Spectral theory and time asymptotics of size-structured two-phase population models

This work provides a general spectral analysis of size-structured two-phase population models. Systematic functional analytic results are given. We deal first with the case of finite maximal size. We characterize the irreducibility of the corresponding $L^{1}$ semigroup in terms of properties of the different parameters of the system. We characterize also the spectral gap property of the semigroup. It turns out that the irreducibility of the semigroup implies the existence of the spectral gap. In particular, we provide a general criterion for asynchronous exponential growth. We show also how to deal with time asymptotics in case of lack of irreducibility. Finally, we extend the theory to the case of infinite maximal size.Comment: 36 page

On L1 compactness in transport theory

A shorter version of this preprint dealing with non-incoming boundary condition only appeared under the title "On L1-spectral theory of neutron transport" in Differential Integral Equations 18 (2005), no. 11, 1221-1242.We give a systematic and nearly optimal treatment of the compact- ness in connection with the L1 spectral theory of neutron transport equations on both n-dimensional torus and spatial domains with nite volume and nonincoming boundary conditions. Some L1 averaging lemmas are also given

On the leading eigenvalue of neutron transport models

AbstractWe give variational characterizations of the leading eigenvalue of neutron transport-like operators. The proofs rely on sub- and super-eigenvalues. Various bounds of the leading eigenvalue are derived

Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations

We provide a honesty theory of substochastic evolution families in real abstract state space, extending to an non-autonomous setting the result obtained for $C_0$-semigroups in our recent contribution \textit{[On perturbed substochastic semigroups in abstract state spaces, \textit{Z. Anal. Anwend.} \textbf{30}, 457--495, 2011]}. The link with the honesty theory of perturbed substochastic semigroups is established. Several applications to non-autonomous linear kinetic equations (linear Boltzmann equation and fragmentation equation) are provided

Rates of convergence to equilibrium states in the stochastic theory of neutron transport

We study a class of nonlinear equations arising in the stochastic theory of neutron transport. After proving existence and uniqueness of the solution, we consider the large-time behaviour of the solution and give explicit rates of convergence of the solution towards the asymptotic state

Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: a semigroup approach in L1-spaces.

International audienceWe investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) $L^{1}$-spaces. We deal with both the cases of hard and soft potentials (with angular cut-off). For hard potentials, we provide a new proof of the fact that, in weighted $L^{1}$-spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak-compactness arguments combined with recent results of the second author on positive semigroups in $L^{1}$-spaces. For soft potentials, in $L^{1}$-spaces, we exploits the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap

Spectral properties of general advection operators and weighted translation semigroups

We investigate the spectral properties of a class of weighted shift semigroups associated to abstract transport equations with a Lipschitz--continuous vector field with no--reentry boundary conditions. We illustrate our results with various examples taken from collisionless kinetic theory