37 research outputs found
A Hamilton-Jacobi approach for front propagation in kinetic equations
In this paper we use the theory of viscosity solutions for Hamilton-Jacobi
equations to study propagation phenomena in kinetic equations. We perform the
hydrodynamic limit of some kinetic models thanks to an adapted WKB ansatz. Our
models describe particles moving according to a velocity-jump process, and
proliferating thanks to a reaction term of monostable type. The scattering
operator is supposed to satisfy a maximum principle. When the velocity space is
bounded, we show, under suitable hypotheses, that the phase converges towards
the viscosity solution of some constrained Hamilton-Jacobi equation which
effective Hamiltonian is obtained solving a suitable eigenvalue problem in the
velocity space. In the case of unbounded velocities, the non-solvability of the
spectral problem can lead to different behavior. In particular, a front
acceleration phenomena can occur. Nevertheless, we expect that when the
spectral problem is solvable one can extend the convergence result
A kinetic eikonal equation
We analyse the linear kinetic transport equation with a BGK relaxation
operator. We study the large scale hyperbolic limit (t,x)\to (t/\eps,x/\eps).
We derive a new type of limiting Hamilton-Jacobi equation, which is analogous
to the classical eikonal equation derived from the heat equation with small
diffusivity. We prove well-posedness of the phase problem and convergence
towards the viscosity solution of the Hamilton-Jacobi equation. This is a
preliminary work before analysing the propagation of reaction fronts in kinetic
equations
The Bramson delay in the non-local Fisher-KPP equation
We consider the non-local Fisher-KPP equation modeling a population with
individuals competing with each other for resources with a strength related to
their distance, and obtain the asymptotics for the position of the invasion
front starting from a localized population. Depending on the behavior of the
competition kernel at infinity, the location of the front is either , as in the local case, or for some
explicit . Our main tools here are a local-in-time Harnack
inequality and an analysis of the linearized problem with a suitable moving
Dirichlet boundary condition. Our analysis also yields, for any
, examples of Fisher-KPP type non-linearities such
that the front for the local Fisher-KPP equation with reaction term
is at
Super-linear spreading in local and non-local cane toads equations
In this paper, we show super-linear propagation in a nonlocal
reaction-diffusion-mutation equation modeling the invasion of cane toads in
Australia that has attracted attention recently from the mathematical point of
view. The population of toads is structured by a phenotypical trait that
governs the spatial diffusion. In this paper, we are concerned with the case
when the diffusivity can take unbounded values, and we prove that the
population spreads as . We also get the sharp rate of spreading in a
related local model
Exponential decay to equilibrium for a fibre lay-down process on a moving conveyor belt
We show existence and uniqueness of a stationary state for a kinetic
Fokker-Planck equation modelling the fibre lay-down process in the production
of non-woven textiles. Following a micro-macro decomposition, we use
hypocoercivity techniques to show exponential convergence to equilibrium with
an explicit rate assuming the conveyor belt moves slow enough. This work is an
extension of (Dolbeault et al., 2013), where the authors consider the case of a
stationary conveyor belt. Adding the movement of the belt, the global Gibbs
state is not known explicitly. We thus derive a more general hypocoercivity
estimate from which existence, uniqueness and exponential convergence can be
derived. To treat the same class of potentials as in (Dolbeault et al., 2013),
we make use of an additional weight function following the Lyapunov functional
approach in (Kolb et al., 2013)
Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts
In this paper, we study the existence and stability of travelling wave
solutions of a kinetic reaction-transport equation. The model describes
particles moving according to a velocity-jump process, and proliferating thanks
to a reaction term of monostable type. The boundedness of the velocity set
appears to be a necessary and sufficient condition for the existence of
positive travelling waves. The minimal speed of propagation of waves is
obtained from an explicit dispersion relation. We construct the waves using a
technique of sub- and supersolutions and prove their \eb{weak} stability in a
weighted space. In case of an unbounded velocity set, we prove a
superlinear spreading. It appears that the rate of spreading depends on the
decay at infinity of the velocity distribution. In the case of a Gaussian
distribution, we prove that the front spreads as
The Bramson delay in the non-local Fisher-KPP equation
We consider the non-local Fisher-KPP equation modeling a population with
individuals competing with each other for resources with a strength related to
their distance, and obtain the asymptotics for the position of the invasion
front starting from a localized population. Depending on the behavior of the
competition kernel at infinity, the location of the front is either , as in the local case, or for some
explicit . Our main tools here are alocal-in-time Harnack
inequality and an analysis of the linearized problem with a suitable moving
Dirichlet boundary condition. Our analysis also yields, for any
, examples of Fisher-KPP type non-linearities such
that the front for the localFisher-KPP equation with reaction term
is at
A Hamilton-Jacobi approach for a model of population structured by space and trait
International audienceWe study a non-local parabolic Lotka-Volterra type equation describing a population structured by a space variable x 2 Rd and a phenotypical trait 2 . Considering diffusion, mutations and space-local competition between the individuals, we analyze the asymptotic (long- time/long-range in the x variable) exponential behavior of the solutions. Using some kind of real phase WKB ansatz, we prove that the propagation of the population in space can be described by a Hamilton-Jacobi equation with obstacle which is independent of . The effective Hamiltonian is derived from an eigenvalue problem. The main difficulties are the lack of regularity estimates in the space variable, and the lack of comparison principle due to the non-local term
Travelling waves for the cane toads equation with bounded traits.
20 pagesIn this paper, we study propagation in a nonlocal reaction-diffusion-mutation model describing the invasion of cane toads in Australia. The population of toads is structured by a space variable and a phenotypical trait and the space-diffusivity depends on the trait. We use a Schauder topological degree argument for the construction of some travelling wave solutions of the model. The speed of the wave is obtained after solving a suitable spectral problem in the trait variable. An eigenvector arising from this eigenvalue problem gives the flavor of the profile at the edge of the front. The major difficulty is to obtain uniform bounds despite the combination of non local terms and an heterogeneous diffusivity