Research On The Existence Of Weak Solutions For A Class Of Anisotropic Non-Newtonian Boussinesq Equations

Non-Newtonian fluids are widely used in industry,nature and our daily life.Both domestic and international researchers are increasingly focused on the study and use of nonNewtonian fluids.We can increase our understanding of non-Newtonian fluids by studying the well-posed solution.In general,the incompressible non-Newtonian Boussinesq equations can be used to describe the thermal convection of high viscosity fluids.This system of higher order nonlinear partial differential equations is a complex model formed by the coupling of fluid velocity field and temperature field.It is not only an important tool in the fields of astrophysics,geophysics and atmospheric science,but also the basis for the study of atmospheric and oceanic hydrodynamics and mathematical hydrodynamics.It provides us with a more accurate simulation method.This topic will discuss this problem in depth in order to better understand some properties of weak solutions of Boussinesq equation system and lay a foundation for future research.This paper mainly studies two kinds of anisotropic non-Newtonian Boussinesq equations and it is divided into four chapters.In the first chapter,we provide an overview of the background of this research and the current state of non-Newtonian flow equation research both domestically and abroad.In the second chapter,we primarily introduce a few markings and conclusions employed in this paper’s proofing process.In the third chapter,we consider the following initial boundary value problem of anisotropic non-Newtonian Boussinesq equations with perturbations in three-dimensional space:Where Ω ? R3 is a bounded region with sufficiently smooth boundary,QT=Ω×[0,T],FT=?Ω×[0,T].The unknown vector function u=(u1,u2,u3)represents the fluid velocity,θ=θ(x,t)represents temperature.π represents pressure,f=(f1,f2,f3),g=g(x)are given external force terms,Diu=(?iu1,?iu2,?1u3).The exponent of qi is a known constant,satisfying 1<qi<∞,(i=1,2,3).Firstly,provide the corresponding function space,use Galerkin method to construct approximate solutions of equation systems,estimate the approximate solution,take the limit of the approximate solution.Finally,use the monotonicity method and the compactness theorem to prove the existence of weak solution.In the last chapter,we consider the following initial boundary value problems of anisotropic non-Newtonian Boussinesq equations with perturbations in three-dimensional space:Where the exponent of σi,qi is a given constant,satisfying 1<qi<∞,1<σi<∞(i=1,2,3).｛e1,…,e3} is the standard base in R3.κi is a non-negative real constant.we thoroughly examine the impact of the perturbation term on the uniform estimation,clarify the link between the indicators,and obtain priori estimation results.Then we use the compactness method and monotonicity method to prove the corresponding convergence results.