# Refined Description and Stability for Singular Solutions of the 2D Keller‐Segel System

@article{Collot2019RefinedDA, title={Refined Description and Stability for Singular Solutions of the 2D Keller‐Segel System}, author={Charles Collot and Tej-eddine Ghoul and Nader Masmoudi and Van Tien Nguyen}, journal={arXiv: Analysis of PDEs}, year={2019} }

We construct solutions to the two dimensional parabolic-elliptic Keller-Segel model for chemotaxis that blow up in finite time $T$. The solution is decomposed as the sum of a stationary state concentrated at scale $\lambda$ and of a perturbation. We rely on a detailed spectral analysis for the linearized dynamics in the parabolic neighbourhood of the singularity performed by the authors, providing a refined expansion of the perturbation. Our main result is the construction of a stable dynamics…

## 10 Citations

Spectral analysis for singularity formation of the two dimensional Keller-Segel system.

- Mathematics
- 2019

We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables,…

Long-time dynamics of classical Patlak-Keller-Segel equation

- Mathematics
- 2020

When the spatial dimension $n =2$, it has been well-known that a global mild solution to classical Patlak-Keller-Segel equation (PKS equation for short) exists if and only if its initial total mass…

Infinite time blow-up in the Keller-Segel system: existence and stability

- Mathematics
- 2019

The simplest version of the parabolic-elliptic Patlak-Keller-Segel system in the two-dimensional Euclidean space has an 8π critical mass which corresponds to the exact threshold between finite-time…

Blowup solutions for the shadow limit model of singular Gierer-Meinhardt system with critical parameters

- Mathematics
- 2021

Abstract. We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the…

Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher

- Mathematics
- 2021

We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time…

Global existence of free-energy solutions to the 2D Patlak--Keller--Segel--Navier--Stokes system with critical and subcritical mass

- Mathematics
- 2021

We consider a coupled Patlak–Keller–Segel–Navier–Stokes system in R2 that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and…

La ecuaci\'on de Keller-Segel

- Mathematics
- 2021

The purpose of this work is the study of chemotaxis and how to model it through the equations of Keller-Segel. Chemotaxis is a natural process which induces the organisms to direct their movement…

Non-existence of some approximately self-similar singularities for the Landau, Vlasov-Poisson-Landau, and Boltzmann equations

- Mathematics
- 2021

We consider the homogeneous and inhomogeneous Landau equation for very soft and Coulomb potentials and show that approximate Type I self-similar blow-up solutions do not exist under mild decay…

Sharp equivalent for the blowup profile to the gradient of a solution to the semilinear heat equation

- Mathematics
- 2021

In this paper, we consider the standard semilinear heat equation ∂tu = ∆u+ |u|p−1u, p > 1. The determination of the (believed to be) generic blowup profile is well-established in the literature, with…

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