34 research outputs found
The one-dimensional Keller-Segel model with fractional diffusion of cells
We investigate the one-dimensional Keller-Segel model where the diffusion is
replaced by a non-local operator, namely the fractional diffusion with exponent
. We prove some features related to the classical
two-dimensional Keller-Segel system: blow-up may or may not occur depending on
the initial data. More precisely a singularity appears in finite time when
and the initial configuration of cells is sufficiently concentrated.
On the opposite, global existence holds true for if the initial
density is small enough in the sense of the norm.Comment: 12 page
A note on the Chern-Simons-Dirac equations in the Coulomb gauge
We prove that the Chern-Simons-Dirac equations in the Coulomb gauge are
locally well-posed from initial data in H^s with s > 1/4 . To study nonlinear
Wave or Dirac equations at this regularity generally requires the presence of
null structure. The novel point here is that we make no use of the null
structure of the system. Instead we exploit the additional elliptic structure
in the Coulomb gauge together with the bilinear Strichartz estimates of
Klainerman-Tataru.Comment: Preliminary version. Final version will appear in Discrete and
Continuous Dynamical Systems - Series
Mathematical description of bacterial traveling pulses
The Keller-Segel system has been widely proposed as a model for bacterial
waves driven by chemotactic processes. Current experiments on {\em E. coli}
have shown precise structure of traveling pulses. We present here an
alternative mathematical description of traveling pulses at a macroscopic
scale. This modeling task is complemented with numerical simulations in
accordance with the experimental observations. Our model is derived from an
accurate kinetic description of the mesoscopic run-and-tumble process performed
by bacteria. This model can account for recent experimental observations with
{\em E. coli}. Qualitative agreements include the asymmetry of the pulse and
transition in the collective behaviour (clustered motion versus dispersion). In
addition we can capture quantitatively the main characteristics of the pulse
such as the speed and the relative size of tails. This work opens several
experimental and theoretical perspectives. Coefficients at the macroscopic
level are derived from considerations at the cellular scale. For instance the
stiffness of the signal integration process turns out to have a strong effect
on collective motion. Furthermore the bottom-up scaling allows to perform
preliminary mathematical analysis and write efficient numerical schemes. This
model is intended as a predictive tool for the investigation of bacterial
collective motion
Magnetic Monopoles, Electric Neutrality and the Static Maxwell-Dirac Equations
We study the full Maxwell-Dirac equations: Dirac field with minimally coupled
electromagnetic field and Maxwell field with Dirac current as source. Our
particular interest is the static case in which the Dirac current is purely
time-like -- the "electron" is at rest in some Lorentz frame. In this case we
prove two theorems under rather general assumptions. Firstly, that if the
system is also stationary (time independent in some gauge) then the system as a
whole must have vanishing total charge, i.e. it must be electrically neutral.
In fact, the theorem only requires that the system be {\em asymptotically}
stationary and static. Secondly, we show, in the axially symmetric case, that
if there are external Coulomb fields then these must necessarily be
magnetically charged -- all Coulomb external sources are electrically charged
magnetic monopoles
Solitary waves in the Nonlinear Dirac Equation
In the present work, we consider the existence, stability, and dynamics of
solitary waves in the nonlinear Dirac equation. We start by introducing the
Soler model of self-interacting spinors, and discuss its localized waveforms in
one, two, and three spatial dimensions and the equations they satisfy. We
present the associated explicit solutions in one dimension and numerically
obtain their analogues in higher dimensions. The stability is subsequently
discussed from a theoretical perspective and then complemented with numerical
computations. Finally, the dynamics of the solutions is explored and compared
to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger
equation. A few special topics are also explored, including the discrete
variant of the nonlinear Dirac equation and its solitary wave properties, as
well as the PT-symmetric variant of the model