Spain. 23.10.2020. Last Tuesday on October 20th, 2020, our team member Jon Asier Bárcena Petisco Postdoctoral researcher from UAM will be talking about “Averaged dynamics and control for heat equations with random diffusion” as part of the CCM – Chair of Computational Mathematics seminars at the University of Deusto, Bilbao – Basque Country, Spain.
This talk included how the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient, assumed to be constant, is allowed to take the null value. First we prove that the averaged dynamics is analytic. This allows to show that, most often, the averaged dynamics enjoys the property of unique continuation and is approximately controllable. We then determine if the averaged dynamics is actually null controllable or not depending on how the density of averaging behaves when the diffusivity vanishes. In the critical density threshold the dynamics of the average is similar to the \frac{1}{2}-fractional Laplacian, which is well-known to be critical in the context of the controllability of fractional diffusion processes. Null controllability then fails (resp. holds) when the density weights more (resp. less) in the null diffusivity regime than in this critical regime.
