Related papers: Vanishing cross-diffusion limit in a Keller-Segel …
The global-in-time existence of weak solutions to a degenerate Cahn-Hilliard cross-diffusion system with singular potential in a bounded domain with no-flux boundary conditions is proved. The model consists of two coupled parabolic…
The Keller-Segel equations are widely used for describing chemotaxis in biology. Recently, a new fully discrete scheme for this model was proposed in [46], mass conservation, positivity and energy decay were proved for the proposed scheme,…
The qualitative study of solutions to the coupled parabolic-elliptic chemotaxis system with nonlinear diffusion for two species will be considered in the whole Euclidean space $\mathbb{R}^d$ ($d\geq 3$). It was proven in \cite{CK2021-ANA}…
This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel in the full space. We derive a critical mass threshold below which global existence is ensured. Using carefully energy…
A transient Poisson-Nernst-Planck system with steric effects is analyzed in a bounded domain with no-flux boundary conditions for the ion concentrations and mixed Dirichlet-Neumann boundary conditions for the electric potential. The steric…
We investigate a class of aggregation-diffusion equations with strongly singular kernels and weak (fractional) dissipation in the presence of an incompressible flow. Without the flow the equations are supercritical in the sense that the…
Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \begin{equation}\tag{$\ast$} \label{ks0} \left\{ \begin{aligned} u_t =&\; \Delta u - \nabla \cdot(u \nabla v) \quad in {\mathbb R}^2\times(0,\infty),\\ v…
A Keller-Segel model describes macroscopic dynamics of bacterial colonies and biological cells. Bacteria secret chemical which attracts other bacteria so that they move towards chemical gradient creating nonlocal attraction between…
We show the weak convergence, up to extraction of a subsequence, of the empirical measure for the Keller-Segel system of particles in both subcritical and critical cases, for general initial conditions. This particle system consists of $N$…
This paper deals with the fully parabolic 1d chemotaxis system with diffusion 1/(1 + u). We prove that the above mentioned nonlinearity, despite being a natural candidate, is not critical. It means that for such a diffusion any initial…
The long-wavelength, weak-dispersion limit of the discrete nonlinear Schr\"odinger equation with long-range dispersion is analytically considered. This continuum approximation is carried out irrespective of the dispersion range and hence…
We consider the parabolic-elliptic Keller-Segel system in dimensions $d \geq 3$, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An…
We derive two forms of conditional a posteriori error estimates for a finite volume scheme approximating the parabolic-elliptic Keller-Segel system. The estimates control the error in the $L^\infty(0,T, L^2(\Omega))$- and…
In the present article we consider several issues concerning the doubly parabolic Keller-Segel system in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. More specifically, we analyze the global…
We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error…
Existence of global finite-time bounded entropy solutions to a parabolic-parabolic system proposed in [16] is established in bounded domains under no-flux boundary conditions for nonnegative bounded initial data. This modification of the…
This paper deals with a sub-critical Keller-Segel equation. Starting from the stochastic particle system associated with it, we show well-posedness results and the propagation of chaos property. More precisely, we show that the empirical…
This paper is devoted to the design and analysis of a numerical algorithm for approximating solutions of a degenerate cross-diffusion system, which models particular instances of taxis-type migration processes under local sensing…
The flux-limited Keller--Segel system \begin{align*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v), \\[] v_t = \Delta v - v + u^{\theta} \end{cases} \end{align*} is considered under homogeneous Neumann…
In this work we numerically study the diffusive limit of run & tumble kinetic models for cell motion due to chemotaxis by means of asymptotic preserving schemes. It is well-known that the diffusive limit of these models leads to the…