Applied Math Colloquium: Dylan McKnight (Nebraska)

Abstract: One of the first methods traditionally taught in Ordinary Differential Equations is the integrating factor method. Much later, one learns the variation of parameters formula for linear systems of ODEs. Both methods involve solving an equation of the form $y_t = Ay$ by appealing to an exponential function of the form $y(t) = e^(At)y_0$ for initial data $y_0$. 

In this talk, we will extend this idea to systems of partial differential equations via the $C_0$-semigroup. In the course of this, we will introduce Fluid-Structure Interaction, which is at minimum a system of PDEs that couple a parabolic fluid to a hyperbolic structure. Such systems have applications in medicine, biology, and engineering. Time permitting, we will conclude our discussion by discussing the regularity of a semigroup, and how this allows us to answer whether a coupled parabolic-hyperbolic system is more parabolic or hyperbolic. We will then provide some numerical examples to reinforce this intuition.