EULER-POISSON EQUATIONS WITH ALIGNEMENT AND RELATED PROBLEMS

我的研究将围绕处理偏微分方程(PDE)的非线性分析工具展开。这一领域被认为是不发达的，特别是与上个世纪吸引了大多数数学分析研究人员的线性偏微分方程组理论相比。我将由陈国全教授和J·卡里洛教授指导。非线性偏微分方程组出现在许多重要的应用中，包括弹性、几何、金融和生物问题。这些问题中的许多都需要量身定做的理论来处理，因为数学对象应该符合问题的物理性质。因此，还需要做大量的工作。我将重点介绍的Euler-Poisson方程就是一个例子。它起源于对多体系统的研究。当描述一个由许多粒子组成的系统的动力学时，人们通常使用一个(可能只是部分)耦合的常微分方程组。这些描述了系统在时间上的演化，系统被称为基于个人的模型(IBM)。如果粒子的数量非常多，那么要得到这样的常微分方程组的解往往是不切实际的，甚至是不可能的。因此，人们可以希望，通过使粒子数在适当意义上趋于无穷大而得到的相关连续模型，可以得到一些有用的近似。我们得到的PDE模型描述了与系统相关的宏观因素。关于初值、参数和相互作用函数的严格条件仍然需要确定，以证明全局时间光滑解或有限时间爆破或渐近行为。这个项目属于EPSRC数学分析、数学生物学和连续介质力学的研究领域。这个问题的一个重要应用将是生物系统。例如，我将重点介绍Cucker-Smer模型“F.Cucker和S.Smer，集群中的紧急行为，IEEE Trans.Autom Control 52(2007)852”(另见“S.-Y.Ha和E.Tadmor，从粒子到集群的动力学和流体动力学描述，Kine.Relat.Models 1(2008)415-435”)，该模型由无压欧拉方程组成，并增加了对齐项。它对应于对IBM的连续描述，该描述通过向动力学添加对齐术语来描述生物实体对齐的趋势。对于一般的潜在情况，人们知之甚少。利用非线性分析技术，以及广义函数空间，我希望澄清这些问题。我将要研究的另一个相关问题可以在这里找到：《气态恒星动力学中产生的具有大球对称初值的可压缩欧拉-泊松方程的整体解》，Arxiv，2021。我将对与渐近行为相关的问题特别感兴趣。在有界变差空间(或甚至散度度量场，见“G-Q.Chen和H.FRID，散度度量场和双曲型守恒律，Arch.有理机械分析147(1999)”)中寻求解，我还可以确定爆破条件。此外，我可以研究瓦瑟斯坦空间方法中的梯度流：L.Ambrosio，N.Gigli和G.Savare，梯度流，Birkhäuser Basel，2008年。此外，我对几何学中出现的非线性问题感兴趣，特别是在弹性和广义相对论领域，以及在与最优运输的交叉点上。通过在PDE中添加适当的随机项，我可以为物理、生物和金融中出现的其他问题建立良好的模型。需要进一步的技术来处理这类SPDEs。

My research will be centered around Nonlinear Analysis tools to deal with Partial Differential Equations (PDEs). This field is considered underdeveloped, especially when compared to the theory of Linear PDEs, which attracted most of the researchers in Mathematical Analysis in the past century. I will be supervised by Professors G.-Q. Chen and J. Carrillo.Nonlinear PDEs arise in numerous important applications, including problems in Elasticity, Geometry, Finance, and Biology. Many of these problems require tailored theories to be dealt with, as the mathematical objects should fit the physicality of the problem. Therefore, plenty of work is yet required to be done. The Euler-Poisson equation with alignment I will be focusing on is an example of this. It arises from the study of a many-body system. When describing the dynamics of a system of many particles, which could be biological cells, molecules in a fluid or even galaxies in the universe, one usually uses a system of (maybe only partially) coupled ordinary differential equations. These describe the evolution in time of the system, and the system is referred to as individuals based model (IBM). If the number of particles is very high, then it is often impractical or even impossible to obtain a solution to such system of ODEs. Therefore, one could hope that some useful approximation could be obtained from the associated continuous model obtained by letting the number of particles tend to infinity in an appropriate sense. The PDE model we obtain describes macroscopical associated to the system. Rigorous conditions on the initial values, on the parameters and the interaction functions still need to be determined to prove global in-time smooth solutions or finite time blow-ups or asymptotic behavior. This project falls within the EPSRC Mathematical Analysis, Mathematical Biology and Continuum Mechanics research areas.An important application of this problem would be Biological Systems. For instance, I will be focusing on the Cucker-Smale model "F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control 52 (2007) 852" (see also "S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models 1 (2008) 415-435"), which consists of a pressureless Euler equation with an added alignment term. It corresponds to a continuous description of the IBM obtained by adding an alignment term to the dynamics describes the tendency of biological entities to align. Little is known for the general potential case. Using techniques of nonlinear analysis, as well as generalized function spaces, my hope is to clarify these issues. Another related problem I will be working on can be found here "G.-Q. Chen, L. He, Y. Wang, and D. Yuan, Global solutions of the Compressible Euler-Poisson equations with large initial data of spherical symmetry arising from the dynamics of gaseous stars, Arxiv, 2021". I will be particularly interested in questions related to asymptotic behavior. Seeking solutions in the Bounded Variation Spaces (or even Divergence Measure Fields, see ", G.-Q. Chen and H. Frid , Divergence-Measure Fields and Hyperbolic Conservation Laws, Arch. Rational Mech. Anal. 147 (1999)"), I could also determine blow-up conditions. Moreover, I could study Gradient Flows in the Wasserstein Space methods "L. Ambrosio, N. Gigli and G. Savare, Gradient Flows, Birkhäuser Basel, 2008" to tackle similar problems. Furthermore, I am interested Nonlinear problems arising in Geometry, particularly in the fields of elasticity and General Relativity and at the intersection with Optimal Transport. By adding an appropriate Stochastic term to the PDE, I could obtain good models for additional problems arising in Physics, Biology and Finance. Further techniques are required to deal with this sort of SPDEs.