard equations in the given topology.Finally,by the Topsoe’s compactness theorem,we prove that a Vishik-Fursikov measure of the B(?)nard equations is approximated by a sequence of α-Vishik-Fursikov measures of the B(?)nard-α equations.For the 3D incompressible micropolar B(?)nard equations with large Prandtl number,we define an eventual time-average measure.We examine the existence of an eventual time-average measure by means of Kakutani-Riesz representation theorem.By defining an eventual invariant measure and by applying the properties of generalized Banach limits as well as continuous dynamical systems,we prove that eventual time-averaged measures are stationary statistical solutions.Finally,with the aid of the Prokhorov’s compactness theorem,we verify that when the micropolar parameter K approaches zero,a sequence of eventual time-average measures of the micropolar B(?)nard equations approximates a stationary statistical solution of the limit model with K= 0.For the 3D MHD equations,through the Topsoe’s compactness theorem we prove that a strong stationary statistical solution can be obtained by taking the limit in a sequence of stationary statistical solutions of the 3D MHD-Voigt equations.Meanwhile,we show that the sequence of global attractors,which are also supports of invariant measures,of the MHD-Voigt equations is upper semicontinuous with the parameters α and β in the sense of Hausdorff semidistance.Finally,by the Urysohn’s lemma,we prove that any limit measure of a sequence of invariant measures of the MHD-Voigt equations is a strong stationary statistical solution supported on the weak attractor of the MHD equations.