EULER-POISSON EQUATIONS WITH ALIGNEMENT AND RELATED PROBLEMS

具有对准的欧拉-泊松方程及相关问题

• 批准号: 2580841
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2580841
• 关键词: EULER; POISSON; EQUATIONS; ALIGNEMENT; RELATED

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.

我的研究将集中在非线性分析工具上，以处理部分微分方程（PDE）。该领域被认为是欠发达的，尤其是与线性PDE的理论相比，该领域吸引了过去一个世纪的数学分析中的大多数研究人员。我将受到G.-Q教授的监督Chen和J. Carrillo.Nonlinalear PDE在许多重要的应用中都出现，包括弹性，几何，金融和生物学问题。这些问题中的许多都需要处理量身定制的理论，因为数学对象应符合问题的物理性。因此，还需要做很多工作。我将重点关注的具有对齐方式的Euler-Poisson方程就是一个例子。它源于对多体系统的研究。当描述许多颗粒的系统的动力学，即可能是生物细胞，宇宙中流体甚至星系中的分子时，通常会使用（也许仅部分）耦合的普通微分方程的系统。这些描述了系统时间的演变，该系统被称为基于个人的模型（IBM）。如果颗粒的数量很高，则通常不切实际甚至不可能获得这种ODES系统的解决方案。因此，人们可以希望通过在适当的意义上让颗粒的数量倾向于无穷大，从而获得了一些有用的近似值。我们获得的PDE模型描述了与系统相关的宏观。在初始值，参数和相互作用函数上的严格条件仍然需要确定以证明全局平滑解决方案或有限的时间爆破或不对称行为。该项目属于EPSRC数学分析，数学生物学和连续力学研究领域。该问题的重要应用将是生物系统。例如，我将重点介绍cucker-smale模型“ F. Cucker和S. Smale，羊群中的紧急行为，IEEETrans。Autom。Control。52（2007）852”（另请参见“ S.-Y. Ha和E. Tadmor”，从粒子到动力学和水的动力学描述。带有附加对齐项的无压力Euler方程。它对应于通过在动力学中添加一个对齐项来描述生物学实体对齐的趋势的连续描述。一般的潜在情况闻名很少。我希望使用非线性分析的技术以及通用的功能空间，希望澄清这些问题。我将要解决的另一个相关问题可以在此处找到。我将对与渐近行为有关的问题特别感兴趣。在有界变化空间中寻求解决方案（甚至是差异测量场，请参见“，G.-Q.-Q. Chen和H. Frid，Divergence-Measure-Meary-Mealisher-Mealion-Mealion-Mealister Wirds and Suplybolic Conservation Laws，Arch。Arch。ConalitationMech。147（1999）”（1999）”（1999年）”，我还可以确定爆炸条件。此外，我还可以在blightiment中进行升级。梯度流动，伯克霍瑟·巴塞尔（BirkhäuserBasel），2008年“要解决类似的问题。此外，我感兴趣的非线性问题在几何形状中引起的问题，尤其是在弹性和一般相对性的领域，以及与最佳运输的交点。通过将适当的随机术语添加到PDE中，我可以在PEDE中添加适当的随机术语，以便在物理学中获得其他问题。