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 & 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., (). Google Scholar

[2]

T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann,, Acta Math., 60 (1933), 91. doi: 10.1007/BF02398270. Google Scholar

[3]

C. Cercignani, The Boltzmann Equation and Its Applications,, Applied Mathematical Sciences, (1988). Google Scholar

[4]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Applied Mathematical Sciences, (1994). doi: 10.1007/978-1-4419-8524-8. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00222-004-0389-9. Google Scholar

[7]

H. Grad, Principles of the kinetic theory of gases,, in Handbuch der Physik (herausgegeben von S. Flügge), (1958), 205. Google Scholar

[8]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem,, preprint., (). Google Scholar

[9]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains,, Arch. Ration. Mech. Anal., 197 (2010), 713. doi: 10.1007/s00205-009-0285-y. Google Scholar

[10]

C. Mouhot, Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions,, Comm. Partial Differential Equations, 30 (2005), 881. doi: 10.1081/PDE-200059299. Google Scholar

[11]

A. Pulvirenti and B. Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation,, Comm. Math. Phys., 183 (1997), 145. doi: 10.1007/BF02509799. Google Scholar

[12]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar

show all references

References:
[1]

M. Briant, Instantaneous filling of the vacuum for the full Boltzmann equation in bounded domains,, preprint., (). Google Scholar

[2]

T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann,, Acta Math., 60 (1933), 91. doi: 10.1007/BF02398270. Google Scholar

[3]

C. Cercignani, The Boltzmann Equation and Its Applications,, Applied Mathematical Sciences, (1988). Google Scholar

[4]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Applied Mathematical Sciences, (1994). doi: 10.1007/978-1-4419-8524-8. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00222-004-0389-9. Google Scholar

[7]

H. Grad, Principles of the kinetic theory of gases,, in Handbuch der Physik (herausgegeben von S. Flügge), (1958), 205. Google Scholar

[8]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem,, preprint., (). Google Scholar

[9]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains,, Arch. Ration. Mech. Anal., 197 (2010), 713. doi: 10.1007/s00205-009-0285-y. Google Scholar

[10]

C. Mouhot, Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions,, Comm. Partial Differential Equations, 30 (2005), 881. doi: 10.1081/PDE-200059299. Google Scholar

[11]

A. Pulvirenti and B. Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation,, Comm. Math. Phys., 183 (1997), 145. doi: 10.1007/BF02509799. Google Scholar

[12]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar

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