-
Previous Article
Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$
- KRM Home
- This Issue
-
Next Article
Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis
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), 599-661.
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), 71-95, 97-109.
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), 343-377.
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), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[5] |
R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189.
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), 3204-3234.
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), 291-328.
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), 1497-1546. |
[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), 189-236.
doi: 10.1007/s00220-007-0366-4. |
[10] |
R. T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 771-847.
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), 5744-5749.
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), 2349-2384.
doi: 10.1016/j.aim.2011.05.005. |
[14] |
Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.
doi: 10.1007/s00220-002-0729-9. |
[15] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[16] |
Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[17] |
Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320. |
[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), 515-535. |
[19] |
R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567.
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), 529-597.
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), 417-429.
doi: 10.1080/03605300500361545. |
[22] |
Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
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), 383-415. |
[24] |
M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117, Springer-Verlag, New York, 1997. |
[25] |
S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.
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), 477-519 (57-99).
doi: 10.2977/prims/1195183569. |
[27] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics," Vol. I, North-Holland, Amsterdam, (2002), 71-305. |
[28] |
C. Villani, "Hypocoercivity," Mem. Amer. Math. Soc., 202 (2009), iv+141. |
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), 599-661.
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), 71-95, 97-109.
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), 343-377.
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), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[5] |
R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189.
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), 3204-3234.
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), 291-328.
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), 1497-1546. |
[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), 189-236.
doi: 10.1007/s00220-007-0366-4. |
[10] |
R. T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 771-847.
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), 5744-5749.
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), 2349-2384.
doi: 10.1016/j.aim.2011.05.005. |
[14] |
Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.
doi: 10.1007/s00220-002-0729-9. |
[15] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[16] |
Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[17] |
Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320. |
[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), 515-535. |
[19] |
R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567.
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), 529-597.
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), 417-429.
doi: 10.1080/03605300500361545. |
[22] |
Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
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), 383-415. |
[24] |
M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117, Springer-Verlag, New York, 1997. |
[25] |
S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.
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), 477-519 (57-99).
doi: 10.2977/prims/1195183569. |
[27] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics," Vol. I, North-Holland, Amsterdam, (2002), 71-305. |
[28] |
C. Villani, "Hypocoercivity," Mem. Amer. Math. Soc., 202 (2009), iv+141. |
[1] |
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 |
[2] |
Ricardo Weder, Dimitri Yafaev. Inverse scattering at a fixed energy for long-range potentials. Inverse Problems and Imaging, 2007, 1 (1) : 217-224. doi: 10.3934/ipi.2007.1.217 |
[3] |
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 |
[4] |
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 |
[5] |
Yingzhe Fan, Yuanjie Lei. The Boltzmann equation with frictional force for very soft potentials in the whole space. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4303-4329. doi: 10.3934/dcds.2019174 |
[6] |
Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235 |
[7] |
Wei-Xi Li, Lvqiao Liu. Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off. Kinetic and Related Models, 2020, 13 (5) : 1029-1046. doi: 10.3934/krm.2020036 |
[8] |
Zheng-an Yao, Yu-Long Zhou. High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials. Kinetic and Related Models, 2020, 13 (3) : 435-478. doi: 10.3934/krm.2020015 |
[9] |
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 |
[10] |
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 |
[11] |
Robert M. Strain, Keya Zhu. Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$. Kinetic and Related Models, 2012, 5 (2) : 383-415. doi: 10.3934/krm.2012.5.383 |
[12] |
Radjesvarane Alexandre, Jie Liao, Chunjin Lin. Some a priori estimates for the homogeneous Landau equation with soft potentials. Kinetic and Related Models, 2015, 8 (4) : 617-650. doi: 10.3934/krm.2015.8.617 |
[13] |
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 |
[14] |
Juan Kalemkerian, Andrés Sosa. Long-range dependence in the volatility of returns in Uruguayan sovereign debt indices. Journal of Dynamics and Games, 2020, 7 (3) : 225-237. doi: 10.3934/jdg.2020016 |
[15] |
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential. Kinetic and Related Models, 2011, 4 (4) : 919-934. doi: 10.3934/krm.2011.4.919 |
[16] |
Laurence Cherfils, Stefania Gatti, Alain Miranville. Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2261-2290. doi: 10.3934/cpaa.2012.11.2261 |
[17] |
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 |
[18] |
Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328 |
[19] |
Renjun Duan, Shuangqian Liu, Tong Yang, Huijiang Zhao. Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials. Kinetic and Related Models, 2013, 6 (1) : 159-204. doi: 10.3934/krm.2013.6.159 |
[20] |
Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705 |
2020 Impact Factor: 1.432
Tools
Metrics
Other articles
by authors
[Back to Top]