Boltzmann Model for Viscoelastic Particles: Asymptotic Behavior, Pointwise Lower Bounds and Regularity

1. Alonso R.J.: Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data. Indiana Univ. Math. J. 58, 999–1022 (2009)

   Article  MATH  MathSciNet  Google Scholar 

2. Alonso R.J., Lods B.: Free cooling and high-energy tails of granular gases with variable restitution coefficient. SIAM J. Math. Anal. 42, 2499–2538 (2010)

   Article  MATH  MathSciNet  Google Scholar 

3. Alonso R.J., Lods B.: Two proofs of Haff’s law for dissipative gases: the use of entropy and the weakly inelastic regime. J. Math. Anal. Appl. 397, 260–275 (2013)

   Article  MATH  MathSciNet  Google Scholar 

4. Alonso R.J., Lods B.: Uniqueness and regularity of steady states of the Boltzmann equation for viscoelastic hard-spheres driven by a thermal bath. Commun. Math. Sci. 11, 851–906 (2013)

   Article  MATH  MathSciNet  Google Scholar 

5. Bobylev A.V., Carrillo J.A., Gamba I.M.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys. 98, 743–773 (2000)

   Article  MATH  MathSciNet  Google Scholar 

6. Bobylev A.V., Gamba I.M., Panferov N.: Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. J. Stat. Phys. 116, 1651–1682 (2004)

   Article  ADS  MATH  MathSciNet  Google Scholar 

7. Bouchut F., Desvillettes L.: A proof of the smoothing properties of the positive part of Boltzmann’s kernel. Rev. Mat. Iberoam. 14, 47–61 (1998)

   Article  MATH  MathSciNet  Google Scholar 

8. Brilliantov N.V., Pöschel T.: Kinetic Theory of Granular Gases. Oxford University Press, Oxford (2004)

   Book  MATH  Google Scholar 

9. Carleman T.: Problèmes mathématiques dans la théorie cinétique des gaz. Publ. Sci. Inst. Mittag-Leffler. 2, Uppsala (1957)

   MATH  Google Scholar 

10. Gamba I., Panferov V., Villani C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246, 503–541 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

11. Mischler S., Mouhot C., Rodriguez Ricard M.: Cooling process for inelastic Boltzmann equations for hard-spheres, Part~I: the Cauchy problem. J. Stat. Phys. 124, 655–702 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

12. Mischler S., Mouhot C.: Cooling process for inelastic Boltzmann equations for hard-spheres, Part~II: self-similar solution and tail behavior. J. Stat. Phys. 124, 655–702 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

13. Mischler S., Mouhot C.: Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard-spheres. Commun. Math. Phys. 288, 431–502 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

14. Mischler S., Mouhot C.: Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discret. Contin. Dyn. Syst. A 24, 159–185 (2009)

    Article  MATH  MathSciNet  Google Scholar 

15. Mouhot C., Villani C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Ration. Mech. Anal. 173, 169–212 (2004)

    Article  MATH  MathSciNet  Google Scholar 

16. Pulvirenti A., Wennberg B.: A Maxwellian lower bound for solutions to the Boltzmann equation. Commun. Math. Phys. 183, 145–160 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

17. Schwager T., Pöschel T.: Coefficient of normal restitution of viscous particles and cooling rate of granular gases. Phys. Rev. E 57, 650–654 (1998)

    Article  ADS  Google Scholar 

18. Toscani G., Villani C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203, 667–706 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

19. Villani C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234, 455–490 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

20. Wennberg B.: Regularity in the Boltzmann equation and the Radon transform. Commun. Partial Differ. Equ. 19, 2057–2074 (1994)

    Article  MATH  MathSciNet  Google Scholar 

Download references