-
Previous Article
A hierarchy of models related to nanoflows and surface diffusion
- KRM Home
- This Issue
-
Next Article
Bounded solutions of the Boltzmann equation in the whole space
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 |
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. |
[1] |
Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1005-1013. doi: 10.3934/dcdss.2019068 |
[3] |
Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic and Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020 |
[4] |
Lvqiao Liu, Hao Wang. Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3113-3136. doi: 10.3934/cpaa.2020135 |
[5] |
Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic and Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645 |
[6] |
Yong-Kum Cho, Hera Yun. On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials. Kinetic and Related Models, 2012, 5 (4) : 769-786. doi: 10.3934/krm.2012.5.769 |
[7] |
Pierre Gervais. A spectral study of the linearized Boltzmann operator in $ L^2 $-spaces with polynomial and Gaussian weights. Kinetic and Related Models, 2021, 14 (4) : 725-747. doi: 10.3934/krm.2021022 |
[8] |
Simona Fornaro, Federica Gregorio, Abdelaziz Rhandi. Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2357-2372. doi: 10.3934/cpaa.2016040 |
[9] |
Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2213-2219. doi: 10.3934/cpaa.2012.11.2213 |
[10] |
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. |
[11] |
Mario Pulvirenti, Sergio Simonella. On the cardinality of collisional clusters for hard spheres at low density. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3903-3914. doi: 10.3934/dcds.2021021 |
[12] |
Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115 |
[13] |
Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967 |
[14] |
Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157 |
[15] |
Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic and Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405 |
[16] |
Fei Meng, Fang Liu. On the inelastic Boltzmann equation for soft potentials with diffusion. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5197-5217. doi: 10.3934/cpaa.2020233 |
[17] |
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 and Continuous Dynamical Systems - S, 2015, 8 (3) : 475-496. doi: 10.3934/dcdss.2015.8.475 |
[18] |
Koya Nishimura. Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1769-1782. doi: 10.3934/cpaa.2019083 |
[19] |
Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations and Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365 |
[20] |
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 and Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715 |
2020 Impact Factor: 1.432
Tools
Metrics
Other articles
by authors
[Back to Top]