第一章 绪论  Chapter I. Introduction

心血管疾病具有发病急, 病情发展快, 致残风险高等特点. 有数据显示, 死于心血管疾病的人数正逐年上升, 心血管疾病已成为全球死亡的主要原因. 因此, 人们迫切希望找到研究血液在循环系统中流动的相关模型, 从而起到防治心血管疾病的作用. 为此, 国内外研究人员提出了许多生物数学模型并对其进行了充分的研究, 其中以下模型 引起了众多学者的关注:
Cardiovascular diseases (CVDs) are characterized by rapid onset, rapid progression and high risk of disability. Data show that the number of deaths from cardiovascular diseases is increasing year by year, and cardiovascular diseases have become a major cause of death worldwide. Therefore, there is an urgent need to find a relevant model to study the flow of blood in the circulatory system, so as to play a role in the prevention and treatment of cardiovascular diseases. To this end, many biomathematical models have been proposed and well studied by researchers in China and abroad, among which the following model has attracted the attention of many scholars.

其中, 表示横截面积, 表示流速, 这里 定义为平均轴向速度 穿过半径为 的血管横截面, 即
where denotes the cross-sectional area, denotes the flow velocity, and here is defined as the mean axial velocity across the cross-section of the vessel with radius , i.e.

和 都是正常数, 分别代表流体密度和摩擦系数. 此外, 压力项被表示为:
and are positive constants representing the fluid density and friction coefficient, respectively. In addition, the pressure term is given as

其中常数 表示参考的横截面积, 表示恒定的外部压力, 表示容器壁的刚度, 最后, 表示应力应变的线性/非线性强度.

模型 (1-1) 主要用于描述人体血管系统的复杂生理现象, 自提出以来就引起了研究者的极大兴趣, 并对其展开一系列研究. 下面, 我们主要介绍一些关于该模型解的存在性和稳定性方面的重要研究成果. 在初始扰动足够小的条件下, Čanić 和 Kim 在文献 [3] 中证明了该模型的解整体存在. 和 Čanić 在文献 [18] 中研究了血管流模型的 Cauchy 问题, 并探究了阻尼项 对解的影响. 后来, 和 Zhao 在文献 [20] 中研究了具有压力项 (1-3) 的血管流模型的初边值问题, 证明了当初始值较大时, 弱嫡解在时间上是整体存在的; 此外, 作者还证明了随着时间的推移, 弱摘解将以指数衰减的速率收玫到常状态. 另一方面, Li 和 Zhao 在文献 [19] 中证明了对于给定接近常状态的光滑初值, 该模型初边值问题存在唯一的整体光滑解, 并且光滑解的渐近行为与文献 [20] 相同. 此外, 文献 [8] 考虑了 Windkessel 型边界条件与该模型的耦合, 证明了经典解是局部存在且唯一的. 之后,

Wei, Yao 和 Zhu 在文献 [39] 中证明了该模型 Cauchy 问题的解随时间渐近收敛于稀疏波. 关于血管流模型数值模拟方面的研究结果, 可以参考文献 [1, 2, 4, 6, 9, 10, 27, 28,35-37].然而据我们所知, 目前对该模型初边值问题解的渐近行为的研究还非常有限. 这也正是本文所关注的问题.
Wei, Yao and Zhu proved in [39] that the solution of the Cauchy problem of this model converges asymptotically to sparse waves with time. The results of numerical simulations of the vascular flow model can be found in [1, 2, 4, 6, 9, 10, 27, 28,35-37]. However, to the best of our knowledge, the asymptotic behavior of the solutions to the initial margin problem of this model is still very limited. This is the concern of this paper.

为了后续更方便地研究, 我们不妨令 并使用变量 , 则方程 被改写成如下形式:
For the sake of subsequent study, we may order and use the variable , then the equation is rewritten in the following form.

这里 , 其中 且 . 不失一般性, 我们取 . 通过分析不难发现在拉格朗日坐标系下研究该模型解的渐近行为更方便. 为此,我们引入拉格朗日坐标变换:
Here , where and . Without loss of generality, we take . It is easy to see that it is more convenient to study the asymptotic behavior of the solutions of this model in the Lagrangian coordinate system. For this reason, we introduce the Lagrangian coordinate transformation.

这里 满足以下积分曲线 Here satisfies the following integral curve

为了简化记号, 我们使用变量 代替变量 , 并引入新变量 . 此时 变为如下形式:
To simplify the notation, we use the variable instead of the variable and introduce the new variable . At this point, takes the following form.

其初值条件为 Its initial value condition is

Dirichlet 边界条件为 The Dirichlet boundary conditions are

这里 和 都是常数. 是关于 的光滑函数并且满足 和 .
Here and are constants. are smooth functions with respect to and satisfy and .

在本文中, 我们将考虑模型 (1-5)-(1-7) 的解的整体存在性和渐近行为. 通过分析方程的结构, 我们不难发现该方程与带阻尼的可压缩欧拉方程组有着类似的结构. 因此,接下来我们有必要回顾一些关于带阻尼的可压缩欧拉方程组的相关结果, 并由此给出我
In this paper, we will consider the overall existence and asymptotic behavior of the solutions of models (1-5)-(1-7). By analyzing the structure of the equations, it is easy to find that the equations have a similar structure to the system of compressible Euler equations with damping. Therefore, it is necessary to review some related results on the system of compressible Euler equations with damping, and thus to give me the following results
们的主要工作. 对于带阻尼的可压缩欧拉方程组  Our main work. For the system of compressible Euler equations with damping

其 Cauchy 问题解的渐近行为已经被许多学者广泛研究 (参见 ). 其中, Hsiao 和 Liu 在文献 [13] 中首次证明了该模型的解整体存在且依时间收玫到如下系统的解
The asymptotic behavior of solutions to the Cauchy problem has been extensively studied by many scholars (see ). Among them, Hsiao and Liu in [13] proved for the first time that the solution of this model exists in its entirety and converges in time to the solution of the following system

同时, 作者还得到了解的收玫速率 . 接着, Nishihara 在文献 [30] 中利用更精细的能量估计方法将解的收玫速率提高到 . 后来, Nishihara, Wang 和 Yang 在文献 [33] 中利用引入近似格林函数的方法, 将解的收玫速率改进到 . 对于上述模型 Cauchy 问题的其它相关结果, 可以参见文献 及其参考文献.
At the same time, the authors also obtained the rate of the solution . Then, in [30], Nishihara used a more refined energy estimation method to improve the rate of the solution to . Later, in [33], Nishihara, Wang and Yang improved the solution rate to by introducing an approximate Green's function. Other related results for the above modeled Cauchy problem can be found in and its references.

对于半空间上初边值问题, 已经有许多学者研究了该模型解的渐近行为(参见 . Nishihara 和 Yang 在文献 [34] 中考虑了具有 Dirichlet 边界的情形, 通过初值在线性扩散波 周围做小扰动得到了解的全局存在性，该线性扩散波满足
The asymptotic behavior of the solutions of this model has been studied by many scholars for the problem of initial marginals on a half-space (see . In [34], Nishihara and Yang considered the case with Dirichlet boundary, and obtained the global existence of the solution by making small perturbations of the initial values around a linear diffusion wave , which satisfies

同时, 作者还得到了解的收玫速率 . 后来, Marcati, Mei 和 Rubino 在文献 [25] 中证明该模型的解依时间渐近收玫到非线性扩散波 , 其满足
At the same time, the authors also obtained the closing rate of the solution . Later, in [25], Marcati, Mei and Rubino proved that the solution of this model asymptotically converges in time to a nonlinear diffusion wave , which satisfies the following criteria

并将收玫速率提高到 . 对于上述模型初边值问题的其它相关研究, 我们参见文献 及其参考文献. 本文的研究过程借鉴了上述文献中有关非线性扩散波收玫理论的研究方法.
and increase the rate to . For other related studies on the initial margin problem of the above model, we refer to and its references. In this paper, we draw on the above literature to study the nonlinear diffusion wave harvesting theory.

本文将研究初边值问题 (1-5)-(1-7) 解的整体存在性和渐近行为. 首先, 受文献 [25]
In this paper, we will study the overall existence and asymptotic behavior of solutions to the initial margin problem (1-5)-(1-7). First of all, the literature [25] has shown the existence and asymptotic behavior of
的启发, 我们猜测该模型的解 会随时间渐近收玫到非线性扩散波 .接着, 通过计算我们发现 和 在无穷远处存在差值. 为了消除这种差值, 我们构造了校正函数 . 之后, 构造了扰动函数 并对初边值问题 (1-5)-(1-7) 进行了转化. 然后, 在初始扰动满足某些小性假设条件下, 我们证明了该模型初边值问题的解整体存在且依时间收玫到非线性扩散波. 在证明方法上, 我们首先利用能量估计方法得到解的整体存在性, 接着利用加权能量估计和格林函数相结合的方法得到了解的最优收玫率.
Inspired by the model, we guess that the solution of the model will asymptotically converge to the nonlinear diffusion wave over time. Then, we find that there is a difference between and at infinity. In order to eliminate this difference, we construct a correction function . After that, the perturbation function is constructed and transformed to the initial margin problem (1-5)-(1-7). Then, under certain smallness assumptions on the initial perturbations, we prove that the solution of the modeled initial margin problem exists in its entirety and converges to a nonlinear diffusion wave in time. In the methodology, we first use energy estimation to obtain the existence of the solution as a whole, and then use a combination of weighted energy estimation and Green's function to obtain the optimal rate of collection of the solution.

与之前的研究结果相比, 本文的难点与创新主要体现在以下两个方面:
Compared with the results of previous studies, the difficulties and innovations of this paper are mainly in the following two aspects.

一方面, 文献 [25] 在 充分小的条件下得到解的收玫率, 而本文在 和 充分小以及 的条件下获得主要结果. 显然, 我们的条件更弱. 一方面是因为我们引入了一组新的校正函数 , 从而避免了提 的假设. 另一方面是因为我们运用格林函数和加权能量估计相结合的方法, 从而避免了仅通过分析解的积分表达式来得到收玫率, 进而对初始值要求降低.
On the one hand, the literature [25] obtains the rose rate of the solution under the condition that is sufficiently small, whereas in this paper, we obtain the main result under the condition that and are sufficiently small and . Obviously, our conditions are weaker. This is partly due to the fact that we introduce a new set of correction functions , thus avoiding the assumption of . On the other hand, it is because we use a combination of Green's function and weighted energy estimation, which avoids the need to analyze only the integral expression of the solution to obtain the rosette rate, which in turn requires less initial values.

另一方面, 在能量估计的过程中, 由于方程结构的复杂性和校正函数的特殊性, 我们需要额外处理扰动方程中由 带来的困难项. 例如: (见 ), (见 ), (见 ) 和 见 (3-136)). 事实上前三个坏项没有对应的好项来吸收它们, 但值得注意的是, 这类项的系数具有指数衰减的性质, 可以通过 Gronwall 不等式对其进行估计. 对于第四个坏项, 我们需要对其关于 进行分部积分, 从而避免由 带来的估计困难.
On the other hand, in the process of energy estimation, due to the complexity of the structure of the equations and the specificity of the correction function, we need to additionally deal with the difficult terms in the perturbation equations brought about by . For example: (see ), (see ), (see ) and see (3-136)). In fact, the first three bad terms do not have good counterparts to absorb them, but it is worth noting that the coefficients of such terms are exponentially decaying in nature and can be estimated by Gronwall's inequality. For the fourth bad term, we need to integrate it partially with respect to to avoid the estimation difficulty caused by .