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 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 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).[-]

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 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.[-]

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).[-]

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 Euler-Korteweg system corresponds to compressible, inviscid fluids with capillary forces. It can be used to model diffuse interfaces. Mathematically it reads as the Euler equations with a third order dispersive perturbation corresponding to the capillary tensor.

In dimension one there exists traveling waves with equal or different limit at infinity, respectively solitons and kinks. Their stability is ruled by a simple criterion a la Grillakis-Shatah-Strauss. This talk is devoted to the construction of multiple traveling waves, namely global solutions that converge as $t\rightarrow \infty $ to a profile made of several (stable) traveling waves. The waves constructed have both solitons and kinks. Multiple traveling waves play a peculiar role in the dynamics of dispersive equations, as they correspond to solutions that follow in some sense a purely nonlinear evolution.[-]

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.[-]