# American Institute of Mathematical Sciences

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

## Gain of integrability for the Boltzmann collisional operator

 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

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.
Citation: Ricardo J. Alonso, Irene M. Gamba. Gain of integrability for the Boltzmann collisional operator. Kinetic & 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.  Google Scholar [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.  Google Scholar [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., ().   Google Scholar [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.  Google Scholar [5] R. Alonso and I. M. Gamba, Revision on classical solutions to the Cauchy Boltzmann problem for soft potentials, submitted for publication (2010). Google Scholar [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., ().   Google Scholar [7] T. Carleman, "Problèmes Mathématiques dans la Théorie Cinétique des Gaz," Publ. Sci. Inst. Mittag-Leffler, 2. Almqvist and Wiksell, Uppsala, 1957.  Google Scholar [8] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Appl. Math. Sci. Springer-Verlag, Berlin, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] L. D. Landau and E. M. Lifshitz, "Mechanics," third ed. A course of theoretical physics. Vol. 1, Pergamon Press, Oxford, 1976.  Google Scholar [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.  Google Scholar [16] C. D. Sogge and E. M. Stein, Averages of functions over hypersurfaces in $\mathbbR^n$. Averages over hypersurfaces II, Invent. Math., 82 (1985), 543-556 and 86 (1986), 233-242.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm.. Part. Diff. Eqs., 19 (1994), 2057-2074. doi: 10.1080/03605309408821082.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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., ().   Google Scholar [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.  Google Scholar [5] R. Alonso and I. M. Gamba, Revision on classical solutions to the Cauchy Boltzmann problem for soft potentials, submitted for publication (2010). Google Scholar [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., ().   Google Scholar [7] T. Carleman, "Problèmes Mathématiques dans la Théorie Cinétique des Gaz," Publ. Sci. Inst. Mittag-Leffler, 2. Almqvist and Wiksell, Uppsala, 1957.  Google Scholar [8] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Appl. Math. Sci. Springer-Verlag, Berlin, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] L. D. Landau and E. M. Lifshitz, "Mechanics," third ed. A course of theoretical physics. Vol. 1, Pergamon Press, Oxford, 1976.  Google Scholar [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.  Google Scholar [16] C. D. Sogge and E. M. Stein, Averages of functions over hypersurfaces in $\mathbbR^n$. Averages over hypersurfaces II, Invent. Math., 82 (1985), 543-556 and 86 (1986), 233-242.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm.. Part. Diff. Eqs., 19 (1994), 2057-2074. doi: 10.1080/03605309408821082.  Google Scholar
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