1,720,988 research outputs found
An Lp -Approach to the Well-Posedness of Transport Equations Associated with a Regular Field: Part I
No full textSubstochastic semigroups for transport equations with conservative boundary conditions
No full textWe consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a strongly continuous semigroup V(t). With techniques borrowed from the additive perturbation theory of substochasic semigroups, we describe precisely the domain of A and provide necessary and sufficient conditions ensuring V(t) to be stochastic. We apply these results to examples from kinetic theory
Long time behavior of nonautonomous Fokker-Planck equations and the cooling of granular gases
No full text
On a Mathematical Model of Immune Competition
Full text linkThis work deals with the qualitative analysis of a nonlinear integro-differential model of immune competition with special attention to the dynamics of tumor cells contrasted by the immune system. The analysis gives evidence of how initial conditions and parameters influence the asymptotic behavior of the solutions
A Kac Model for Kinetic Annihilation
No full textIn this paper, we consider the stochastic dynamics of a finite system of particles in a finite volume (Kac-like particle system) which annihilate with probability α∈ (0,1) or collide elastically with probability 1 - α. We first establish the well-posedness of the particle system which exhibits no conserved quantities. We rigorously prove that, in some mean-field limit, a suitable hierarchy of kinetic equations is recovered for which tensorized solution to the homogenous Boltzmann with annihilation is a solution. For bounded collision kernels, this shows in particular that propagation of chaos holds true. Furthermore, we make conjectures about the limit behaviour of the particle system when hard-sphere interactions are taken into account
Kinetic Description of a Rayleigh Gas with Annihilation
No full textIn this paper, we consider the dynamics of a tagged point particle in a gas of moving hard-spheres that are non-interacting among each other. This model is known as the ideal Rayleigh gas. We add to this model the possibility of annihilation (ideal Rayleigh gas with annihilation), requiring that each obstacle is either annihilating or elastic, which determines whether the tagged particle is elastically reflected or removed from the system. We provide a rigorous derivation of a linear Boltzmann equation with annihilation from this particle model in the Boltzmann–Grad limit. Moreover, we give explicit estimates for the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. The estimates show that the system can be approximated by the Boltzmann equation on an algebraically long time scale in the scaling parameter
Integral representation of the linear Boltzmann operator for granular gas dynamics with applications
No full text19 pages, to appear in Journal of Statistical Physics.International audienceWe investigate the properties of the collision operator associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of the collision operator in an Hilbert space setting, generalizing results from T. Carleman to granular gases. In the same way, we obtain from this integral representation of the gain operator that the semigroup in L^1(\R \times \R,\d \x \otimes \d\v) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from the first author
On the kinetic theory for active particles: A model for tumor-immune system competition
No full textThis paper deals with the qualitative analysis of a model describing the competition between tumor and immune cells. Such competition is characterized by proliferation-destruction phenomena and the interacting entities are characterized by a microscopic state which is modified by interactions. The model also includes the description of the natural trend of immune cells to reach a healthy or sentinel level, even when they have been involved in the competition with the tumor cells. The model is developed in the mathematical framework of the kinetic theory for active particles
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