EULER-POISSON EQUATIONS WITH ALIGNEMENT AND RELATED PROBLEMS

我的研究将集中在非线性分析工具来处理偏微分方程（PDEs）。这个领域被认为是不发达的，特别是与线性偏微分方程理论相比，线性偏微分方程理论在过去的一个世纪里吸引了大多数数学分析研究人员。我将由g - q教授指导。Chen和J. Carrillo。非线性偏微分方程在许多重要的应用中出现，包括弹性、几何、金融和生物学问题。许多这样的问题需要专门的理论来处理，因为数学对象应该适合问题的物理性质。因此，还有大量的工作要做。我将重点讨论的带对齐的欧拉-泊松方程就是一个例子。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，这些粒子可能是生物细胞、流体中的分子，甚至是宇宙中的星系，人们通常使用（可能只是部分）耦合的常微分方程系统。这些描述了系统在时间上的演变，系统被称为基于个体的模型（IBM）。如果粒子数量非常高，那么通常是不切实际的，甚至不可能获得这种ode系统的解决方案。因此，我们可以希望从相关的连续模型中得到一些有用的近似，这些模型是通过在适当的意义上让粒子的数量趋于无穷大而得到的。我们得到的PDE模型描述了与系统相关的宏观。为了证明全局及时光滑解或有限时间爆破或渐近行为，还需要确定初始值、参数和相互作用函数的严格条件。该项目属于EPSRC数学分析、数学生物学和连续介质力学研究领域。这个问题的一个重要应用是生物系统。例如，我将专注于Cucker- small模型（F. Cucker和S. Smale，羊群中的涌现行为，IEEE译）。奥特曼。Control 52(2007) 852”（另见“s - y”）。《从粒子到动力学和流体力学对植绒的描述》，中国科学院学报。遗传代数。型号1(2008)415-435”)，由一个无压欧拉方程和一个附加的校准项组成。它对应于通过向描述生物实体对齐趋势的动态描述中添加对齐术语而获得的IBM的连续描述。对于一般势情况，我们所知甚少。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。我要解决的另一个相关问题可以在这里找到“g - q”。陈磊，何磊，王勇，袁东，气态恒星动力学中具有大初始数据的可压缩Euler-Poisson方程的整体解，地球物理学报，2021"。我会对与渐近行为有关的问题特别感兴趣。求有界变分空间（甚至散度测度域）中的解，参见“，g - q”。陈和H. Frid，发散-测量场和双曲守恒定律，Arch。合理的机械。肛门。147（1999）”)，我也可以确定爆炸条件。此外，我可以研究Wasserstein空间方法中的梯度流动（L. Ambrosio， N. Gigli和G. Savare, Gradient Flows, Birkhäuser Basel, 2008）来解决类似的问题。此外，我对几何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与最优传输的交叉点。通过在PDE中加入一个适当的随机项，我可以为物理、生物和金融中出现的其他问题获得很好的模型。需要进一步的技术来处理这类spde。

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.