Skip to main content
Log in

Propagation of Smoothness and the Rate of Exponential Convergence to Equilibrium for a Spatially Homogeneous Maxwellian Gas

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract:

We prove an inequality for the gain term in the Boltzmann equation for Maxwellian molecules that implies a uniform bound on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. We then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm. These results are then combined, using interpolation inequalities, to obtain the optimal rate of exponential convergence in the strong L 1 norm, as well as various Sobolev norms. These results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non-linear Boltzmann equation, even for initial data that is far from equilibrium.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Author information

Authors and Affiliations

Authors

Additional information

Received: 8 January 1997 / Accepted: 12 May 1998

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carlen, E., Gabetta, E. & Toscani, G. Propagation of Smoothness and the Rate of Exponential Convergence to Equilibrium for a Spatially Homogeneous Maxwellian Gas. Comm Math Phys 199, 521–546 (1999). https://doi.org/10.1007/s002200050511

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s002200050511

Keywords

Navigation