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. 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. 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. 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)., (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, (1957).

[8]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Appl. Math. Sci. Springer-Verlag, (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. 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. 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. 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.

[13]

T. Gustafsson, Global $L^p$ properties for the spatially homogeneous Boltzmann equation,, Arch. Rat. Mech. Anal., 103 (1988), 1. doi: 10.1007/BF00292919.

[14]

L. D. Landau and E. M. Lifshitz, "Mechanics," third ed., A course of theoretical physics. Vol. \textbf{1}, 1 (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.

[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.

[17]

C. D. Sogge and E. M. Stein, Averages over hypersurfaces. Smoothness of generalized Radon transforms,, J. Analyse Math., 54 (1990), 165. 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. 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. 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. 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. 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. 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)., (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, (1957).

[8]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Appl. Math. Sci. Springer-Verlag, (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. 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. 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. 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.

[13]

T. Gustafsson, Global $L^p$ properties for the spatially homogeneous Boltzmann equation,, Arch. Rat. Mech. Anal., 103 (1988), 1. doi: 10.1007/BF00292919.

[14]

L. D. Landau and E. M. Lifshitz, "Mechanics," third ed., A course of theoretical physics. Vol. \textbf{1}, 1 (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.

[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.

[17]

C. D. Sogge and E. M. Stein, Averages over hypersurfaces. Smoothness of generalized Radon transforms,, J. Analyse Math., 54 (1990), 165. 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. 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. doi: 10.1080/03605309408821082.

[1]

Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049

[2]

Shaofei Wu, Mingqing Wang, Maozhu Jin, Yuntao Zou, Lijun Song. Uniform $L^1$ stability of the inelastic Boltzmann equation with large external force for hard potentials. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1005-1013. doi: 10.3934/dcdss.2019068

[3]

Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic & Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645

[4]

Yong-Kum Cho, Hera Yun. On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials. Kinetic & Related Models, 2012, 5 (4) : 769-786. doi: 10.3934/krm.2012.5.769

[5]

Simona Fornaro, Federica Gregorio, Abdelaziz Rhandi. Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2357-2372. doi: 10.3934/cpaa.2016040

[6]

Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2213-2219. doi: 10.3934/cpaa.2012.11.2213

[7]

Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51.

[8]

Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967

[9]

Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115

[10]

Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157

[11]

Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic & Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405

[12]

Pierre-Étienne Druet. Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 475-496. doi: 10.3934/dcdss.2015.8.475

[13]

Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365

[14]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715

[15]

Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053

[16]

Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115

[17]

Der-Chen Chang, Jie Xiao. $L^q$-Extensions of $L^p$-spaces by fractional diffusion equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905

[18]

Kevin Zumbrun. L resolvent bounds for steady Boltzmann's Equation. Kinetic & Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048

[19]

Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic & Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53

[20]

Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]