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Optimal time decay of the non cut-off Boltzmann equation in the whole space
1. | University of Pennsylvania, Department of Mathematics, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104, United States |
References:
[1] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions,, Arch. Ration. Mech. Anal., 202 (2011), 599.
doi: 10.1007/s00205-011-0432-0. |
[2] |
Russel E. Caflisch, The Boltzmann equation with a soft potential. I, II,, Comm. Math. Phys., 74 (1980), 71.
doi: 10.1007/BF01197579. |
[3] |
Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case,, Arch. Ration. Mech. Anal., 203 (2012), 343.
doi: 10.1007/s00205-011-0482-3. |
[4] |
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. |
[5] |
R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations,, Nonlinearity, 24 (2011), 2165.
doi: 10.1088/0951-7715/24/8/003. |
[6] |
R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_\varepsilon$($H^N_x$),, J. Differential Equations, 244 (2008), 3204.
doi: 10.1016/j.jde.2007.11.006. |
[7] |
R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$,, Arch. Rational Mech. Anal., 199 (2011), 291.
doi: 10.1007/s00205-010-0318-6. |
[8] |
R. Duan and R. M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Pure Appl. Math., 64 (2011), 1497.
|
[9] |
R. Duan, S. Ukai, T. Yang and H. Zhao, Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications,, Comm. Math. Phys., 277 (2008), 189.
doi: 10.1007/s00220-007-0366-4. |
[10] |
R. T. Glassey, "The Cauchy Problem in Kinetic Theory,", Society for Industrial and Applied Mathematics (SIAM), (1996).
|
[11] |
P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off,, J. Amer. Math. Soc., 24 (2011), 771.
doi: 10.1090/S0894-0347-2011-00697-8. |
[12] |
P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions,, Proc. Nat. Acad. Sci. U. S. A., 107 (2010), 5744.
doi: 10.1073/pnas.1001185107. |
[13] |
P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production,, Advances in Math., 227 (2011), 2349.
doi: 10.1016/j.aim.2011.05.005. |
[14] |
Y. Guo, The Landau equation in a periodic box,, Comm. Math. Phys., 231 (2002), 391.
doi: 10.1007/s00220-002-0729-9. |
[15] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians,, Invent. Math., 153 (2003), 593.
doi: 10.1007/s00222-003-0301-z. |
[16] |
Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081.
doi: 10.1512/iumj.2004.53.2574. |
[17] |
Shuichi Kawashima, The Boltzmann equation and thirteen moments,, Japan J. Appl. Math., 7 (1990), 301.
|
[18] |
C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff,, J. Math. Pures Appl. (9), 87 (2007), 515.
|
[19] |
R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Math. Phys., 268 (2006), 543.
doi: 10.1007/s00220-006-0109-y. |
[20] |
Robert M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft-potentials,, Comm. Math. Phys., 300 (2010), 529.
doi: 10.1007/s00220-010-1129-1. |
[21] |
Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian,, Comm. Partial Differential Equations, 31 (2006), 417.
doi: 10.1080/03605300500361545. |
[22] |
Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian,, Arch. Ration. Mech. Anal., 187 (2008), 287.
doi: 10.1007/s00205-007-0067-3. |
[23] |
R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $mathbbR^3_x$,, Kinetic and Related Models, 5 (2012), 383. Google Scholar |
[24] |
M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations,", Applied Mathematical Sciences, 117 (1997).
|
[25] |
S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179.
doi: 10.3792/pja/1195519027. |
[26] |
Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential,, Publ. Res. Inst. Math. Sci., 18 (1982), 477.
doi: 10.2977/prims/1195183569. |
[27] |
C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71.
|
[28] |
C. Villani, "Hypocoercivity,", Mem. Amer. Math. Soc., 202 (2009).
|
show all references
References:
[1] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions,, Arch. Ration. Mech. Anal., 202 (2011), 599.
doi: 10.1007/s00205-011-0432-0. |
[2] |
Russel E. Caflisch, The Boltzmann equation with a soft potential. I, II,, Comm. Math. Phys., 74 (1980), 71.
doi: 10.1007/BF01197579. |
[3] |
Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case,, Arch. Ration. Mech. Anal., 203 (2012), 343.
doi: 10.1007/s00205-011-0482-3. |
[4] |
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. |
[5] |
R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations,, Nonlinearity, 24 (2011), 2165.
doi: 10.1088/0951-7715/24/8/003. |
[6] |
R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_\varepsilon$($H^N_x$),, J. Differential Equations, 244 (2008), 3204.
doi: 10.1016/j.jde.2007.11.006. |
[7] |
R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$,, Arch. Rational Mech. Anal., 199 (2011), 291.
doi: 10.1007/s00205-010-0318-6. |
[8] |
R. Duan and R. M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Pure Appl. Math., 64 (2011), 1497.
|
[9] |
R. Duan, S. Ukai, T. Yang and H. Zhao, Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications,, Comm. Math. Phys., 277 (2008), 189.
doi: 10.1007/s00220-007-0366-4. |
[10] |
R. T. Glassey, "The Cauchy Problem in Kinetic Theory,", Society for Industrial and Applied Mathematics (SIAM), (1996).
|
[11] |
P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off,, J. Amer. Math. Soc., 24 (2011), 771.
doi: 10.1090/S0894-0347-2011-00697-8. |
[12] |
P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions,, Proc. Nat. Acad. Sci. U. S. A., 107 (2010), 5744.
doi: 10.1073/pnas.1001185107. |
[13] |
P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production,, Advances in Math., 227 (2011), 2349.
doi: 10.1016/j.aim.2011.05.005. |
[14] |
Y. Guo, The Landau equation in a periodic box,, Comm. Math. Phys., 231 (2002), 391.
doi: 10.1007/s00220-002-0729-9. |
[15] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians,, Invent. Math., 153 (2003), 593.
doi: 10.1007/s00222-003-0301-z. |
[16] |
Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081.
doi: 10.1512/iumj.2004.53.2574. |
[17] |
Shuichi Kawashima, The Boltzmann equation and thirteen moments,, Japan J. Appl. Math., 7 (1990), 301.
|
[18] |
C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff,, J. Math. Pures Appl. (9), 87 (2007), 515.
|
[19] |
R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Math. Phys., 268 (2006), 543.
doi: 10.1007/s00220-006-0109-y. |
[20] |
Robert M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft-potentials,, Comm. Math. Phys., 300 (2010), 529.
doi: 10.1007/s00220-010-1129-1. |
[21] |
Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian,, Comm. Partial Differential Equations, 31 (2006), 417.
doi: 10.1080/03605300500361545. |
[22] |
Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian,, Arch. Ration. Mech. Anal., 187 (2008), 287.
doi: 10.1007/s00205-007-0067-3. |
[23] |
R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $mathbbR^3_x$,, Kinetic and Related Models, 5 (2012), 383. Google Scholar |
[24] |
M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations,", Applied Mathematical Sciences, 117 (1997).
|
[25] |
S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179.
doi: 10.3792/pja/1195519027. |
[26] |
Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential,, Publ. Res. Inst. Math. Sci., 18 (1982), 477.
doi: 10.2977/prims/1195183569. |
[27] |
C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71.
|
[28] |
C. Villani, "Hypocoercivity,", Mem. Amer. Math. Soc., 202 (2009).
|
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