Local Existence Of Local Energy Weak Solutions For Viscoelastic Navier-Stokes Equations

In this paper,we mainly study the local existence of local energy weak solutions of viscoelastic Navier-Stokes equations in three-dimensional space.First,we give the definition of local energy weak solutions,In order to prove the local existence of the local energy weak solutions of viscoelastic Navier-Stokes equations,we polish and localize the viscoelastic Navier-Stokes equations and obtain modified equations.By using Banach fixed point theorem,we prove the existence of mild solutions of the modified equations and the local energy norm of this mild solutions is uniformly bounded with respect to the parameter.By the uniform boundedness of the local energy norm,we can prove that the generation time of the sequence of mild solutions has a positive lower bound.By means of compactness argument and diagonal rule,we obtain the local existence of the local energy weak solutions of the original equations.In addition,we select the appropriate test functions for the modified equations to obtain the local energy equation7 By using the prior estimates of the sequence of approximate solutions,we get the energy inequality satisfied by the local energy weak solutions of the original equations.Finally,we study the spatial decay estimates of the local energy weak solutions at infinity,and find that the local energy weak solutions of the viscoelastic Navier-Stokes equations also decay at infinity when the initial value decay.This paper is mainly divided into four chaptersIn Chapter One,we introduce the research background and the physical meaning of viscoelastic Navier-Stokes equations.Next,we give the research status of viscoelastic Navier-Stokes equations and the meaning of the local energy weak solutions of viscoelastic Navier-Stokes equations,and finally give the main results of this paperIn Chapter Two,we give some symbols,inequalities and theorems,the definition of local energy space and so on.We review the basic properties of the Green function corresponding to the heat operator et? and Stokes operator et?P,and give the Luloc p-Luloc q estimates of heat kernel and Oseen kernelIn Chapter Three,through the Luloc p-Luloc q estimates of heat kernel and Oseen kernel and Banach fixed point theorem to prove the existence of mild solutions of the modified equations,and we prove that the energy norm of the mild solutions is uniformly bounded with respect to parameter.By means of uniformly bounded and compact argument,the local existence of local energy weak solutions of viscoelastic Navier-Stokes equations is proved.In Chapter Four,we study the spatial decay estimates of local energy weak solutions at infinity.Assuming that the initial value has no decay and has decay,we consider the decay property of the local energy weak solutions of viscoelastic Navier-Stokes equations.