CMX Lunch Seminar
My main research topic is the Hypocoercivity for Kinetic-Reaction equations. With Prof. Christian Schmeiser we investigate the laws that rule the convergence to the equilibrium in chemical systems where there are both transport and chemical reactions. In particular we are considering systems coupled with parabolic equation describing the temperature of the background. On the flow of this we are interested also on the energy balance. I also worked with Prof. Giulio Schimperna on existence and regularities results for coupled Navier-Stokes and Allen-Cahn systems.
We consider the thermalization of a gas towards a Maxwellian velocity distribution which depends locally on the temperature of the background. The exchange of kinetic and thermal energy between the gas and the background drives the system towards a global equilibrium with constant temperature. The heat flow is governed by the Fourier's law.
Mathematically we consider a coupled system of nonlinear kinetic and heat equations where in both cases we add a term that describes the energy exchange. For this problem we are able to prove existence of the solution in 1D, exponential convergence to the equilibrium through a hypocoercivity technique, macroscopic limit toward a cross-diffusion system. In the last two cases a perturbative approach is taken into account.
It's worth noticing that also without heat conductivity we can show the temperature diffusion thanks to the transport of energy. It is also interesting to show that the thermalization is highly influenced by the background temperature. All these aspects have been investigated also from a numerical viewpoint in order to provide simulations in 2D.