Title: Global existence and blowup of solutions of stochastic Keller–Segel-type equation
Abstract: In this talk I will consider a stochastic Keller–Segel-type equation, perturbed with random noise. For random pertubations in divergence form, the equation has a global weak solution for small initial data. This result is consistent with the deterministic case. However, in stark contrast with the deterministic case, if the noise is not in a divergence form, the solution has a finite time blowup with nonzero probability for any nonzero initial data. This is a joint project with Alex Misiats (VCU).

Title: Asymptotic Behaviors For The Compressible Euler System With Nonlinear Velocity Alignment
Abstract: We consider the pressureless compressible Euler system with a family of nonlinear velocity alignment. The system is a nonlinear extension of the Euler – alignment system in collective dynamics. I will show the asymptotic emergent phenomena of the system: alignment and flocking. Different types of the nonlinearity and the nonlocal communication protocols are investigated, resulting a variety of different asymptotic behaviors.

Title: Doubly Non-Local Cahn Hilliard Equation with a Fractional Time Derivative
Abstract:We consider a doubly nonlocal Cahn-Hilliard equation (dnCHE) which describes phase separation of a binary system but replace the classical time derivative with a Caputo fractional time derivative. In doing so, this modification can be used to model dynamic processes in which particles are thought to have some ‘memory’. We establish both the existence and uniqueness of a solution to this modified equation. Then, using a combination of a forward Euler scheme and a convolution quadrature rule we numerically approximate our mild solution. We establish convergence of mild solutions to that of the mild solution of the dnCHE when the order of the fractional derivative approaches 1.

Title: On the Riccati dynamics of 2D Euler-Poisson equations with attractive forcing
Abstract: The multi-dimensional Euler-Poisson(EP) system describes the dynamic behavior of many important physical flows. In this talk, a Riccati system that governs pressureless two-dimensional EP equations is discussed. The evolution of divergence is governed by the Riccati type equation with several nonlinear/nonlocal terms. Among these, the vorticity accelerates divergence while others further amplify the blow-up behavior of a flow. The growth of these blow-up amplifying terms are related to the Riesz transform of density, which lacks a uniform bound makes it difficult to study global solutions of the multi-dimensional EP system. We show that the Riccati system can afford to have global solutions, as long as the growth rate of blow-up amplifying terms is not higher than exponential, and admits global smooth solutions for a large set of initial configurations. Several recent works in a similar vein will be reviewed.