# American Institute of Mathematical Sciences

June  2015, 8(2): 281-308. doi: 10.3934/krm.2015.8.281

## Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions

 1 Division of Applied Mathematics, Brown University, Box F, 182 George Street, Providence, RI 02912, United States

Received  October 2014 Revised  January 2015 Published  March 2015

We prove the immediate appearance of an exponential lower bound, uniform in time and space, for continuous mild solutions to the full Boltzmann equation in a $C^2$ convex bounded domain with the physical Maxwellian diffusion boundary conditions, under the sole assumption of regularity of the solution. We investigate a wide range of collision kernels, with and without Grad's angular cutoff assumption. In particular, the lower bound is proven to be Maxwellian in the case of cutoff collision kernels. Moreover, these results are entirely constructive if the initial distribution contains no vacuum, with explicit constants depending only on the a priori bounds on the solution.
Citation: Marc Briant. Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions. Kinetic and Related Models, 2015, 8 (2) : 281-308. doi: 10.3934/krm.2015.8.281
##### References:
 [1] M. Briant, Instantaneous filling of the vacuum for the full Boltzmann equation in bounded domains, preprint. [2] T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146. doi: 10.1007/BF02398270. [3] C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. [4] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [5] L. Desvillettesand and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42. [6] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9. [7] H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin, 1958, 205-294. [8] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, preprint. [9] 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. [10] C. Mouhot, Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions, Comm. Partial Differential Equations, 30 (2005), 881-917. doi: 10.1081/PDE-200059299. [11] A. Pulvirenti and B. Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation, Comm. Math. Phys., 183 (1997), 145-160. doi: 10.1007/BF02509799. [12] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, 2002, 71-305. doi: 10.1016/S1874-5792(02)80004-0.

show all references

##### References:
 [1] M. Briant, Instantaneous filling of the vacuum for the full Boltzmann equation in bounded domains, preprint. [2] T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146. doi: 10.1007/BF02398270. [3] C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. [4] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [5] L. Desvillettesand and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42. [6] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9. [7] H. Grad, Principles of the kinetic theory of gases, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin, 1958, 205-294. [8] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, preprint. [9] 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. [10] C. Mouhot, Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions, Comm. Partial Differential Equations, 30 (2005), 881-917. doi: 10.1081/PDE-200059299. [11] A. Pulvirenti and B. Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation, Comm. Math. Phys., 183 (1997), 145-160. doi: 10.1007/BF02509799. [12] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, 2002, 71-305. doi: 10.1016/S1874-5792(02)80004-0.
 [1] Ling-Bing He, Jie Ji, Ling-Xuan Shao. Lower bound for the Boltzmann equation whose regularity grows tempered with time. Kinetic and Related Models, 2021, 14 (4) : 705-724. doi: 10.3934/krm.2021020 [2] Xinkuan Chai. The Boltzmann equation near Maxwellian in the whole space. Communications on Pure and Applied Analysis, 2011, 10 (2) : 435-458. doi: 10.3934/cpaa.2011.10.435 [3] Srimanta Bhattacharya, Sushmita Ruj, Bimal Roy. Combinatorial batch codes: A lower bound and optimal constructions. Advances in Mathematics of Communications, 2012, 6 (2) : 165-174. doi: 10.3934/amc.2012.6.165 [4] Alexander V. Bobylev, Irene M. Gamba. Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules. Kinetic and Related Models, 2017, 10 (3) : 573-585. doi: 10.3934/krm.2017023 [5] Yoshinori Morimoto. A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules. Kinetic and Related Models, 2012, 5 (3) : 551-561. doi: 10.3934/krm.2012.5.551 [6] Florent Foucaud, Tero Laihonen, Aline Parreau. An improved lower bound for $(1,\leq 2)$-identifying codes in the king grid. Advances in Mathematics of Communications, 2014, 8 (1) : 35-52. doi: 10.3934/amc.2014.8.35 [7] Geng Chen, Ronghua Pan, Shengguo Zhu. A polygonal scheme and the lower bound on density for the isentropic gas dynamics. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4259-4277. doi: 10.3934/dcds.2019172 [8] Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55 [9] Pablo Blanc. A lower bound for the principal eigenvalue of fully nonlinear elliptic operators. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3613-3623. doi: 10.3934/cpaa.2020158 [10] Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024 [11] Luís Simão Ferreira. A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. Kinetic and Related Models, 2022, 15 (1) : 91-117. doi: 10.3934/krm.2021045 [12] Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinetic and Related Models, 2017, 10 (4) : 901-924. doi: 10.3934/krm.2017036 [13] Shijin Deng, Weike Wang, Shih-Hsien Yu. Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary. Networks and Heterogeneous Media, 2007, 2 (3) : 383-395. doi: 10.3934/nhm.2007.2.383 [14] Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97 [15] Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure and Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269 [16] Claude Carlet, Brahim Merabet. Asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions. Advances in Mathematics of Communications, 2013, 7 (2) : 197-217. doi: 10.3934/amc.2013.7.197 [17] Nam Yul Yu. A Fourier transform approach for improving the Levenshtein's lower bound on aperiodic correlation of binary sequences. Advances in Mathematics of Communications, 2014, 8 (2) : 209-222. doi: 10.3934/amc.2014.8.209 [18] Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681 [19] Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic and Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1 [20] Yoshinori Morimoto, Chao-Jiang Xu. Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules. Kinetic and Related Models, 2020, 13 (5) : 951-978. doi: 10.3934/krm.2020033

2021 Impact Factor: 1.398