【作者】 宋玉坤；

【导师】 袁洪君；

【作者基本信息】 吉林大学 ， 基础数学， 2014， 博士

【摘要】 流体力学是力学的一门分支,是研究流体现象以及相关力学行为的科学.目前为止,流体力学和其他学科之间的相互浸透和融合,已经形成了许多分支,很多物理学家和数学家都在致力于这方面的研究.经典的牛顿流体力学认为,在平行流动中,剪切力与剪切速率是成正比的,其比例系数称为粘性系数.在此基础上,可以得到著名的Navier-Stokes方程.近年来,人们对非牛顿流体的重要特性有了越来越多的认识.人们发现,日常生活中存在着大量的不服从牛顿常粘度定律的流体,即非牛顿流体.在自然界中非牛顿流体是普遍存在的.对于这类流体,在应力作用下,它将连续改变其运动状态,它的本构关系与牛顿常粘度定律有显著区别.然而,目前关于这方面的结果还很少,因此对非牛顿流的研究是很有必要的.本文我们主要讨论了几类可压缩非牛顿流.在第二章中,我们研究了一维有界区间上的带有位势的可压缩非牛顿流具有下列初边值条件其中ΩT=I×(0,T),I=(0,1),初始密度ρ0≥0,4/3<p,q<2是给定的常数.未知变量ρ,u,P,(?)中分别代表密度、速度、压力和非牛顿重力位势.由于方程具有强耦合性、奇异性,同时我们允许真空的出现,这些都给我们直接证明带来了很大的困难.为了克服此困难我们先对具有奇异性的粘性项进行了正则化处理,证明无真空情形下解的存在性,再通过取极限的过程回到原问题,进而证明了问题强解的存在唯一性.具体定理如下：定理0.0.1.假设(ρ0,u0,f)满足下列条件且满足函数g∈L2(I),使得初始值满足下列相容性条件则存在一个时间T*∈(0,+∞),问题(1)-(2)存在唯一强解(ρ,u,(?)),并具有下列性质:在第三章中,我们研究了带有粒子作用的非牛顿流剪切变稠流具有下列初边值条件其中未知量ρ,u,η,P(ρ)=aργ分别代表流体密度、速度、混合物中粒子的密度和压力,a>0,γ>1,μ0>0,p>2,给定函数(?)(x)代表外部势能(例如重力和浮力).特殊地,如果中是重力势,则中=cx,其中c是重力常数.λ>0是粘性系数,β≠0是一个常数.对上述问题,我们证明了强解的存在唯一性.主要结果叙述如下：定理0.0.2.令μ0>0是一个正常数且(?)∈C2(Ω),假设初值(ρ0,u0,η0)满足下面条件以及相容性条件其中g∈L2((Ω).则存在一个时间T*∈(0,+∞),问题(5)-(6)存在唯一的强解(ρ,u,η),且满足在第四章中,我们研究了带有粒子作用的非牛顿流剪切变稀流具有下列初边值条件其中未知量ρ,u,η,P(ρ)=aργ分别代表流体密度、速度、混合物中粒子的密度和压力,a>0,γ>1,4/3<p<2,给定函数(?)(x)代表外部势能(例如重力和浮力).特殊地,如果中是重力势,则中=cx,其中c是重力常数.λ>0是粘性系数,β≠0是一个常数.对上述问题,由于方程具有强耦合性、奇异性,类似于第二章的证明方法,我们先对具有奇异性的粘性项进行了正则化处理,证明无真空情形下解的存在性,再通极限过程证明了问题强解的存在唯一性.我们得到如下结果：定理0.0.3.令(?)∈C2((Ω),假设初值(ρ0,u0,η0)满足下面条件以及相容性条件其中g∈L2((Ω).则存在一个时间T*∈(0,+∞),使得问题(9)-(10)存在唯一的强解(ρ,u,η),且满足更多还原

【Abstract】 Fluid mechanics is a branch of physics which studies the phenomenon and behaviorof fluids in the associate field. So far, Fluid mechanics has formed many branches betweenother sciences, many physicists and mathematicians have devoted in this field.In classical newtonian mechanics, in parallel flow, shear stress is proportional to theshear rate. The coefcient of proportionality is called viscous coefcient. On this basis,one can get the famous Navier-Stokes equation.In recent years, with the growing awareness of the important characteristics of nonNewtonian fluid, one found that, in daily life, there are a lot of fluids do not obey theconstant viscosity of Newtonian law, namely the non Newtonian fluid. Non-Newtonianfluid is universal in nature. For this type of fluid, under stress, will continue to changeits state of motion, its constitutive relationship with Newton viscosity law have obviousdiferences. However, there are few results about this type of fluids. So it is necessary tostudy the non Newtonian fluids.This thesis is devoted to study several classes of compressible non-Newtonian fluid models. In chapter2, we consider the following system in one-dimensional bounded intervalwith the initial and boundary conditionswhere ΩT=I x (0,T),I=(0,1), the initial density p0≥0,4/3＜p, q<2are given constants.The unknown variables p,u,P,Φ denote the density, velocity, pressure and the non-Newtonian gravitational potential, respectively.Since the second equation (13) has singularity, and the vacuum may appear, We first regularized the viscous term, then by using the iterative method we prove the local existence and uniqueness of strong solutions based on some compatibility condition. We obtain the following theorem:Theorem0.0.1. If (p0,u0, f) satisfies the following conditions and if there is a function g E L2(I), such that the initial data satisfy the following compatibility condition: Then there exist a time T*∈(0,+∞) and a unique strong solution (p, u,Φ) to (13)-(14) such thatIn chapter3,we consider the following system in one-dimensional bounded intervalwith the initial and boundary conditionswhere p,u,δη,P(p)=apγ denotes the fluid density, velocity, the density of particle in the mixture and pressure respectively, a>0,γ>1,μ0>0,p>2, the given function Φ(χ) denotes the external potential (typically incorporating gravity and buoyancy). In particular, if Φ is the gravitational potential, then Φ=cx, where c is the gravitational constant.γ>0is the viscosity coefficient and β≠0is a constant.We obtain the local existence and uniqueness of strong solutions based on some compatibility condition. Our main result is stated in the following theoremTheorem0.0.2. Let μ0>0be a positive constant and Φ∈C2(Ω)，and assume that the initial data(p0,u0,η0) satisfy the following conditions and the compatibility condition ((u0x2+μ0)(p-2)/2u0χ)χ-(p(ρ0)+η0)χ-η0Φχ=ρ0(g+βΦχ),(19) for some g∈L2(Ω).Then there exist a T*∈(0,+∞) and unique strong solution (ρ,u,η)to(17)-(18such thanIn chapter4,we consider the following system in one-dimensional bounded interval with the initial and boundary conditionswhere p,u,η,P(ρ)=aργ denotes the fluid density,velocity,the density of particle in the mixture and pressure respectively,a>0,γ>1,4/3<p<2,the given function Φ(χ)denotes the external potential(typically incorporating gravity and buoyancy).In particular,if Φ is the gravitational potential,then Φ=cχ,where c is the gravitational constant.λ>0is the viscosity coefficient and β≠0is a constant.Similar to the method in chapter2,Firstly we regularized the viscous term,then by using the iterative method, the local existence and uniqueness of strong solutions was proved.we have the following theorem:Theorem0.0.3. LetΦ∈C2(Ω),and assume that the initial data (po,uo,ηo) satisfy the following conditions and the compatibility condition for some g∈L2(Ω). Then there exist a time T*∈(0,+∞) and a unique strong solution (p,u,η) to (21)-(22)such that更多还原