372 research outputs found
Global existence results for complex hyperbolic models of bacterial chemotaxis
Bacteria are able to respond to environmental signals by changing their rules
of movement. When we take into account chemical signals in the environment,
this behaviour is often called chemotaxis. At the individual-level, chemotaxis
consists of several steps. First, the cell detects the extracellular signal
using receptors on its membrane. Then, the cell processes the signal
information through the intracellular signal transduction network, and finally
it responds by altering its motile behaviour accordingly. At the population
level, chemotaxis can lead to aggregation of bacteria, travelling waves or
pattern formation, and the important task is to explain the population-level
behaviour in terms of individual-based models. It has been previously shown
that the transport equation framework is suitable for connecting different
levels of modelling of bacterial chemotaxis. In this paper, we couple the
transport equation for bacteria with the (parabolic/elliptic) equation for the
extracellular signals. We prove global existence of solutions for the general
hyperbolic chemotaxis models of cells which process the information about the
extracellular signal through the intracellular biochemical network and interact
by altering the extracellular signal as well. The conditions for global
existence in terms of the properties of the signal transduction model are
given.Comment: 22 pages, submitted to Discrete and Continuous Dynamical Systems
Series
Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions
We describe the structure of solutions of the kinetic Fokker-Planck equations
in domains with boundaries near the singular set in one-space dimension. We
study in particular the behaviour of the solutions of this equation for
inelastic boundary conditions which are characterized by means of a coefficient
describing the amount of energy lost in the collisions of the particles
with the boundaries of the domain. A peculiar feature of this problem is the
onset of a critical exponent rc which follows from the analysis of McKean (cf.
[26]) of the properties of the stochastic process associated to the
Fokker-Planck equation under consideration. In this paper, we prove rigorously
that the solutions of the considered problem are nonunique if and
unique if . In particular, this nonuniqueness explains the
different behaviours found in the physics literature for numerical simulations
of the stochastic differential equation associated to the Fokker-Planck
equation. In the proof of the results of this paper we use several asymptotic
formulas and computations in the companion paper [16].Comment: 64 pages, 1 figure. Previous version has been split into tw
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