Partial Differential Equations Seminar

We study a very general class of first-order linear hyperbolic
systems that both become weakly hyperbolic and contain singular
lower-order coefficients at a single time t = 0. In "critical" weakly
hyperbolic settings, it is well-known that solutions lose a finite
amount of regularity at t = 0. Here, we both improve upon the analysis
in the weakly hyperbolic setting, and we extend this analysis to systems
containing critically singular coefficients, which may also exhibit
modified asymptotics and regularity loss at t = 0.

In particular, we give precise quantifications for (1) the asymptotics
of solutions as t approaches 0, (2) the scattering problem of solving
the system with asymptotic data at t = 0, and (3) the loss of regularity
due to the degeneracies at t = 0. Finally, we discuss a wide range of
applications for these results, including weakly hyperbolic wave
equations (and equations of higher order), as well as equations arising
from relativity and cosmology (e.g. at big bang singularities).

This is joint work with Bolys Sabitbek (Ghent).

The aim of this talk is to discuss a finite-volume scheme for the aggregation-diffusion family of equations with non-linear mobility
∂tρ = ∇ · (m(ρ)∇(U′(ρ) + V + W ∗ ρ)) in bounded domains with no-flux conditions. We will present basic properties of the scheme: existence, decay of a free, and comparison principle (where applicable); and a convergence-by-compactness result for the saturation case where m(0) = m(1) = 0, under general assumptions on m,U, V , and W. The results are joint works published in [1, 2]. At the end of the talk, we will discuss an extension to the Porous-Medium Equation with non-local pressure that corresponds to m(ρ) = ρm, U, V = 0 and W(x) = c|x|^−d−2s.

This project is joint work with Jose Carrillo (University of Oxford). 
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