Kolloquium Holzegel/Seis/Weber

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.

We start by demonstrating that the interplay between advection and diffusion in the incompressible porous media equation with diffusion -- a dissipative version of the classical active scalar problem -- can lead to enhanced dissipation. Subsequently, we derive a scaling limit that perfectly balances these two physical mechanisms. The high degeneracy of the limiting equation prevents us from proving existence of weak solutions in the distributional form. To address this challenge, we use the gradient flow structure of the equation to define weak solutions within a robust "geometric" framework and show that the solution space is compact. The talk is based on the joint work with Felix Otto and Bian Wu.

In this talk, we study the rigidity properties of a differential inclusion in linearized elasticity. We will introduce and discuss a set K of three diagonal strains that are pairwise incompatible but with non-trivial (symmetrized) rank-1-convex hull. This is called a T3 structure. We first prove that K is rigid at the level of exact solutions, namely that Lipschitz maps whose gradient is locally in K must be affine. After this, we study the scaling behaviour of the corresponding singularly-perturbed elastic energy, giving quantitative information on the flexibility of K in the sense of approximate solutions. This is based on a joint work with R. Indergand, D. Kochmann, A. Rüland, and C. Zillinger.

The Boltzmann equation is one of the central equations in statistical mechanics and models the evolution of a gas through particle interactions. In recent years, groundbreaking work by Imbert and Silvestre has led to a conditional regularity theory for periodic solutions of the Boltzmann equation. They established that any possible singularity of a periodic solution to the Boltzmann equation must be visible macroscopically. A major open challenge is whether such a theory can be extended to bounded domains with physically relevant boundary conditions. In this talk, I will first give an accessible overview of the conditional regularity program by Imbert and Silvestre, highlighting its main ideas and implications. As a first step toward understanding the boundary case, I will then discuss the smoothness of solutions to linear kinetic Fokker-Planck equations in domains with specular reflection condition. While the interior regularity of such equations is well understood, their behavior near the boundary has remained open, even in the simplest case of Kolmogorov?s equation. Finally, I will report on recent joint work with Xavier Ros-Oton, in which we establish sharp boundary regularity results for this class of equations.