Abstract
We prove the global existence of the unique mild solution for the Cauchy problem of the cut-off Boltzmann equation for soft potential model \(\gamma =2-N\) with initial data small in \(L^N_{x,v}\) where \(N=2,3\) is the dimension. The proof relies on the existing inhomogeneous Strichartz estimates for the kinetic equation by Ovcharov (SIAM J Math Anal 43(3):1282–1310, 2011) and convolution-like estimates for the gain term of the Boltzmann collision operator by Alonso et al. (Commun Math Phys 298:293–322, 2010). The global dynamics of the solution is also characterized by showing that the small global solution scatters with respect to the kinetic transport operator in \(L^N_{x,v}\). Also the connection between function spaces and cut-off soft potential model \(-N<\gamma <2-N\) is characterized in the local well-posedness result for the Cauchy problem with large initial data.
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Acknowledgements
J.-C. Jiang was supported in part by National Sci-Tech Grant MOST 105-2115-M-007-005, Mathematics Research Promotion Center and National Center for Theoretical Sciences. The authors would like to thank anonymous referees for helpful comments.
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He, L., Jiang, JC. Well-posedness and Scattering for the Boltzmann Equations: Soft Potential with Cut-off. J Stat Phys 168, 470–481 (2017). https://doi.org/10.1007/s10955-017-1807-x
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DOI: https://doi.org/10.1007/s10955-017-1807-x