Journal
JOURNAL OF FUNCTIONAL ANALYSIS
Volume 262, Issue 3, Pages 915-1010Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2011.10.007
Keywords
Boltzmann equation; Coercivity estimate; Non-cutoff cross-sections; Global existence; Non-isotropic norm; Soft potential
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Funding
- Zhiyuan foundation
- Shanghai Jiao Tong University
- Japan Society of the Promotion of Science [22540187]
- Fundamental Research Funds for the Central Universities
- General Research Fund of Hong Kong, CityU [103109]
- Wuhan University
- City University of Hong Kong
- Kyoto University
- Rouen University
- Grants-in-Aid for Scientific Research [22540187] Funding Source: KAKEN
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It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces. (C) 2011 Elsevier Inc. All rights reserved.
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