# American Institute of Mathematical Sciences

December  2018, 11(6): 1301-1331. doi: 10.3934/krm.2018051

## Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces

 1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China 2 Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japan

* Corresponding author: Renjun Duan

Received  October 2017 Published  June 2018

In this paper the Boltzmann equation near global Maxwellians is studied in the $d$-dimensional whole space. A unique global-in-time mild solution to the Cauchy problem of the equation is established in a Chemin-Lerner type space with respect to the phase variable $(x,v)$. Both hard and soft potentials with angular cutoff are considered. The new function space for global well-posedness is introduced to essentially treat the case of soft potentials, and the key point is that the velocity variable is taken in the weighted supremum norm, and the space variable is in the $s$-order Besov space with $s≥ d/2$ including the spatially critical regularity. The proof is based on the time-decay properties of solutions to the linearized equation together with the bootstrap argument. Particularly, the linear analysis in case of hard potentials is due to the semigroup theory, where the extra time-decay plays a role in coping with initial data in $L^2$ with respect to the space variable. In case of soft potentials, for the time-decay of linear equations we borrow the results based on the pure energy method and further extend them to those in $L^∞$ framework through the technique of $L^2-L^∞$ interplay. In contrast to hard potentials, $L^1$ integrability in $x$ of initial data is necessary for soft potentials in order to obtain global solutions to the nonlinear Cauchy problem.

Citation: Renjun Duan, Shota Sakamoto. Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces. Kinetic & Related Models, 2018, 11 (6) : 1301-1331. doi: 10.3934/krm.2018051
##### References:
 [1] R. Alexandre, Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Ⅰ, Global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010. doi: 10.1016/j.jfa.2011.10.007. Google Scholar [2] D. Arsénio and N. Masmoudi, A new approach to velocity averaging lemmas in Besov spaces, J. Math. Pures Appl. (9), 101 (2014), 495-551. doi: 10.1016/j.matpur.2013.06.012. Google Scholar [3] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differen-tial Equations, Grundlehren der Mathematischen Wissenschaften 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar [4] R. E. Caflisch, The Boltzmann equation with a soft potential. Ⅰ. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95. doi: 10.1007/BF01197579. Google Scholar [5] R. E. Caflisch, The Boltzmann equation with a soft potential. Ⅱ. Nonlinear, spatially-periodic, Comm. Math. Phys., 74 (1980), 97-109. doi: 10.1007/BF01197752. Google Scholar [6] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math. (2), 130 (1989), 321-366. doi: 10.2307/1971423. Google Scholar [7] R. J. Duan, F. M. Huang, Y. Wang and T. Yang, Global well-posedness of the Boltzmann equation with large amplitude initial data, Arch. Ration. Mech. Anal., 225 (2017), 375-424. doi: 10.1007/s00205-017-1107-2. Google Scholar [8] R. J. Duan, S. Q. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann equation, Arch. Ration. Mech. Appl., 220 (2016), 711-745. doi: 10.1007/s00205-015-0940-4. Google Scholar [9] R. J. Duan, T. Yang and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Models Methods Appl. Sci., 23 (2013), 979-1028. doi: 10.1142/S0218202513500012. Google Scholar [10] R. J. Duan and Y. Wang, The Boltzmann equation with large-amplitude initial data in bounded domains, preprint, arXiv: 1703.07978.Google Scholar [11] Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353. doi: 10.1007/s00205-003-0262-9. Google Scholar [12] Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y. Google Scholar [13] Y. Morimoto and S. Sakamoto, Global solutions in the critical Besov space for the non-cutoff Boltzmann equation, J. Differential Equations, 261 (2016), 4073-4134. doi: 10.1016/j.jde.2016.06.017. Google Scholar [14] V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbb{R}^n_x$, Adv. Math., 261 (2014), 274-332. doi: 10.1016/j.aim.2014.04.012. Google Scholar [15] R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinet. Relat. Models, 5 (2012), 583-613. doi: 10.3934/krm.2012.5.583. Google Scholar [16] H. Tang and Z. R. Liu, On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces, Discrete Contin. Dyn. Syst., 36 (2016), 2229-2256. doi: 10.3934/dcds.2016.36.2229. Google Scholar [17] S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027. Google Scholar [18] S. Ukai, Solution of the Boltzmann equation, in Patterns and Waves (eds. T. Nishida, M. Mimura and H. Fujii), Stud. Math. Appl., 18, North-Holland, Amsterdam, (1986), 37-96. doi: 10.1016/S0168-2024(08)70128-0. Google Scholar [19] S. Ukai and K. Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci., 18 (1982), 477-519 (57-99). doi: 10.2977/prims/1195183569. Google Scholar [20] S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^∞_β$: global and time-periodic solutions, Anal. Appl. (Singap.), 4 (2006), 263-310. doi: 10.1142/S0219530506000784. Google Scholar

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##### References:
 [1] R. Alexandre, Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Ⅰ, Global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010. doi: 10.1016/j.jfa.2011.10.007. Google Scholar [2] D. Arsénio and N. Masmoudi, A new approach to velocity averaging lemmas in Besov spaces, J. Math. Pures Appl. (9), 101 (2014), 495-551. doi: 10.1016/j.matpur.2013.06.012. Google Scholar [3] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differen-tial Equations, Grundlehren der Mathematischen Wissenschaften 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar [4] R. E. Caflisch, The Boltzmann equation with a soft potential. Ⅰ. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95. doi: 10.1007/BF01197579. Google Scholar [5] R. E. Caflisch, The Boltzmann equation with a soft potential. Ⅱ. Nonlinear, spatially-periodic, Comm. Math. Phys., 74 (1980), 97-109. doi: 10.1007/BF01197752. Google Scholar [6] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math. (2), 130 (1989), 321-366. doi: 10.2307/1971423. Google Scholar [7] R. J. Duan, F. M. Huang, Y. Wang and T. Yang, Global well-posedness of the Boltzmann equation with large amplitude initial data, Arch. Ration. Mech. Anal., 225 (2017), 375-424. doi: 10.1007/s00205-017-1107-2. Google Scholar [8] R. J. Duan, S. Q. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann equation, Arch. Ration. Mech. Appl., 220 (2016), 711-745. doi: 10.1007/s00205-015-0940-4. Google Scholar [9] R. J. Duan, T. Yang and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Models Methods Appl. Sci., 23 (2013), 979-1028. doi: 10.1142/S0218202513500012. Google Scholar [10] R. J. Duan and Y. Wang, The Boltzmann equation with large-amplitude initial data in bounded domains, preprint, arXiv: 1703.07978.Google Scholar [11] Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353. doi: 10.1007/s00205-003-0262-9. Google Scholar [12] Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y. Google Scholar [13] Y. Morimoto and S. Sakamoto, Global solutions in the critical Besov space for the non-cutoff Boltzmann equation, J. Differential Equations, 261 (2016), 4073-4134. doi: 10.1016/j.jde.2016.06.017. Google Scholar [14] V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbb{R}^n_x$, Adv. Math., 261 (2014), 274-332. doi: 10.1016/j.aim.2014.04.012. Google Scholar [15] R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinet. Relat. Models, 5 (2012), 583-613. doi: 10.3934/krm.2012.5.583. Google Scholar [16] H. Tang and Z. R. Liu, On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces, Discrete Contin. Dyn. Syst., 36 (2016), 2229-2256. doi: 10.3934/dcds.2016.36.2229. Google Scholar [17] S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027. Google Scholar [18] S. Ukai, Solution of the Boltzmann equation, in Patterns and Waves (eds. T. Nishida, M. Mimura and H. Fujii), Stud. Math. Appl., 18, North-Holland, Amsterdam, (1986), 37-96. doi: 10.1016/S0168-2024(08)70128-0. Google Scholar [19] S. Ukai and K. Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci., 18 (1982), 477-519 (57-99). doi: 10.2977/prims/1195183569. Google Scholar [20] S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^∞_β$: global and time-periodic solutions, Anal. Appl. (Singap.), 4 (2006), 263-310. doi: 10.1142/S0219530506000784. Google Scholar
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