Gain of integrability for the Boltzmann collisional operator

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March 2011, 4(1): 41-51. doi: 10.3934/krm.2011.4.41

Gain of integrability for the Boltzmann collisional operator

Ricardo J. Alonso ^1, and Irene M. Gamba ^2,

1.	Dept. of Computational & Applied Mathematics, Rice University, Houston, TX 77005-1892, United States
2.	Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Texas 78712, United States

Received December 2010 Revised December 2010 Published January 2011

Abstract
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In this short note we revisit the gain of integrability property of the gain part of the Boltzmann collision operator. This property implies the $W^{l,r}_k$ regularity propagation for solutions of the associated space homogeneous initial value problem. We present a new method to prove the gain of integrability that simplifies the technicalities of previous approaches by avoiding the argument of gain of regularity estimates for the gain collisional integral. In addition our method calculates explicit constants involved in the estimates.

Keywords: Boltzmann equation for hard potentials; Collisional Operator; Regularity in $L^p$ spaces..

Mathematics Subject Classification: 76P05, 35D0.

Citation: Ricardo J. Alonso, Irene M. Gamba. Gain of integrability for the Boltzmann collisional operator. Kinetic and Related Models, 2011, 4 (1) : 41-51. doi: 10.3934/krm.2011.4.41

References:

[1]	R. Alonso and E. Carneiro, Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform, Adv. Math., 223 (2010), 511-528. doi: 10.1016/j.aim.2009.08.017.
[2]	R. Alonso, E. Carneiro and I. M. Gamba, Convolution inequalities for the Boltzmann collision operator, Comm. Math. Physics, 298 (2010), 293-322. doi: 10.1007/s00220-010-1065-0.
[3]	R. Alonso, J. A. Canizo, I. M. Gamba, C. Mohout and S. Mischler, The Homogeneous Boltzmann equation for hard potentials with a cold thermostat, work in progress.
[4]	R. Alonso and I. M. Gamba, Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section, J. Stat. Phys., 137 (2009), 1147-1165. doi: 10.1007/s10955-009-9873-3.
[5]	R. Alonso and I. M. Gamba, Revision on classical solutions to the Cauchy Boltzmann problem for soft potentials, submitted for publication (2010).
[6]	R. Alonso, I. M. Gamba and S. H. Tharkabhushaman, Accuracy and consistency of Lagrangian based conservative spectral method for space-homogeneous Boltzmann equation, work in progress.
[7]	T. Carleman, "Problèmes Mathématiques dans la Théorie Cinétique des Gaz," Publ. Sci. Inst. Mittag-Leffler, 2. Almqvist and Wiksell, Uppsala, 1957.
[8]	C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Appl. Math. Sci. Springer-Verlag, Berlin, 1994.
[9]	I. M. Gamba, V. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys., 246 (2004), 503-541. doi: 10.1007/s00220-004-1051-5.
[10]	I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellians bounds for the spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal., 194 (2009), 253-282. doi: 10.1007/s00205-009-0250-9.
[11]	I. M. Gamba and S. H. Tharkabhushaman, Spectral-Lagrangian based methods applied to computation of non-equilibrium statistical states, Jour. Comp. Phys., 228 (2009), 2012-2036. doi: 10.1016/j.jcp.2008.09.033.
[12]	I. M. Gamba and S. H. Tharkabhushanam, Shock and boundary structure formation by spectral-lagrangian methods for the inhomogeneous Boltzmann transport equation, Jour. Comp. Math., 28 (2010), 430-460.
[13]	T. Gustafsson, Global $L^p$ properties for the spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal., 103 (1988), 1-38. doi: 10.1007/BF00292919.
[14]	L. D. Landau and E. M. Lifshitz, "Mechanics," third ed. A course of theoretical physics. Vol. 1, Pergamon Press, Oxford, 1976.
[15]	P.-L. Lions, Compactness in Boltzmann equation via Fourier integral operators and applications I, II, III, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461, 539-584.
[16]	C. D. Sogge and E. M. Stein, Averages of functions over hypersurfaces in $\mathbb{R}^{N}$. Averages over hypersurfaces II, Invent. Math., 82 (1985), 543-556 and 86 (1986), 233-242.
[17]	C. D. Sogge and E. M. Stein, Averages over hypersurfaces. Smoothness of generalized Radon transforms, J. Analyse Math., 54 (1990), 165-188. doi: 10.1007/BF02796147.
[18]	C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Rat. Mech. Anal., 173 (2004), 169-212. doi: 10.1007/s00205-004-0316-7.
[19]	B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm.. Part. Diff. Eqs., 19 (1994), 2057-2074. doi: 10.1080/03605309408821082.

show all references

References:

[1]	R. Alonso and E. Carneiro, Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform, Adv. Math., 223 (2010), 511-528. doi: 10.1016/j.aim.2009.08.017.
[2]	R. Alonso, E. Carneiro and I. M. Gamba, Convolution inequalities for the Boltzmann collision operator, Comm. Math. Physics, 298 (2010), 293-322. doi: 10.1007/s00220-010-1065-0.
[3]	R. Alonso, J. A. Canizo, I. M. Gamba, C. Mohout and S. Mischler, The Homogeneous Boltzmann equation for hard potentials with a cold thermostat, work in progress.
[4]	R. Alonso and I. M. Gamba, Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section, J. Stat. Phys., 137 (2009), 1147-1165. doi: 10.1007/s10955-009-9873-3.
[5]	R. Alonso and I. M. Gamba, Revision on classical solutions to the Cauchy Boltzmann problem for soft potentials, submitted for publication (2010).
[6]	R. Alonso, I. M. Gamba and S. H. Tharkabhushaman, Accuracy and consistency of Lagrangian based conservative spectral method for space-homogeneous Boltzmann equation, work in progress.
[7]	T. Carleman, "Problèmes Mathématiques dans la Théorie Cinétique des Gaz," Publ. Sci. Inst. Mittag-Leffler, 2. Almqvist and Wiksell, Uppsala, 1957.
[8]	C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Appl. Math. Sci. Springer-Verlag, Berlin, 1994.
[9]	I. M. Gamba, V. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys., 246 (2004), 503-541. doi: 10.1007/s00220-004-1051-5.
[10]	I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellians bounds for the spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal., 194 (2009), 253-282. doi: 10.1007/s00205-009-0250-9.
[11]	I. M. Gamba and S. H. Tharkabhushaman, Spectral-Lagrangian based methods applied to computation of non-equilibrium statistical states, Jour. Comp. Phys., 228 (2009), 2012-2036. doi: 10.1016/j.jcp.2008.09.033.
[12]	I. M. Gamba and S. H. Tharkabhushanam, Shock and boundary structure formation by spectral-lagrangian methods for the inhomogeneous Boltzmann transport equation, Jour. Comp. Math., 28 (2010), 430-460.
[13]	T. Gustafsson, Global $L^p$ properties for the spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal., 103 (1988), 1-38. doi: 10.1007/BF00292919.
[14]	L. D. Landau and E. M. Lifshitz, "Mechanics," third ed. A course of theoretical physics. Vol. 1, Pergamon Press, Oxford, 1976.
[15]	P.-L. Lions, Compactness in Boltzmann equation via Fourier integral operators and applications I, II, III, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461, 539-584.
[16]	C. D. Sogge and E. M. Stein, Averages of functions over hypersurfaces in $\mathbb{R}^{N}$. Averages over hypersurfaces II, Invent. Math., 82 (1985), 543-556 and 86 (1986), 233-242.
[17]	C. D. Sogge and E. M. Stein, Averages over hypersurfaces. Smoothness of generalized Radon transforms, J. Analyse Math., 54 (1990), 165-188. doi: 10.1007/BF02796147.
[18]	C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Rat. Mech. Anal., 173 (2004), 169-212. doi: 10.1007/s00205-004-0316-7.
[19]	B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm.. Part. Diff. Eqs., 19 (1994), 2057-2074. doi: 10.1080/03605309408821082.

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