The Keller-Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemoattractant concentration. This paper deals with the rate of convergence towards a unique stationary state in self-similar variables, which describes the intermediate asymptotics of the solutions in the original variables. Although it is known that solutions globally exist for any mass less 8π , a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar variables.

We study a two-dimensional system in solid rotation at constant angular velocity driven by a self-consistent three-dimensional gravitational field. We prove the existence of stationary solutions of such a flat system in the rotating frame as long as the angular velocity does not exceed some critical value which depends on the mass. The solutions can be seen as stationary solutions of a kinetic equation with a relaxation-time collision kernel forcing the convergence to the polytropic gas solutions, or as stationary solutions of an extremely simplified drift-diffusion model, which is derived from the kinetic equation by formally taking a diffusion limit. In both cases, the solutions are critical points of a free energy functional, and can be seen as localized minimizers in an appropriate sense.

Embryo–maternal cross-talk has emerged as a vitally important process for embryo development and implantation, which is driven by secreted factors and extracellular vesicles (EVs). The EV cargo of bioactive molecules significantly influences target cells and primes them for critical stages of reproductive biology, including embryo development, adhesion, and implantation. Recent research has suggested that EVs and their cargo represent a powerful non-invasive tool that can be leveraged to assess embryo and maternal tissue quality during assisted reproduction treatments. Here, we review the current scientific literature regarding the intercellular cross-talk between embryos and maternal tissues from fertilization to implantation, focusing on human biology and signaling mechanisms identified in animal models.

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