| This thesis will study the well-posedness problems of two types of fluid dynamics equations,both of which are coupled models based on the Navier-Stokes equations:namely the liquid crystal flow model and the fluid-structure interaction model.The liquid crystal flow model represents an intermediate state between solid and liquid within the same space,which not only exhibits fluid-like mobility,but also maintains the solid-specific properties.The fluid-structure interaction model involves fluid motion inside a bounded space while the exterior is governed by the motion of an elastic body,with an interaction between them at the interface.These two types of classical models are among the mainstream research topics in fluid dynamics systems.For the nematic liquid crystal flow model,we focus on studying the trivial solution problem under steady-state conditions,namely the Liouville theorem.For the nonlinear Boussinesq fluid-structure interaction model,we focus on studying the existence and uniqueness of solutions.We will introduce the main content of this dissertation as follows:Chapter 1:Firstly,a brief introduction to the basic background of the NavierStokes equations and the research progress on their trivial solutions is provided.The advancements in the study of nematic liquid crystal fluid models and fluidstructure interaction models are introduced,along with the definitions of function spaces,common inequalities,and theorems required in this article.Chapter 2:We will prove the Liouville theorem for three-dimensional steadystate nematic liquid crystal fluid equations,including the Liouville theorem for the steady incompressible nematic liquid crystal flow and the steady compressible nematic liquid crystal flow.Although the nematic liquid crystal flow model takes Navier-Stokes equation as the basic structure,it is also coupled with a nonlinear supercritical term and a harmonic mapping structure.Studying the harmonic mapping equation itself is a challenge,with many scholars often starting from a geometric perspective to construct solutions.When studying the harmonic mapping equation,it is essential not only to focus on its geometric structure but also to delve into the analysis from a mathematical theoretical perspective,as these comprehensive research methods help to more fully understand the essence and impact of the harmonic mapping equation.Chapter 3:For the nonlinear Boussinesq fluid-structure interaction model,we hope to prove the existence and uniqueness of weak solutions by the method of Galerkin approximation and compact theorem.This model originates from fluidstructure interaction models in biology,where the liquid flows inside the elastic solid region and is entirely controlled by the infinitesimal displacement of the elastic solid.The motion between them mainly involves high-frequency small displacement oscillations,which are considered to be steady at the boundary.This nonlinear Boussinesq fluid-structure interaction model maintains the continuity of velocity,heat transfer,and stress tensor normal vectors on the coupled boundary.Chapter 4:For the nonlinear Boussinesq fluid-structure interaction model,we establish the global existence of strong solutions with large initial data in the two-dimensional case by proposing appropriate compatibility conditions on the boundary and using the method of finite differences.Chapter 5:Summarizing the well-posedness issues of the two types of fluid dynamics systems studied in this thesis and proposing further research questions. |
