Low Mach Number Limit For The Compressible Micropolar Fluids System

Micropolar fluids are fluids with microstructure characterized by an asymmetric stress tensor,The micropolar fluid models were firstly proposed by A.C.Eringen in 1966.Generally speaking,there are two kind of micropolar fluid models:incompressible micropolar fluid models and compressible micropolar fluid models.The mathematical theory for incompressible micropolar fluids has been developed in a rather satisfactory way so far.However,the compressible micropolar fluids are more complex than the incompressible ones and only a few results are available.The doctoral thesis is dedicated to the low Mach number limit for the 3D compressible micropolar fluids in the whole space or torus and in a bounded domain.Our purpose is to establish the relationship between the compressible micropolar fluids model and the incompressible micropolar fluids model.Firstly,we focus on the global existence and low Mach number limit for compressible micropolar fluids model in a bounded domain of R3.Then,we discuss the relationship between the weak solution of compressible micropolar fluids model and the strong solution of incompressible micropolar fluids model with general data in the whole space.Finally,we study the low Mach number limit for full compressible micropolar fluids model with well-prepared data in the whole space or torus.This paper is divided into four chapters.In Chapter 1,we devote to the thesis research background briefly and give some relevant results.We recall the mathematical theory in the existing literature on the subject of compressible micropolar fluids model.In addition,we introduce the importance of low Mach number limit and give some relevant results.In Chapter 2,we investigate the low Mach number limit for compressible micropolar fluids model in a bounded domain.We find that the compressible micropolar fluids model in a bounded domain with small initial data has a unique strong solution and the solution passes to the limit of incompressible micropolar fluids model.The key point of the global existence is to obtain the uniform estimates of the compressible micropolar fluids system,using integration by parts,Holder inequality and Gronwall-type inequality.We have to deal with the uniform estimates involving higher-order spatial derivatives carefully.Then,the result of low Mach number limit is just an easy application of the uniform estimates and the Arzela-Ascoli theorem.In Chapter 3,we study the low Mach number limit for compressible micropolar fluids model with general data in the whole space.Based on the combination of the classical relative entropy method,Strichartz's estimate of linear wave equations,weak convergence compactness method,and Morrey's inequality,we follow and adapt those methods and ideas to prove that,as the Mach number goes to zero,the weak solutions of the compressible micropolar fluids model with general initial data converge to the strong solution of the ideal incompressible micropolar fluids model.Furthermore,the rate of convergence is also obtained.In Chapter 4,we concern with the low Mach number limit for the full compressible micropolar fluids model with well-prepared initial data.The key point issue is to obtain the uniform estimates of the error system and apply convergence-stability lemma and Gronwall-type inequality.For well-prepared initial data,we prove rigorously that the solutions of the compressible micropolar fluids model converge to that of the incompressible micropolar fluids model as the Mach number tends to zero.