EULER-POISSON EQUATIONS WITH ALIGNEMENT AND RELATED PROBLEMS

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.