Strong convergence towards homogeneous cooling states for dissipative Maxwell models

  • Eric A. Carlen

    Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA
  • José A. Carrillo

    Institució Catalana de Recerca i Estudis Avançats and Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
  • Maria C. Carvalho

    CMAF and Departamento de Matematica da Faculdade de Ciências da Universidade de Lisboa, 1640-003 Lisboa, Portugal


We show the propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for small inelasticity. This result together with the weak convergence towards the homogeneous cooling state present in the literature implies the strong convergence in Sobolev norms and in the L1L^{1} norm towards it depending on the regularity of the initial data. The strategy of the proof is based on a precise control of the growth of the Fisher information for the inelastic Boltzmann equation. Moreover, as an application we obtain a bound in the L1L^{1} distance between the homogeneous cooling state and the corresponding Maxwellian distribution vanishing as the inelasticity goes to zero.

Cite this article

Eric A. Carlen, José A. Carrillo, Maria C. Carvalho, Strong convergence towards homogeneous cooling states for dissipative Maxwell models. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1675–1700

DOI 10.1016/J.ANIHPC.2008.10.005