# American Institute of Mathematical Sciences

July  2019, 18(4): 1769-1782. doi: 10.3934/cpaa.2019083

## Global existence for the Boltzmann equation in $L^r_v L^\infty_t L^\infty_x$ spaces

 Independent scholar

Received  June 2018 Revised  June 2018 Published  January 2019

We study the Boltzmann equation near a global Maxwellian. We prove the global existence of a unique mild solution with initial data which belong to the $L^r_v L^\infty_x$ spaces where $r \in (1,\infty]$ by using the excess conservation laws and entropy inequality introduced in [5].

Citation: Koya Nishimura. Global existence for the Boltzmann equation in $L^r_v L^\infty_t L^\infty_x$ spaces. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1769-1782. doi: 10.3934/cpaa.2019083
##### References:
 [1] R. Duan, F. Huang, Y. Wang and T. Yang, Global well-posedness of the Boltzmann equation with large Amplitude initial data, Arch. Rational Mech. Anal., 225 (2017), 375-424. doi: 10.1007/s00205-017-1107-2. Google Scholar [2] Robert T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477. Google Scholar [3] Yan Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., Vol. LV (2002), 1104-1135. doi: 10.1002/cpa.10040. Google Scholar [4] Yan Guo, Decay and continuity of Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y. Google Scholar [5] Yan Guo, Bounded solutions for the Boltzmann equation, Quart. Appl. Math., LXVIII (2010), 143-148. doi: 10.1090/S0033-569X-09-01180-4. Google Scholar [6] M. Strain Robert and Keya Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$, Kinetic and Related Models, 5 (2012), 383-415. doi: 10.3934/krm.2012.5.383. Google Scholar [7] S. Ukai and T. Yang, The Boltzmann equation in the space $L^2 \cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications, 4 (2006), 263-310. doi: 10.1142/S0219530506000784. Google Scholar

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##### References:
 [1] R. Duan, F. Huang, Y. Wang and T. Yang, Global well-posedness of the Boltzmann equation with large Amplitude initial data, Arch. Rational Mech. Anal., 225 (2017), 375-424. doi: 10.1007/s00205-017-1107-2. Google Scholar [2] Robert T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477. Google Scholar [3] Yan Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., Vol. LV (2002), 1104-1135. doi: 10.1002/cpa.10040. Google Scholar [4] Yan Guo, Decay and continuity of Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y. Google Scholar [5] Yan Guo, Bounded solutions for the Boltzmann equation, Quart. Appl. Math., LXVIII (2010), 143-148. doi: 10.1090/S0033-569X-09-01180-4. Google Scholar [6] M. Strain Robert and Keya Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$, Kinetic and Related Models, 5 (2012), 383-415. doi: 10.3934/krm.2012.5.383. Google Scholar [7] S. Ukai and T. Yang, The Boltzmann equation in the space $L^2 \cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications, 4 (2006), 263-310. doi: 10.1142/S0219530506000784. Google Scholar
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