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

References

SHOWING 1-10 OF 62 REFERENCES
Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system
Abstract.We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every $$u_0 \in L^1 (\mathbb {R}^2)$$ . The local existence time is characterized for
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions
The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion
On strongly anisotropic type II blow up
We consider the energy super critical 4 dimensional semilinear heat equation $$\partial_tu=\Delta u+|u|^{p-1}u, \ \ x\in \Bbb R^4, \ \ p>5.$$ Let $\Phi(r)$ be a three dimensional radial self similar
Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model
We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions $\rhohls$, $\lambda>0$, with thick tails
Type II Blow Up for the Four Dimensional Energy Critical Semi Linear Heat Equation
We consider the energy critical four dimensional semi linear heat equation \partial tu-\Deltau-u3 = 0. We show the existence of type II finite time blow up solutions and give a sharp description of
SELF-SIMILAR SOLUTIONS TO A NONLINEAR PARABOLIC-ELLIPTIC SYSTEM
We study the forward self-similar solutions to a parabolic-elliptic system $$ u_t = \Delta u - \nabla \cdot (u\nabla v),\quad 0 = \Delta v + u $$ in the whole space $\bf R^2$. First it is proved that
Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow
We exhibit a stable finite time blow up regime for the 1-corotational energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$ which reduces to the
Infinite time aggregation for the critical Patlak-Keller-Segel model in ℝ2
We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R2. Under the hypotheses of integrable initial data with finite second moment and entropy, we
On the formation of singularities in the critical $O(3)$ $\sigma$-model
We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from ℝ 2+1 Minkowski space into the sphere S 2 . We establish rigorously and
Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow
We consider the energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$. For initial data with corotational symmetry, the evolution reduces to the
...
1
2
3
4
5
...