本博士学位论文研究了 Euler 队列方程 Cauchy 问题的整体适定性和解的渐近行为. 这里所研究的方程是带压力 (等熵或者等温的情形) 和强奇异性队列项的. 此方程是经典的Cucker–Smale 模型的宏观表述, 用于刻画生态学、社会学、物理学等学科中多个体运动的集群行为. 本文主要研究内容包括以下部分:
一、研究了队列项的奇异性指标在区间 (1, 2) 时, Euler 队列方程 Cauchy 问题的整体适定性和渐近性. 为了获得封闭的先验估计, 本文通过计算交叉项 (即密度与速度的内积), 充分开发了方程线性部分的光滑效应. 在估计压力项时, 为了克服密度正则性不足的困难, 引入密度的非线性变换降低了正则性的需求. 为了估计队列项的非线性部分, 利用Littlewood–Paley 理论和分数阶 Laplace 算子的性质, 建立了临界 Besov 空间上的交换子估计. 克服以上困难后, 在临界 Besov 空间上证明了当初值在平衡态 (1, 0) (密度为 1, 速度为0) 附近时, 高维 ( ⩾ 2) 带压力的 Euler 队列方程整体解的存在唯一性. 为了获得加权能量估计, 开发了一些加权型仿积估计. 通过选取合适的时间权函数来建立加权能量估计, 证明了高维方程整体解的时间衰减估计, 且此衰减率与线性化方程低频部分的最优衰减率一致.
二、研究了队列项的奇异性指标在区间 (0, 1] 时, Euler 队列方程 Cauchy 问题的整体适定性和最优衰减估计. 通过对方程的线性部分作谱分析, 本文阐明了解在高频和低频的光滑效应. 为了得到密度的光滑效应, 使用了交叉项估计的方法. 为了处理质量和动量方程中非线性项的导数损失, 引入了密度的非线性变换 (不同于第一部分). 通过建立解的先验估计, 得到了任意维带压力的 Euler 队列方程当初值在平衡态附近时整体解的存在唯一性. 此外, 为了研究线性化方程的最优衰减估计, 分析了其对应 Green 矩阵的一些基本性质. 基于 Green 矩阵的性质, 获得了高维 ( ⩾ 2) 方程整体解的时间衰减上界估计. 再结合方程的结构, 得到了整体解的衰减下界估计. 由于衰减率的上界与下界相同, 因此所得衰减率为最优的.
三、研究了队列项的奇异性指标在区间 (3/2, 2) 时, 一维 Euler 队列方程的解关于稀疏波的渐近稳定性. 在计算过程中, 为了与 Euler 队列方程的解相匹配, 引入了一类稀疏波的光滑逼近波, 并且借助一些调和分析工具改进了光滑稀疏波在更一般的 Sobolev 空间中的估计. 在建立扰动方程的能量估计过程中, 使用了分数阶 Laplace 算子和 Besov 空间的一些性质. 此外, 为了充分开发方程的正则效应, 运用了加权能量估计的方法. 从而当初值在稀疏波附近且稀疏波强度很小时, 证明了 Euler 队列方程解的存在唯一性, 并且当时间趋于无穷时, 对应的解收敛到稀疏波. 由于稀疏波的性质不足以提供更多的正则性, 故只有当队列项的奇异性指标在区间 (3/2, 2) 时, 才可以证明稀疏波是渐近稳定的.

This thesis is devoted to the study of the global well­posedness theory and the asymptotic behavior for the Euler alignment system with pressures (isothermal or isentropic flow) and a strongly singular alignment term. This system is the macroscopic representation of the celebrated Cucker–Smale model, which describes the collective motions for M­agent in ecology, sociology or physics and so on. This thesis is mainly divided into the following parts:
The first part studies the global well­posedness and asymptotic behavior of Cauchy problem for the Euler alignment system when the order of the singular alignment term is in the interval (1, 2). In order to obtain a closed a priori estimates, we calculate the cross terms (i.e., the inner product of density and velocity) to develop the smoothing effect of the linear part of the system. In the process, in order to overcome the lack of control on pressure terms from the loss of regularity for density, this paper introduces a nonlinear transformation, which can reduce the need for regularity of density. Besides that, in order to bound the nonlinear part of the alignment term, using the Littlewood–Paley theory and the properties of fractional Laplacian, we establish a commutator estimate in critical Besov space. After overcoming the above difficulties, we prove the existence and uniqueness of the global solutions for Euler alignment system with pressure and singular alignment when the initial data is close to the equilibrium state (1, 0) (i.e. the density is 1 and velocity is 0) in multidimensional ( ⩾ 2) critical Besov space. In order to obtain weighted energy estimates, this paper develops some weighted paraproduct estimates. By choosing an appropriate time weight function and establishing a weighted energy estimate, we obtain the time decay estimates for the global solutions of this system in multidimensional space and the decay rates agree with the low-frequency part of the linearized system.
The second part is devoted to the research of the global well­posedness and optimal decay estimate of Cauchy problem for Euler alignment system with the order of the singular alignment term in the interval (0, 1]. Based on the spectral analysis of the linear part of the system, we expect the smoothing effects of the solutions in the high and low frequencies. In order to get the smoothing effect of density, we consider the cross term. In addition, in order to deal with the derivative losses of nonlinear terms in the mass and momentum equations, this paper introduces a new nonlinear transformation of density (which is different from the previous part). By establishing a priori estimates of the solutions, we obtain the existence and uniqueness of the global solutions of the Euler alignment system with a pressure term when the initial data close to the equilibrium state in all dimension. In addition, in order to study the optimal decay estimates of linearized system, we analyze some basic properties of the Green matrix. Taking advantage of the properties of the Green matrix, this paper obtains the time decay upper bounds for the global solutions of this system in multidimensional ( ⩾ 2) space. By the structure of the system, we get the lower bounds of the global solutions. The decay rates of the upper bounds are the same as the lower bounds, so the obtained decay rates are optimal.
The last part studies the asymptotic stability of the rarefaction wave for the one­dimensional Euler alignment system with the order of the singular alignment term in the interval (3/2, 2). During the calculation process, in order to match the solution of the Euler alignment system, we introduce a smooth approximate rarefaction wave and develop some promoted estimates for the smooth approximate rarefaction wave in many general Sobolev spaces by some harmonic analysis tools. In the process of building energy estimates for the perturbation system, we use some properties of fractional Laplacian and Besov spaces. In addition, in order to develop the smoothing effect of the system, this paper introduces a weighted energy method. Thus, when the initial data approach to rarefaction wave and the rarefaction wave strength is small enough, we prove the existence and uniqueness of the solutions of the Euler alignment system and the corresponding solutions converge toward the rarefaction wave as time tends to infinity. Because rarefaction waves are not sufficient to provide more regularity, we only prove the asymptotic stability of the rarefaction wave when the order of singular alignment term is in the interval (3/2, 2).