Mini-workshop: “Analysis, Numerics and Control”

Date: Fri. November 11, 2022
Organized by: FAU DCN-AvH, Chair for Dynamics, Control and Numerics – Alexander von Humboldt Professorship at FAU Erlangen-Nürnberg (Germany)
Title: Mini-workshop “Analysis, Numerics and Control

Title: Stabilization results for the KdV equation
Speaker: Hugo Parada
Affiliation: Visiting PhD Student from the Laboratory Jean Kunztmann, Grenoble (France)

Abstract.The Korteweg-de Vries (KdV) equation, was introduced as a model to describe the propagation of long water waves in a channel. This nonlinear third order dispersive equation has been many studied in the past years from different points of view, in particular its controllability and stabilization properties. In this talk, we focus on two problems. In the first part, we study the case where the KdV equation is in the presence of a boundary time-varying delay, and we show an exponential stability result. Then, we pass to the KdV equation posed in a star network with bounded and unbounded lengths. In this case we show the exponential stability by acting with damping terms not necessarily in all the branches. This talk is based on joint works with E. Crépeau, C. Prieur, J. Valein and C. Timimoun.

Title: Relaxation approximation and asymptotic stability of stratified solutions to the Incompressible Porous Media equation
Speaker: Dr. Timothée Crin-Barat
Affiliation: FAU DCN-AvH Chair for Dynamics, Control and Numerics – Alexander von Humboldt Professorship (Germany)

Abstract. In this joint work with Roberta Bianchini and Marius Paicu, we address the existence of stably stratified solutions to the two-dimensional Boussinesq equations with damped vorticity. We justify its nonlinear asymptotic stability for initial perturbations in H^s \cap H^{1-\tau} for s \gt 3 and 0 \lt \tau \lt 1. In addition, uniform estimates with respect to the damping parameter allow us to establish the strong relaxation limit of the Boussinesq system towards the Incompressible Porous Media equation (IPM) under a suitable scaling. And, as a byproduct, we deduce the global well-posedness of (IPM) in the same regularity setting. A crucial point of our analysis is the use of an anisotropic Littlewood-Paley decomposition to derive new bounds on the vorticity.


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