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.

我的研究将围绕非线性分析工具来处理偏微分方程（PDE）。这一领域被认为是欠发达的，特别是当与线性偏微分方程理论相比，吸引了大多数研究人员在数学分析在过去的世纪。我将由G教授指导。Q.非线性偏微分方程出现在许多重要的应用中，包括弹性、几何、金融和生物学问题。许多这些问题需要量身定制的理论来处理，因为数学对象应该适合问题的物理性。因此，还有大量的工作要做。我将重点讨论的欧拉-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，这些粒子可以是生物细胞、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变，并且该系统被称为基于个体的模型（IBM）。如果粒子的数目非常大，那么得到这样的常微分方程组的解通常是不切实际的，甚至是不可能的。因此，人们希望通过让粒子数在适当的意义上趋于无穷大，可以从相关的连续模型中得到一些有用的近似。我们得到的偏微分方程模型描述了与系统相关的宏观情况。初值、参数和相互作用函数的严格条件仍然需要确定，以证明全局及时光滑解或有限时间爆破或渐近行为。这个项目福尔斯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是生物系统。例如，我将专注于Cucker-Smale模型“F。Cucker和S. Smale，Emergent behavior in flocks，IEEE Trans. Autom. Control 52（2007）852”（也参见“S.- Y. Ha和E. Tadmor，从粒子到动力学和流体动力学描述植绒，Kinet。相对模型1（2008）415-435”），由添加了对齐项的无压欧拉方程组成。它对应于一个连续的描述的IBM通过添加一个对齐项的动态描述的生物实体的趋势对齐。对于一般的潜在情况知之甚少。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。另一个相关的问题，我将工作可以在这里找到“G。Q.陈湖，澳-地他，Y. Wang和D. Yuan，Global solutions of the Compressible Euler-Poisson equations with large initial data of spherical symmetry arising arising from the dynamics of gasic stars，Arxiv，2021”.我将对与渐近行为有关的问题特别感兴趣。在有界变差空间（甚至是发散测度域，见“，G.- Q. Chen和H.发散测度场与双曲守恒律，《建筑理性力学分析》。147（1999）”），我也可以确定爆破条件。此外，我可以研究Wasserstein空间方法中的梯度流“L。Ambrosio，N. Gigli和G. Savare，Gradient Flows，Birkhäuser巴塞尔，2008”来解决类似的问题。此外，我感兴趣的几何中出现的非线性问题，特别是在弹性和广义相对论领域，并在最优运输的交叉点。通过添加一个适当的随机项的偏微分方程，我可以获得良好的模型，在物理学，生物学和金融中出现的其他问题。需要进一步的技术来处理这种SPDE。