Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Global classical solutions of the Boltzmann equation without angular cut-off


Authors: Philip T. Gressman and Robert M. Strain
Journal: J. Amer. Math. Soc. 24 (2011), 771-847
MSC (2010): Primary 35Q20, 35R11, 76P05, 82C40, 35H20, 35B65, 26A33
Published electronically: March 18, 2011
MathSciNet review: 2784329
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $ r^{-(p-1)}$ with $ p>2$, for initial perturbations of the Maxwellian equilibrium states, as announced in an earlier paper by the authors. We more generally cover collision kernels with parameters $ s\in (0,1)$ and $ \gamma$ satisfying $ \gamma > -n$ in arbitrary dimensions $ \mathbb{T}^n \times \mathbb{R}^n$ with $ n\ge 2$. Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann $ H$-theorem. When $ \gamma \ge -2s$, we have exponential time decay to the Maxwellian equilibrium states. When $ \gamma <-2s$, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $ \gamma \ge -2s$, as conjectured by Mouhot and Strain. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 35Q20, 35R11, 76P05, 82C40, 35H20, 35B65, 26A33

Retrieve articles in all journals with MSC (2010): 35Q20, 35R11, 76P05, 82C40, 35H20, 35B65, 26A33


Additional Information

Philip T. Gressman
Affiliation: Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Email: gressman@math.upenn.edu

Robert M. Strain
Affiliation: Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Email: strain@math.upenn.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2011-00697-8
PII: S 0894-0347(2011)00697-8
Keywords: Kinetic theory, Boltzmann equation, long-range interaction, non-cut-off, soft potentials, hard potentials, fractional derivatives, anisotropy, harmonic analysis.
Received by editor(s): February 15, 2010
Received by editor(s) in revised form: July 22, 2010, and January 21, 2011
Published electronically: March 18, 2011
Additional Notes: The first author was partially supported by the NSF grant DMS-0850791 and an Alfred P. Sloan Foundation Research Fellowship.
The second author was partially supported by the NSF grant DMS-0901463 and an Alfred P. Sloan Foundation Research Fellowship.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.