Stability of the essential spectrum for 2D--transport models with Maxwell boundary conditions

We discuss the spectral properties of collisional semigroups associated to various models from transport theory by exploiting the links between the so-called resolvent approach and the semigroup approach. Precisely, we show that the essential spectrum of the full transport semigroup coincides with that of the collisionless transport semigroup in any $L^p$--spaces $(1 <p < \infty)$ for three 2D--transport models with Maxwell--boundary conditions.Comment: 23 page

Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases

We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases by showing that the Homogenous Cooling State attracts any solution at an algebraic rate.Проаналізовано асимптотичну поведінку лінійних рівнянь Фоккера - Планка з коефіцієнтами, залежними від часу. Показано, що за явно обчислюваних умов відбувається релаксація до розподілу Максвелла з залежною від часу температурою. Цей результат застосовано до вивчення броунівського руху в гранульованих газах і показано, що однорідний охолоджуючий стан притягує будь-який розв'язок з алгебраїчною швидкістю

On a Mathematical Model of Immune Competition

This 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

On a mathematical model of immune competition

AbstractThis 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

Convergence to self-similarity for ballistic annihilation dynamics

International audienceWe consider the spatially homogeneous Boltzmann equation for ballistic annihilation in dimension d \geq 2. Such model describes a system of ballistic hard spheres that, at the moment of interaction, either annihilate with probability α ∈ (0, 1) or collide elastically with probability 1 − α. Such equation is highly dissipative in the sense that all observables, hence solutions, vanish as time progresses. Following a contribution , by two of the authors, considering well posedness of the steady self-similar profile in the regime of small annihilation rate α ≪ 1, we prove here that such self-similar profile is the intermediate asymptotic attractor to the annihilation dynamics with explicit universal algebraic rate. This settles the issue about universality of the annihilation rate for this model brought in the applied literature

Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

International audienceThis paper deals with collisionless transport equationsin bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega$, orthogonallyinvariant velocity measure \bm{m}(\d v) with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing andincoming fluxes. Under very general conditions, such equations are governedby stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $$ We give a general criterion of irreducibility of $$ and we show that, under very natural assumptions, if an invariant densityexists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ converges strongly (notsimply in Cesar\`o means) to its ergodic projection. We show also that if noinvariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ is\emph{sweeping} in the sense that, for any density $\varphi$, the total mass of $$ concentrates near suitable sets of zero measure as $$ We show also a general weak compactness theoremwhich provides a basis for a general theory on existence of invariantdensities. This theorem is based on a series of results on smoothness andtransversality of the dynamical flow associated to $\left( U_{\mathsf{H}}(t)\right) _{t\geq0}.

Translations in the exponential Orlicz space with Gaussian weight

We study the continuity of space translations on non-parametric exponential families based on the exponential Orlicz space with Gaussian reference density.Comment: Submitted to GSI 2017, Pari