The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modelled by nonlinear diffusion, and long-range attraction modelled by nonlocal interaction. In this course, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint with Carrillo, Hittmeir and Volzone). Once the symmetry is known, we further investigate whether steady states are unique within the radial class, and show that for a given mass, the uniqueness/non-uniqueness of steady states is determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint with Delgadino and Yan). I'll also discuss some properties on the long-time behavior of aggregation-diffusion equation with linear diffusion (joint with Carrillo, Gomez-Castro and Zeng), and global-wellposedness if Keller-Segel equation when coupled with an active advection term (joint with Hu and Kiselev).[-]

The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modelled by nonlinear diffusion, and long-range attraction modelled by nonlocal interaction. In this course, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint with Carrillo, Hittmeir and Volzone). Once the symmetry is known, we further investigate whether steady states are unique within the radial class, and show that for a given mass, the uniqueness/non-uniqueness of steady states is determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint with Delgadino and Yan). I'll also discuss some properties on the long-time behavior of aggregation-diffusion equation with linear diffusion (joint with Carrillo, Gomez-Castro and Zeng), and global-wellposedness if Keller-Segel equation when coupled with an active advection term (joint with Hu and Kiselev).[-]

As a toy model for the viscous interaction of planar vortices, we consider the solution of the two-dimensional Navier-Stokes equation with singular initial data corresponding to a pair of point vortices with opposite circulations. In the large Reynolds number regime, we construct an approximate solution which takes into account the deformation of the stream lines due to vortex interactions, as well as the corrections to the translation speed of the dipole due to finite size effects. Using energy estimates based on Arnold's variational characterization of equilibria for the Euler equation, we then show that our approximation remains valid over a very long time interval, if the viscosity is sufficiently small. This is a joint work with Michele Dolce (Lausanne), which relies on previous studies in collaboration with Vladimir Sverak (Minneapolis).[-]

The talk will review the motivations, state of the art, recent results, and open questions on four very related PDE models related to phase transitions: Allen-Cahn, Peierls-Nabarro, Minimal surfaces, and Nonlocal Minimal surfaces. We will focus on the study of stable solutions (critical points of the corresponding energy functionals with nonnegative second variation). We will discuss new nonlocal results on stable phase transitions, explaining why the stability assumption gives stronger information in presence of nonlocal interactions. We will also comment on the open problems and obstructions in trying to make the nonlocal estimates robust as the long-range (or nonlocal) interactions become short-range (or local).[-]

I will describe joint work with Dietrich Häfner and Andràs Vasy in which we study the asymptotic behavior of linearized gravitational perturbations of Schwarzschild or slowly rotating Kerr black hole spacetimes. We show that solutions of the linearized Einstein equation decay at an inverse polynomial rate to a stationary solution (given by an infinitesimal variation of the mass and angular momentum of the black hole), plus a pure gauge term. Our proof uses a detailed description of the resolvent of an associated wave equation on symmetric 2-tensors near zero energy.[-]

The Hamiltonian Mean-Field (HMF) model is a 1D simplified version of the gravitational Vlasov-Poisson system. I will present two recent works in collaboration with Mohammed Lemou and Ana Maria Luz. In the first one, we proved the nonlinear stability of steady states for this model, using a technique of generalized Schwarz rearrangements. To be stable, the steady state has to satisfy a criterion. If this criterion is not satisfied, some instabilities can occur: this is the topic of the second work that I will present.[-]

Chemotactic blow up in the context of the Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that when the Keller-Segel equation is coupled with passive advection, blow-up can be prevented if the flow possesses mixing or diffusion-enhancing properties, and its amplitude is sufficiently strong. In this talk, we consider the Keller-Segel equation coupled with an active advection, which is an incompressible flow obeying Darcy's law for incompressible porous media equation and driven by buoyancy force. We prove that in contrast with passive advection, this active advection coupling is capable of suppressing chemotactic blow up at arbitrary small coupling strength: namely, the system always has globally regular solutions. (Joint work with Zhongtian Hu and Alexander Kiselev).[-]

The purpose of this talk is to present two 1d congestion models: a soft congestion model with a singular pressure, and a hard congestion model in which the dynamic is different in the congested and non-congested zone (incompressible vs. compressible dynamic). The hard congested model is the limit of the soft one as the parameter within the singular presure vanishes.
For each model, we prove the existence of traveling waves, and we study their stability. This is a joint work with Charlotte Perrin.[-]

The multiplier approach is applied to a class of port-Hamiltonian systems with boundary dissipation to establish exponential decay. The exponential stability of port-Hamiltonian systems has been studied and sufficient conditions obtained. Here the decay rate $Me^{-\alpha t}$ is established with $M$ and $\alpha$ are in terms of system parameters. This approach is illustrated by several examples, in particular, boundary stabilization of a piezoelectric beam with magnetic effects.[-]