随机非牛顿流解的适定性及其动力系统的研究
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摘要
非牛顿流与牛顿流的最大区别就在于应力张量与变形率张量已经不可以通过Newton线性本构关系表达出来.非牛顿流体普遍存在于自然界和现实生活之中,在化学工业、石油工业、生物力学、冰川学、地质学及血液流变学等领域都有着广泛的应用,因此对非牛顿流体的研究具有很重要的现实意义.本论文主要考察Bellout,Bloom和Necas等人提出的非线性、等温、不可压、双极粘性流模型,研究了该类非牛顿流发展方程在确定(deterministic)和随机(stochastic)外力驱动下解的适定性和长时间行为.
第一章是绪论.着重介绍本文研究的非牛顿流的物理背景、已有结果以及最新进展,并概述了随机微分方程的研究现状.
第二章,主要研究了在L2,H1范数意义下,当粘性系数μ0,1→0时,非牛顿流的解与其极限Navier-Stokes方程的解的收敛性,并分别估计了收敛速度.证明的关键是进行一些范数估计,主要分为两大类：一类是不依赖于粘性系数μ0,μ1的一致估计,另外一类是虽然依赖于粘性系数μ0,μ1,但是当粘性系数μ0,μ1→0时,这些估计仍然是有界的.以上估计都依赖于p,因此依据p的取值对流体剪薄和剪厚两种情况都进行了分析.
第三章,证明了分数次Boussinesq Approximation弱解的存在唯一性和衰减.首先考虑了周期边值问题,通过交换子估计对近似解获得了一些重要的先验估计,同时主要利用Galerkin方法获得了周期边值下弱解的存在唯一性.其次,以上获得的先验估计是不依赖于周期区域Ω的,因而令│Ω│→∞就得到了相应初值问题解的整体存在唯一性.最后,通过Fourier splitting方法讨论了解的衰减.由于分数次扩散算子Λ2α(Λ=(-△)1/2,0<α<1)的出现使得解的衰减估计依赖于α的取值,因此分0<α<1/2,α=1/2,1/2<α<1三种情况分别进行了讨论.
第四章,研究了可加噪音驱动下的随机非牛顿流方程,并且获得了随机吸引子的存在性.首先通过Ornstein-Uhlenbeck过程将随机偏微分方程转化为带随机参数的方程.其次,运用Crauel,Debussche和Flandoli等人提出的方法证明随机动力系统在零时刻有紧的吸收集,则可判定随机吸引子的存在性.最后,通过证明不同能量空间的吸引子的等价性,得到了随机吸引子的正则性.这一,结论暗含了两层意思：一个是随机吸引子不依赖于数学研究所选定的能量空间；另一个是流体的渐近影响最终使得解比初值变得更正则.
第五章,讨论了具有乘积噪声的随机非牛顿流方程,运用Flandoli和Gatarek提出的方法证得鞅解的存在性(或者说随机方程弱解的存在性),这一方法也是确定性偏微分方程紧性方法的随机一般化.首先,证明了在非线性项具有线性增长和连续性条件下,有限维逼近问题鞅解的存在性.其次,用Ito公式得到了近似有限维问题解的范数带有期望的先验估计,然后证明了分布是紧的.第三,近似有限维问题取极限后,则由鞅表示定理可得方程鞅解的存在性.最后指出：如果弱解的唯一性可以得到,则同时可以获得不变测度的存在性.
When differentiates the Newtonian fluids and non-Newtonian fluids, the key is whether the constitutive relation between stress tensor and the velocity gradi-ent can be characterized linearly. Non-Newtonian fluids are commonly found in the natural world and the real lives, there is a wide range of applications in chem-ical industry, petroleum industry, biomechanics, glaciology, geology, hemorheol-ogy, and so on. Therefore, the research of non-Newtonian fluids has a very im-portant practical significance. This dissertation mainly discusses the isothermal nonlinear incompressible bipolar viscosity model proposed by Bellout, Bloom and Necas, and the well-posedness and long-time behavior of solution to non-Newtonian fluids evolutionary equation driven by deterministic and stochastic force is considered.
The dissertation consists of five chapters.
In Chapter 1, we pay attention to introduce the physical background, some already known results and recently development of non-Newtonian fluids, and present some research status of stochastic differential equations.
In Chapter 2, we mainly study the convergence of solution for non-Newtonian fluid to the solution for Navier-Stokes equation under the L2, H1-norm as the viscositiesμ0,μ1→0, and estimate the convergence rate. The key of proof is some norm estimates, some are uniform estimates independent onμ0,μ1,the others may be dependent onμ0,μ1, but whenμ0,μ1→0, these estimates are still bounded. At the same time, because these estimates lie on p, we analysis shear thinning and shear thickening case according to the value of p.
In Chapter 3, we prove the existence, uniqueness, and decay of weak so-lution to fractal Boussinesq Approximation. First, we are concerned with the periodic boundary problem, and obtain some important a prior estimates by the commutator estimate, which are independent on domainΩ, the existence and uniqueness of the weak solution is obtained by Galerkin method. Then, we let│Ω│→∞, the existence and uniqueness of the weak solution for Cauchy prob- lem is easily established. Finally, we make use of the Fourier splitting method to prove the decay of weak solution. Owing to the appearance of the diffusion operator, the decay estimate of solution lies onα. Thus, we discuss it in three cases of 0<α<1/2,α=1/2,1/2<α<1 respectively.
In Chapter 4, the existence of random attractors for stochastic non-Newtonian fluid with additive white noise is proved. Firstly, translating the stochastic differ-ential equation to a one with random coefficients by Ornstein-Uhlenbeck process. Secondly, applying the method proposed by Crauel, Debussche and Flandoli to prove the compact attracting set at time zero, then we can conclude the existence of random attractor. In the end, we verify regularity of the random attractors by showing the equivalence of attractor lying in different energy space, which implies the smoothing effect of the fluids in the sense that solution becomes eventually more regular than the initial data.
In Chapter 5, we discuss the stochastic non-Newtonian fluid driven by mul-tiplicative noise, and adopt a method developed by Flandoli and Gatarek to con-struct martingale solutions (namely, the weak solution in the stochastic sense), this method is a stochastic generalization of compactness methods for deter-ministic partial differential equations. First, existence of martingale solution to approximating finite-dimensional problem whose nonlinear terms satisfy linear growth and continuity condition is obtained. Second, the family of laws is tight, because we obtain some norm estimates with expectation of solution to approx-imating finite-dimensional problem by Ito formula. Third, taking the limit of approximating finite-dimensional problem, the martingale solution of equations can be obtained with the aid of representation theorem for martingale. If the uniqueness of weak solution can be obtained, then the existence of invariant measure is also proved.
第一章是绪论.着重介绍本文研究的非牛顿流的物理背景、已有结果以及最新进展,并概述了随机微分方程的研究现状.
第二章,主要研究了在L2,H1范数意义下,当粘性系数μ0,1→0时,非牛顿流的解与其极限Navier-Stokes方程的解的收敛性,并分别估计了收敛速度.证明的关键是进行一些范数估计,主要分为两大类：一类是不依赖于粘性系数μ0,μ1的一致估计,另外一类是虽然依赖于粘性系数μ0,μ1,但是当粘性系数μ0,μ1→0时,这些估计仍然是有界的.以上估计都依赖于p,因此依据p的取值对流体剪薄和剪厚两种情况都进行了分析.
第三章,证明了分数次Boussinesq Approximation弱解的存在唯一性和衰减.首先考虑了周期边值问题,通过交换子估计对近似解获得了一些重要的先验估计,同时主要利用Galerkin方法获得了周期边值下弱解的存在唯一性.其次,以上获得的先验估计是不依赖于周期区域Ω的,因而令│Ω│→∞就得到了相应初值问题解的整体存在唯一性.最后,通过Fourier splitting方法讨论了解的衰减.由于分数次扩散算子Λ2α(Λ=(-△)1/2,0<α<1)的出现使得解的衰减估计依赖于α的取值,因此分0<α<1/2,α=1/2,1/2<α<1三种情况分别进行了讨论.
第四章,研究了可加噪音驱动下的随机非牛顿流方程,并且获得了随机吸引子的存在性.首先通过Ornstein-Uhlenbeck过程将随机偏微分方程转化为带随机参数的方程.其次,运用Crauel,Debussche和Flandoli等人提出的方法证明随机动力系统在零时刻有紧的吸收集,则可判定随机吸引子的存在性.最后,通过证明不同能量空间的吸引子的等价性,得到了随机吸引子的正则性.这一,结论暗含了两层意思：一个是随机吸引子不依赖于数学研究所选定的能量空间；另一个是流体的渐近影响最终使得解比初值变得更正则.
第五章,讨论了具有乘积噪声的随机非牛顿流方程,运用Flandoli和Gatarek提出的方法证得鞅解的存在性(或者说随机方程弱解的存在性),这一方法也是确定性偏微分方程紧性方法的随机一般化.首先,证明了在非线性项具有线性增长和连续性条件下,有限维逼近问题鞅解的存在性.其次,用Ito公式得到了近似有限维问题解的范数带有期望的先验估计,然后证明了分布是紧的.第三,近似有限维问题取极限后,则由鞅表示定理可得方程鞅解的存在性.最后指出：如果弱解的唯一性可以得到,则同时可以获得不变测度的存在性.
When differentiates the Newtonian fluids and non-Newtonian fluids, the key is whether the constitutive relation between stress tensor and the velocity gradi-ent can be characterized linearly. Non-Newtonian fluids are commonly found in the natural world and the real lives, there is a wide range of applications in chem-ical industry, petroleum industry, biomechanics, glaciology, geology, hemorheol-ogy, and so on. Therefore, the research of non-Newtonian fluids has a very im-portant practical significance. This dissertation mainly discusses the isothermal nonlinear incompressible bipolar viscosity model proposed by Bellout, Bloom and Necas, and the well-posedness and long-time behavior of solution to non-Newtonian fluids evolutionary equation driven by deterministic and stochastic force is considered.
The dissertation consists of five chapters.
In Chapter 1, we pay attention to introduce the physical background, some already known results and recently development of non-Newtonian fluids, and present some research status of stochastic differential equations.
In Chapter 2, we mainly study the convergence of solution for non-Newtonian fluid to the solution for Navier-Stokes equation under the L2, H1-norm as the viscositiesμ0,μ1→0, and estimate the convergence rate. The key of proof is some norm estimates, some are uniform estimates independent onμ0,μ1,the others may be dependent onμ0,μ1, but whenμ0,μ1→0, these estimates are still bounded. At the same time, because these estimates lie on p, we analysis shear thinning and shear thickening case according to the value of p.
In Chapter 3, we prove the existence, uniqueness, and decay of weak so-lution to fractal Boussinesq Approximation. First, we are concerned with the periodic boundary problem, and obtain some important a prior estimates by the commutator estimate, which are independent on domainΩ, the existence and uniqueness of the weak solution is obtained by Galerkin method. Then, we let│Ω│→∞, the existence and uniqueness of the weak solution for Cauchy prob- lem is easily established. Finally, we make use of the Fourier splitting method to prove the decay of weak solution. Owing to the appearance of the diffusion operator, the decay estimate of solution lies onα. Thus, we discuss it in three cases of 0<α<1/2,α=1/2,1/2<α<1 respectively.
In Chapter 4, the existence of random attractors for stochastic non-Newtonian fluid with additive white noise is proved. Firstly, translating the stochastic differ-ential equation to a one with random coefficients by Ornstein-Uhlenbeck process. Secondly, applying the method proposed by Crauel, Debussche and Flandoli to prove the compact attracting set at time zero, then we can conclude the existence of random attractor. In the end, we verify regularity of the random attractors by showing the equivalence of attractor lying in different energy space, which implies the smoothing effect of the fluids in the sense that solution becomes eventually more regular than the initial data.
In Chapter 5, we discuss the stochastic non-Newtonian fluid driven by mul-tiplicative noise, and adopt a method developed by Flandoli and Gatarek to con-struct martingale solutions (namely, the weak solution in the stochastic sense), this method is a stochastic generalization of compactness methods for deter-ministic partial differential equations. First, existence of martingale solution to approximating finite-dimensional problem whose nonlinear terms satisfy linear growth and continuity condition is obtained. Second, the family of laws is tight, because we obtain some norm estimates with expectation of solution to approx-imating finite-dimensional problem by Ito formula. Third, taking the limit of approximating finite-dimensional problem, the martingale solution of equations can be obtained with the aid of representation theorem for martingale. If the uniqueness of weak solution can be obtained, then the existence of invariant measure is also proved.
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