Mathematical models for elastic materials undergoing growth will be considered. The characteristic feature is a multiplicative decomposition of the deformation gradient into an elastic part a growth-related part. Approaches towards the existence of solutions will be discussed in
various settings, including models with and without codimension. This is joint work with Kira Bangert and Julian Blawid.

In this talk, we focus on the existence of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations, thereby providing a rigorous realization of a phenomenon first conjectured by Helmholtz (1858). In the leapfrogging motion, two coaxial vortex rings periodically exchange positions, a striking behavior repeatedly observed in experiments and numerical simulations, yet lacking complete mathematical justification. Our construction relies on a desingularization of two interacting vortex filaments within the contour dynamics formulation, which yields a Hamiltonian description of nearly concentric vortex rings. The main difficulty stems from a singular small-divisor problem arising in the linearized transport dynamics, where the effective time scale degenerates with the ring thickness parameter. To overcome this obstruction, we develop a degenerate KAM-type analysis combined with pseudo-differential operator techniques to control the linearized dynamics around symmetric configurations. Combining these tools with a Nash-Moser iteration scheme, we construct families of nontrivial time-periodic solutions in an almost uniformly translating frame. This establishes the first rigorous construction of classical leapfrogging motion for axisymmetric Euler flows without swirl, with no restriction on the time interval of existence. 

This is a joint work with Zineb Hassainia and Taoufik Hmidi.

In this talk I will discuss recent work, with R. Tione, on the regularity of stationary points for a class of planar polyconvex integrands which are conformally-invariant, a natural assumption in view of geometric applications. We prove that, in two dimensions, stationary points are smooth away from a discrete set. We also show full C^1-regularity for orientation-preserving solutions, which appear naturally in minimization problems of Teichmüller type.

In this talk we describe several aspects related to the theory of some anisotropic parabolic equations. The anisotropy shown in such equations will appear in the form of porous medium, in the fast and porous medium diffusion regime. In particular, we show the existence of selfsimilar fundamental solutions, which is uniquely determined by its mass, and the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. 

The investigation of both models are objects of joint works with F. Feo and J. L. V´azquez.

Consider a scalar conservation law with a spatially discontinuous flux at a single point x = 0, and assume that the flux is uniformly convex when x ̸= 0. I will discuss controllability problems for AB-entropy solutions associated to the so-called (A, B)-interface connection. I will first present a characterization of the set of profiles of AB-entropy solutions at a time horizon T > 0, as fixed points of a backward-forward solution operator. Next, I will address the problem of identifying the set of initial data driven by the corresponding AB-entropy solution to a given target profile ω T, at a time horizon T > 0. These results rely on the introduction of proper concepts of AB-backward solution operator, and AB-genuine/interface characteristics associated to an (A, B)-interface connection, and exploit duality properties of backward/forward shocks for AB-entropy solutions.
 

Based on joint works with Luca Talamini (SISSA-ISAS, Trieste)

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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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).