300 research outputs found
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher
This review is dedicated to recent results on the 2d parabolic-elliptic
Patlak-Keller-Segel model, and on its variant in higher dimensions where the
diffusion is of critical porous medium type. Both of these models have a
critical mass such that the solutions exist globally in time if the mass
is less than and above which there are solutions which blowup in finite
time. The main tools, in particular the free energy, and the idea of the
methods are set out. A number of open questions are also stated.Comment: 26 page
Refined Asymptotics for the subcritical Keller-Segel system and Related Functional Inequalities
We analyze the rate of convergence towards self-similarity for the
subcritical Keller-Segel system in the radially symmetric two-dimensional case
and in the corresponding one-dimensional case for logarithmic interaction. We
measure convergence in Wasserstein distance. The rate of convergence towards
self-similarity does not degenerate as we approach the critical case. As a
byproduct, we obtain a proof of the logarithmic Hardy-Littlewood-Sobolev
inequality in the one dimensional and radially symmetric two dimensional case
based on optimal transport arguments. In addition we prove that the
one-dimensional equation is a contraction with respect to Fourier distance in
the subcritical case
Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations
In this paper we consider a -dimensional () parabolic-elliptic
Keller-Segel equation with a logistic forcing and a fractional diffusion of
order . We prove uniform in time boundedness of its solution
in the supercritical range , where is an explicit
constant depending on parameters of our problem. Furthermore, we establish
sufficient conditions for , where
is the only nontrivial homogeneous solution. Finally, we
provide a uniqueness result
An aggregation equation with a nonlocal flux
In this paper we study an aggregation equation with a general nonlocal flux.
We study the local well-posedness and some conditions ensuring global
existence. We are also interested in the differences arising when the
nonlinearity in the flux changes. Thus, we perform some numerics corresponding
to different convexities for the nonlinearity in the equation
A one-dimensional Keller-Segel equation with a drift issued from the boundary
We investigate in this note the dynamics of a one-dimensional Keller-Segel
type model on the half-line. On the contrary to the classical configuration,
the chemical production term is located on the boundary. We prove, under
suitable assumptions, the following dichotomy which is reminiscent of the
two-dimensional Keller-Segel system. Solutions are global if the mass is below
the critical mass, they blow-up in finite time above the critical mass, and
they converge to some equilibrium at the critical mass. Entropy techniques are
presented which aim at providing quantitative convergence results for the
subcritical case. This note is completed with a brief introduction to a more
realistic model (still one-dimensional).Comment: short version, 8 page
A semilinear parabolic-elliptic chemotaxis system with critical mass in any space dimension
We study radial solutions in a ball of of a semilinear,
parabolic-elliptic Patlak-Keller-Segel system with a nonlinear sensitivity
involving a critical power. For , the latter reduces to the classical
linear model, well-known for its critical mass . We show that a critical
mass phenomenon also occurs for , but with a strongly different
qualitative behaviour. More precisely, if the total mass of cells is smaller or
equal to the critical mass M, then the cell density converges to a regular
steady state with support strictly inside the ball as time goes to infinity. In
the case of the critical mass, this result is nontrivial since there exists a
continuum of stationary solutions and is moreover in sharp contrast with the
case where infinite time blow-up occurs. If the total mass of cells is
larger than M, then all solutions blow up in finite time. This actually follows
from the existence (unlike for ) of a family of self-similar, blowing up
solutions with support strictly inside the ball.Comment: 35 page
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