We quantify the long-time behavior of solutions to the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures. Existence, uniqueness, and precise estimates of overpopulated high energy tails of the self-similar profile proved in ref. 9 are revisited and derived from this new Liapunov functional. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as t →∞ to the self-similar solution in this distance, which metrizes the weak convergence of measures. Moreover, we can relate this Fourier distance to the Euclidean Wasserstein distance or Tanaka functional proving also its exponential convergence towards the homogeneous cooling states. The findings are relevant in the understanding of the conjecture formulated by Ernst and Brito in refs. 15, 16, and complement and improve recent studies on the same problem of Bobylev and Cercignani(9) and Bobylev, Cercignani and one of the authors.(11)

Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model / BISI M.; CARRILLO J.A; TOSCANI G. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 124(2006), pp. 625-653. [10.1007/s10955-006-9035-9]

Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model

BISI, Marzia;
2006

Abstract

We quantify the long-time behavior of solutions to the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures. Existence, uniqueness, and precise estimates of overpopulated high energy tails of the self-similar profile proved in ref. 9 are revisited and derived from this new Liapunov functional. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as t →∞ to the self-similar solution in this distance, which metrizes the weak convergence of measures. Moreover, we can relate this Fourier distance to the Euclidean Wasserstein distance or Tanaka functional proving also its exponential convergence towards the homogeneous cooling states. The findings are relevant in the understanding of the conjecture formulated by Ernst and Brito in refs. 15, 16, and complement and improve recent studies on the same problem of Bobylev and Cercignani(9) and Bobylev, Cercignani and one of the authors.(11)
Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model / BISI M.; CARRILLO J.A; TOSCANI G. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 124(2006), pp. 625-653. [10.1007/s10955-006-9035-9]
File in questo prodotto:
File Dimensione Formato  
JSPBisiCarrilloToscani2.pdf

non disponibili

Tipologia: Documento in Post-print
Licenza: NON PUBBLICO - Accesso privato/ristretto
Dimensione 226.75 kB
Formato Adobe PDF
226.75 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11381/1507536
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 28
  • ???jsp.display-item.citation.isi??? 29
social impact