2011
DOI: 10.1007/s13370-011-0014-1
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New generation theorems in transport theory

Abstract: New generation theorems for singular kinetic equations in L 1 -spaces are given. We show that space homogeneous linear kinetic equations in L 1 spaces and full transport equations with advection termv. ∂ ∂ x in L 1 spaces on arbitrary spatial domains share several generation properties. In particular, in the subcritical case, they share the so-called "closure property" of the generator. We also show, for subcritical equations, that a principle of detailed balance insures this "closure property". We show, in th… Show more

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Cited by 13 publications
(22 citation statements)
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“…Therefore, as soon as α ∈ (0, 1), one deduces from Desch's Theorem (see for instance [26,32]) that L = L N = K + T,…”
Section: Perturbative Argument: the Case α ≃mentioning
confidence: 99%
“…Therefore, as soon as α ∈ (0, 1), one deduces from Desch's Theorem (see for instance [26,32]) that L = L N = K + T,…”
Section: Perturbative Argument: the Case α ≃mentioning
confidence: 99%
“…The construction of the mentioned semigroup S is based on a perturbation technique, developed in [16], and some aspects of the honesty theory [1,10]. Its adaptation to the present context is given in the following three statements.…”
Section: The Stochastic Semigroupmentioning
confidence: 99%
“…The proposed model seems to be the simplest individual-based model that takes into account the basic aspects of the phenomenon which we intended to describe: (a) essential mortality caused by external factors and independent of the interactions inside the population; (b) randomly distributed lifetimes of the population members, at the end of which each of them branches into two progenies; (c) branching independent of the interactions inside the population. The main difficulty of its mathematical study stemmed from the presence of the gradient in the Kolmogorov operator L in (2.35), which is typical for transport problems [10]. A more general version of the proposed model instead of the last summand in (2.35) could contain…”
Section: The Model and Its Studymentioning
confidence: 99%
“…[24,Inequality (5)] is satisfied. Notice however that nothing guarantees that the subcriticality condition [24,Inequality (6)] is met in X. It it not clear if the above is still true for soft potential, i.e.…”
Section: Arguing Then As In the Proof Of Theorem 44 One Conclude Thatmentioning
confidence: 99%