The main organizer of these seminars is Kai Yang.

## 2019 Applied Mathematics Seminars

The Applied Math Seminar takes place Mondays 2-3pm in DM 409A. It incorporates not only topics in applied math but also connections with various other disciplines and areas. For info, contact: Kai Yang (email: yangk@fiu.edu)

## Upcoming Seminars

#### Dynamics of Population Models with Two Physiological Structures

**Speaker:** Hao Kang, Bio-Math, University of Miami**Date:** Oct 14

Abstract: It is well-known for a long time that the age-structure of a population affects the nonlinear dynamics of the species in ecology and the transmission dynamics of infectious diseases in epidemiology. In

modeling specific diseases, the age could be chronological age (the age of the population), infection age (the time elapsed since infection), recovery age (the time elapsed sine the last infection), class age (the

length of time in the present group), etc. Other physiological conditions or physical characteristics such as size, location, status, and movement have also been taken in consideration in population dynamical models.

Recently there are some studies taking into account the combined effects of two physiological characteristics (such as age-age, age-size, age-maturation, age-stage), however there are very few theoretical studies on such models.

In this paper, we consider a scalar population model with two physiological structures and study its fundamental properties and dynamical behaviors. First, the semigroup will be defined based on the solutions and its infinitesimal generator will be determined; then the compactness of the solution trajectories will be analyzed; next spectrum theory will be employed to investigate stability of the zero steady state; when the zero steady state is unstable asynchronous growth of the solutions will be studied; finally we will apply theory of integral semigroup and non-densely defined operators to nonlinear models to establish the principle of linearized stability.

modeling specific diseases, the age could be chronological age (the age of the population), infection age (the time elapsed since infection), recovery age (the time elapsed sine the last infection), class age (the

length of time in the present group), etc. Other physiological conditions or physical characteristics such as size, location, status, and movement have also been taken in consideration in population dynamical models.

Recently there are some studies taking into account the combined effects of two physiological characteristics (such as age-age, age-size, age-maturation, age-stage), however there are very few theoretical studies on such models.

In this paper, we consider a scalar population model with two physiological structures and study its fundamental properties and dynamical behaviors. First, the semigroup will be defined based on the solutions and its infinitesimal generator will be determined; then the compactness of the solution trajectories will be analyzed; next spectrum theory will be employed to investigate stability of the zero steady state; when the zero steady state is unstable asynchronous growth of the solutions will be studied; finally we will apply theory of integral semigroup and non-densely defined operators to nonlinear models to establish the principle of linearized stability.

#### Conservative, positivity preserving and free energy dissipative numerical methods for the Poisson-Nernst-Planck equations

**Speaker:** Professor Zhongming Wang**Date:** Oct 7

We design and analyze some numerical methods for solving the Poisson--Nernst--Planck (PNP) equations. The numerical schemes, including finite difference method and discontinuous Galerkin method, respect three desired properties that are possessed by analytical solutions: I) conservation, II) positivity of solution, and III) free-energy dissipation. Advantages of different types of methods are discussed. Numerical experiments are performed to validate the numerical analysis. An application to an electrochemical charging system is also studied to demonstrate the effectiveness of our schemes in solving realistic problems. This is a joint work with H. Liu, D. Jie and S. Zhou.

#### Threshold solutions for the nonlinear Schrödinger equation

**Speaker:** Lucas Campos**Date:** Sept 23

We consider the energy-subcritical and energy-critical nonlinear focusing Schrödinger equation. In previous works by Holmer-Roudenko in the subcritical setting, and by Kenig-Merle, the mass-energy of the

*ground state*has been shown to be a threshold for the global behavior of solutions to the equation. In the present article, we study the dynamics at the threshold level and classify the corresponding solutions.#### On the stochastic 2D nonlocal Cahn-Hilliard- Navier-Stokes model

**Speaker:** Professor Deugoue**Date:** Sept 9

In this talk, we study a stochastic version of a well-known diffuse interface model. The model consists of the Navier-Stokes equations for the average velocity, nonlinearly coupled with a nonlocal Cahn-Hilliard equation for the order parameter. The system describes the evolution of an incompressible isothermal mixture of binary fluids excited by random forces in a two-dimensional bounded domain. For a fairly general class of random forces, we prove the existence and uniqueness of a variational solution.