Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff
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A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.
KeywordsBoltzmann equation Homogenuous Measure valued solutions Characteristic functions
Mathematics Subject ClassificationPrimary 35Q20 76P05 Secondary 35H20 82B40 82C40
Authors would like to express their hearty gratitude to anonymous referees for many suggestions and advises which improved the submitted manuscript. The research of the first author was supported in part by Grant-in-Aid for Scientific Research No.25400160, Japan Society for the Promotion of Science. The research of the third author was supported in part by the General Research Fund of Hong Kong, CityU No. 11303614.
- 14.Jacob, N.: Pseudo-Differential Operators and Markov Processes. Fourier Analysis and Semigroups, vol. 1. Imperial College Press, London (2001)Google Scholar
- 24.Pulvirenti, A., Wennberg, B.: Lower bounds for the solutions to the Kac and the Boltzmann equation. In: Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), pp. 437-446 (1996)Google Scholar
- 28.Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of Fluid Mathematical Fluid Dynamics. Elsevier Science, New York (2002)Google Scholar
- 29.C. Villani, Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, (2003)Google Scholar