Optimal Estimate of the Spectral Gap for the Degenerate Goldstein-Taylor Model
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Optimal Estimate of the Spectral Gap for the Degenerate Goldstein-Taylor Model Étienne Bernard · Francesco Salvarani
Received: 17 December 2012 / Accepted: 3 August 2013 / Published online: 29 August 2013 © Springer Science+Business Media New York 2013
Abstract In this paper we study the decay to the equilibrium state for the solution of a generalized version of the Goldstein-Taylor system, posed in the one-dimensional torus T = R/Z, by allowing that the nonnegative cross section σ can vanish in a subregion X := {x ∈ T | σ (x) = 0} of the domain with meas (X) ≥ 0 with respect to the Lebesgue measure. We prove that the solution converges in time, with respect to the strong L2 -topology, to its unique equilibrium with an exponential rate whenever meas (T \ X) ≥ 0 and we give an optimal estimate of the spectral gap. Keywords Goldstein-Taylor model · Spectral gap · Degenerate cross section
1 Introduction The investigation about explicit rates of approach to equilibrium in large time, for kinetic equations, is an active field of research and many results have been obtained, both in the linear and in the non linear case. An important concept, in this context, is hypocoercivity. This property appears in many evolution equations which have a conservative part and a dissipative one. Even if the conservative part alone does not induce relaxation and the dissipative one is not sufficient to induce convergence to equilibrium, sometimes the combination of the two parts leads to relaxation. When this situation occurs, the equation is said to be hypocoercive.
É. Bernard Institut Géographique National, Laboratoire de Recherche en Géodésie, Université Paris Diderot, Bâtiment Lamarck A, 5, rue Thomas Mann, Case courrier 7071, 75205 Paris Cedex 13, France e-mail: [email protected]
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F. Salvarani ( ) Dipartimento di Matematica F. Casorati, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy e-mail: [email protected]
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É. Bernard, F. Salvarani
For kinetic equations, the conservative term is usually the free transport operator, which mixes the space and the velocity variables, whereas the dissipative part is a collision operator, whose null space does not depend on the space variable. Furthermore, the key ingredient of many proofs is based upon the independence of the null space of the dissipative operator on the space variable. This allows indeed a local control of the dissipative properties of the equation, and hence the solution is locally “attracted” everywhere toward its local equilibrium (see, for instance, [6, 8, 12, 17]). The situation is, however, quite different in the degenerate case, i.e. when the collision operator can vanish in the spatial domain of the problem (even if the degeneracy happens only at isolated points). In the region of degeneracy, indeed, the null space of the collision operator becomes trivial. This problem has been studied for the first time by Desvillettes and the second author in [5]. In this article, they proved, under very stringent hypotheses on the degeneracy
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