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September  2019, 39(9): 4979-5015. doi: 10.3934/dcds.2019204

## Moments and regularity for a Boltzmann equation via Wigner transform

 Department of Mathematics, University of Texas at Austin, 2515 Speedway C1200, Austin, TX 78712, USA

* Corresponding author: Nataša Pavlović

Received  April 2018 Revised  December 2018 Published  May 2019

Fund Project: T.C.is funded by NSF grants DMS-1151414(CAREER) and DMS-1716198.N.P.is funded in part by NSF grant DMS-1516228.

In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain $\mathbb{R}^d$, $d\geq 2$, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform when $\left< v \right>^\beta f_0 \in L^2_v H^\alpha_x$ for $\min (\alpha,\beta) > \frac{d-1}{2}$. We prove that if $\alpha,\beta$ are large enough, then it is possible to propagate moments in $x$ and derivatives in $v$ (for instance, $\left< x \right>^k \left< \nabla_v \right>^\ell f \in L^\infty_T L^2_{x,v}$ if $f_0$ is nice enough). The mechanism is an exchange of regularity in return for moments of the (inverse) Wigner transform of $f$. We also prove a persistence of regularity result for the scale of Sobolev spaces $H^{\alpha,\beta}$; and, continuity of the solution map in $H^{\alpha,\beta}$. Altogether, these results allow us to conclude non-negativity of solutions, conservation of energy, and the $H$-theorem for sufficiently regular solutions constructed via the Wigner transform. Non-negativity in particular is proven to hold in $H^{\alpha,\beta}$ for any $\alpha,\beta > \frac{d-1}{2}$, without any additional regularity or decay assumptions.

Citation: Thomas Chen, Ryan Denlinger, Nataša Pavlović. Moments and regularity for a Boltzmann equation via Wigner transform. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 4979-5015. doi: 10.3934/dcds.2019204
##### References:
 [1] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.  doi: 10.1007/s00220-011-1242-9. [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Local existence with mild regularity for the Boltzmann equation, Kinetic and Related Models, 6 (2013), 1011-1041.  doi: 10.3934/krm.2013.6.1011. [3] R. Alonso, J. A. Cañizo, I. Gamba and C. Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Comm. Partial Differential Equations, 38 (2013), 155-169.  doi: 10.1080/03605302.2012.715707. [4] R. J. Alonso and I. M. Gamba, Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section, Journal of Statistical Physics, 137 (2009), 1147-1165.  doi: 10.1007/s10955-009-9873-3. [5] D. Arsenio, On the global existence of mild solutions to the Boltzmann equation for small data in $L^D$, Comm. Math. Phys., 302 (2011), 453-476.  doi: 10.1007/s00220-010-1159-8. [6] C. Bardos, I. M. Gamba, F. Golse and C. D. Levermore, Global solutions of the Boltzmann equation over $\mathbb{R}^d$ near global Maxwellians with small mass, Communications in Mathematical Physics, 346 (2016), 435-467.  doi: 10.1007/s00220-016-2687-7. [7] A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Statist. Phys., 88 (1997), 1183-1214.  doi: 10.1007/BF02732431. [8] T. Bodineau, I. Gallagher and L. Saint-Raymond, From hard spheres dynamics to the Stokes-Fourier equations: an ${L}^2$ analysis of the Boltzmann-Grad limit, C. R. Math. Acad. Sci. Paris, 353 (2015), 623–627, arXiv: 1511.03057. doi: 10.1016/j.crma.2015.04.013. [9] T. Bodineau, I. Gallagher and L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard spheres, Invent. math., 203 (2016), 493-553.  doi: 10.1007/s00222-015-0593-9. [10] L. Boudin and L. Desvillettes, On the singularities of the global small solutions of the full Boltzmann equation, Monatshefte für Mathematik, 131 (2000), 91–108. doi: 10.1007/s006050070015. [11] F. Castella and B. Perthame, Estimations de Strichartz pour les équations de transport cinétique, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 535-540. [12] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Verlag, 1994. doi: 10.1007/978-1-4419-8524-8. [13] T. Chen, R. Denlinger and N. Pavlović, Local well-posedness for Boltzmann's equation and the Boltzmann hierarchy via Wigner transform., Commun. Math. Phys., 368 (2019), 427-465.  doi: 10.1007/s00220-019-03307-9. [14] T. Chen and N. Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discr. Contin. Dyn. Syst. A, 27 (2010), 715-739.  doi: 10.3934/dcds.2010.27.715. [15] T. Chen and N. Pavlović, The quintic NLS as the mean field limit of a boson gas with three-body interactions, Journal of Functional Analysis, 260 (2011), 959-997.  doi: 10.1016/j.jfa.2010.11.003. [16] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423. [17] R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_\xi (H^N_x)$, Journal of Differential Equations, 244 (2008), 3204-3234.  doi: 10.1016/j.jde.2007.11.006. [18] I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard spheres and short-range potentials, European Mathematical Society (EMS), Z"urich, 2013. [19] I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282.  doi: 10.1007/s00205-009-0250-9. [20] P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8. [21] Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Archive for Rational Mechanics and Analysis, 169 (2003), 305-353.  doi: 10.1007/s00205-003-0262-9. [22] S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, Journal of Differential Equations, 215 (2005), 178–205, URL http://www.sciencedirect.com/science/article/pii/S0022039604003249. doi: 10.1016/j.jde.2004.07.022. [23] R. Illner and M. Shinbrot, The Boltzmann equation: Global existence for a rare gas in an infinite vacuum, Comm. Math. Phys., 95 (1984), 217–226, URL https://projecteuclid.org:443/euclid.cmp/1103941523. doi: 10.1007/BF01468142. [24] S. Kaniel and M. Shinbrot, The Boltzmann equation: I. Uniqueness and local existence, Communications in Mathematical Physics, 58 (1978), 65-84. [25] F. King, BBGKY Hierarchy for Positive Potentials, PhD thesis, Univ. California, Berkeley, 1975. [26] S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185.  doi: 10.1007/s00220-008-0426-4. [27] O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems, Theory and Applications (ed. J. Moser), vol. 38 of Lecture Notes in Physics, Springer Berlin Heidelberg, 1975, 1–111. [28] X. Lu and Y. Zhang, On nonnegativity of solutions of the Boltzmann equation, Transport Theory and Statistical Physics, 30 (2001), 641-657.  doi: 10.1081/TT-100107420. [29] X. Lu and C. Mouhot, On measure solutions of the Boltzmann equation, part Ⅰ: moment production and stability estimates, J. Differential Equations, 252 (2012), 3305-3363.  doi: 10.1016/j.jde.2011.10.021. [30] J. Polewczak, Classical solutions of the nonlinear Boltzmann equation in all $\mathbb{R}^3$: asymptotic behavior of solutions, J. Stat. Phys., 50 (1988), 611-632.  doi: 10.1007/BF01026493. [31] M. Tasković, R. J. Alonso, I. M. Gamba and N. Pavlović, On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff, SIAM J. Math. Anal., 50 (2018), 834-869.  doi: 10.1137/17M1117926. [32] S. Ukai, On the existence of global solutions of mixed problem for the non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.  doi: 10.3792/pja/1195519027. [33] 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. [34] C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, vol. 1 of Handbook of mathematical fluid dynamics, North-Holland, 2002. doi: 10.1016/S1874-5792(02)80004-0.

show all references

##### References:
 [1] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.  doi: 10.1007/s00220-011-1242-9. [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Local existence with mild regularity for the Boltzmann equation, Kinetic and Related Models, 6 (2013), 1011-1041.  doi: 10.3934/krm.2013.6.1011. [3] R. Alonso, J. A. Cañizo, I. Gamba and C. Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Comm. Partial Differential Equations, 38 (2013), 155-169.  doi: 10.1080/03605302.2012.715707. [4] R. J. Alonso and I. M. Gamba, Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section, Journal of Statistical Physics, 137 (2009), 1147-1165.  doi: 10.1007/s10955-009-9873-3. [5] D. Arsenio, On the global existence of mild solutions to the Boltzmann equation for small data in $L^D$, Comm. Math. Phys., 302 (2011), 453-476.  doi: 10.1007/s00220-010-1159-8. [6] C. Bardos, I. M. Gamba, F. Golse and C. D. Levermore, Global solutions of the Boltzmann equation over $\mathbb{R}^d$ near global Maxwellians with small mass, Communications in Mathematical Physics, 346 (2016), 435-467.  doi: 10.1007/s00220-016-2687-7. [7] A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Statist. Phys., 88 (1997), 1183-1214.  doi: 10.1007/BF02732431. [8] T. Bodineau, I. Gallagher and L. Saint-Raymond, From hard spheres dynamics to the Stokes-Fourier equations: an ${L}^2$ analysis of the Boltzmann-Grad limit, C. R. Math. Acad. Sci. Paris, 353 (2015), 623–627, arXiv: 1511.03057. doi: 10.1016/j.crma.2015.04.013. [9] T. Bodineau, I. Gallagher and L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard spheres, Invent. math., 203 (2016), 493-553.  doi: 10.1007/s00222-015-0593-9. [10] L. Boudin and L. Desvillettes, On the singularities of the global small solutions of the full Boltzmann equation, Monatshefte für Mathematik, 131 (2000), 91–108. doi: 10.1007/s006050070015. [11] F. Castella and B. Perthame, Estimations de Strichartz pour les équations de transport cinétique, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 535-540. [12] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Verlag, 1994. doi: 10.1007/978-1-4419-8524-8. [13] T. Chen, R. Denlinger and N. Pavlović, Local well-posedness for Boltzmann's equation and the Boltzmann hierarchy via Wigner transform., Commun. Math. Phys., 368 (2019), 427-465.  doi: 10.1007/s00220-019-03307-9. [14] T. Chen and N. Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discr. Contin. Dyn. Syst. A, 27 (2010), 715-739.  doi: 10.3934/dcds.2010.27.715. [15] T. Chen and N. Pavlović, The quintic NLS as the mean field limit of a boson gas with three-body interactions, Journal of Functional Analysis, 260 (2011), 959-997.  doi: 10.1016/j.jfa.2010.11.003. [16] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423. [17] R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_\xi (H^N_x)$, Journal of Differential Equations, 244 (2008), 3204-3234.  doi: 10.1016/j.jde.2007.11.006. [18] I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard spheres and short-range potentials, European Mathematical Society (EMS), Z"urich, 2013. [19] I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282.  doi: 10.1007/s00205-009-0250-9. [20] P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8. [21] Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Archive for Rational Mechanics and Analysis, 169 (2003), 305-353.  doi: 10.1007/s00205-003-0262-9. [22] S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, Journal of Differential Equations, 215 (2005), 178–205, URL http://www.sciencedirect.com/science/article/pii/S0022039604003249. doi: 10.1016/j.jde.2004.07.022. [23] R. Illner and M. Shinbrot, The Boltzmann equation: Global existence for a rare gas in an infinite vacuum, Comm. Math. Phys., 95 (1984), 217–226, URL https://projecteuclid.org:443/euclid.cmp/1103941523. doi: 10.1007/BF01468142. [24] S. Kaniel and M. Shinbrot, The Boltzmann equation: I. Uniqueness and local existence, Communications in Mathematical Physics, 58 (1978), 65-84. [25] F. King, BBGKY Hierarchy for Positive Potentials, PhD thesis, Univ. California, Berkeley, 1975. [26] S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185.  doi: 10.1007/s00220-008-0426-4. [27] O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems, Theory and Applications (ed. J. Moser), vol. 38 of Lecture Notes in Physics, Springer Berlin Heidelberg, 1975, 1–111. [28] X. Lu and Y. Zhang, On nonnegativity of solutions of the Boltzmann equation, Transport Theory and Statistical Physics, 30 (2001), 641-657.  doi: 10.1081/TT-100107420. [29] X. Lu and C. Mouhot, On measure solutions of the Boltzmann equation, part Ⅰ: moment production and stability estimates, J. Differential Equations, 252 (2012), 3305-3363.  doi: 10.1016/j.jde.2011.10.021. [30] J. Polewczak, Classical solutions of the nonlinear Boltzmann equation in all $\mathbb{R}^3$: asymptotic behavior of solutions, J. Stat. Phys., 50 (1988), 611-632.  doi: 10.1007/BF01026493. [31] M. Tasković, R. J. Alonso, I. M. Gamba and N. Pavlović, On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff, SIAM J. Math. Anal., 50 (2018), 834-869.  doi: 10.1137/17M1117926. [32] S. Ukai, On the existence of global solutions of mixed problem for the non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.  doi: 10.3792/pja/1195519027. [33] 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. [34] C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, vol. 1 of Handbook of mathematical fluid dynamics, North-Holland, 2002. doi: 10.1016/S1874-5792(02)80004-0.
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