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On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces
1. | Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China |
2. | Department of Mathematics, South China University of Technology, Guangzhou 510640 |
References:
[1] |
R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.
doi: 10.1007/s002050000083. |
[2] |
R. Alexandre R. 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. Rational Mech. Anal., 202 (2011), 599-661.
doi: 10.1007/s00205-011-0432-0. |
[3] |
R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010.
doi: 10.1016/j.jfa.2011.10.007. |
[4] |
R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, Local existence with mild regularity for the Boltzmann equation, Kinet. Relat. Models., 6 (2013), 1011-1041.
doi: 10.3934/krm.2013.6.1011. |
[5] |
D. Arsénio and N. Masmoudi, A new approach to velocity averaging lemmas in Besov spaces, J. Math.Pures Appl., 101 (2014), 495-551.
doi: 10.1016/j.matpur.2013.06.012. |
[6] |
H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[7] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[8] |
J. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998. |
[9] |
J. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Diff. Equs., 121 (1995), 314-328.
doi: 10.1006/jdeq.1995.1131. |
[10] |
R. Duan, The Boltzmann equation near equilibrium states in $R^n$, Methods Appl. Anal., 14 (2007), 227-249.
doi: 10.4310/MAA.2007.v14.n3.a2. |
[11] |
R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189.
doi: 10.1088/0951-7715/24/8/003. |
[12] |
R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: global existence and uniform stability in $L^2(H_x^N)$, J. Diff. Equs., 244 (2008), 3204-3234.
doi: 10.1016/j.jde.2007.11.006. |
[13] |
R. Duan, S. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann equation, arXiv:1310.2727. |
[14] |
R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[15] |
R. Duan and T. Yang, Stability of the one species Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal., 41 (2010), 2353-2387.
doi: 10.1137/090745775. |
[16] |
N. Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, Ann. Appl. Probab., 25 (2015), 860-897.
doi: 10.1214/14-AAP1012. |
[17] |
R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[18] |
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. |
[19] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[20] |
Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.
doi: 10.1007/s00205-003-0262-9. |
[21] |
Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[22] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[23] |
T. Kato, Quasi-linear equations of evolution with applications to partial differential equations, in: Spectral Theory and Differential Equations, in: Lecture Notes in Math., Springer-Verlag, Berlin, 448 (1975), 25-70. |
[24] |
T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Phys. D., 188 (2004), 178-192.
doi: 10.1016/j.physd.2003.07.011. |
[25] |
T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.
doi: 10.1007/s00220-003-1030-2. |
[26] |
H. J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley in Chichester, New York, 1987. |
[27] |
V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbbR_x^n$, Adv. in Math., 261 (2014), 274-332.
doi: 10.1016/j.aim.2014.04.012. |
[28] |
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. |
[29] |
T. Tao, Global well-posedness of the Benjamin-ono equation in $H^1(\mathbbR)$, J. Hyperbolic Diff. Equ., 1 (2004), 27-49.
doi: 10.1142/S0219891604000032. |
[30] |
H. Tang and Z. Liu, Continuous properties of the solution map for the Euler equations, J. Math. Phys., 55 (2014), 031504, 10pp.
doi: 10.1063/1.4867622. |
[31] |
H. Tang and Z. Liu, Well-posedness of the modfied Camassa-Holm equation in Besov spaces, Z. Angew. Math. Phys., 66 (2015), 1559-1580.
doi: 10.1007/s00033-014-0483-9. |
[32] |
H. Tang, Y. Zhao and Z. Liu, A note on the solution map for the periodic Camassa-Holm equation, Appl. Anal., 93 (2014), 1745-1760.
doi: 10.1080/00036811.2013.847923. |
[33] |
H. Tang, S. Shi and Z. Liu, The dependences on initial data for the b-family equation in critical Besov space, Monatsh. Math., 177 (2015), 471-492.
doi: 10.1007/s00605-014-0673-8. |
[34] |
H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[35] |
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. |
[36] |
S. Ukai, Solutions of the Boltzmann equation, Patterns and waves, Stud. Math. Appl., North-Holland, Amsterdam, 18 (1986), 37-96.
doi: 10.1016/S0168-2024(08)70128-0. |
[37] |
S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Anal. Appl. (Singap.), 4 (2006), 263-310.
doi: 10.1142/S0219530506000784. |
[38] |
S. Ukai, T. Yang and H. Zhao, Global solutions to the Boltzmann equation with external forces, Anal. Appl., 3 (2005), 157-193.
doi: 10.1142/S0219530505000522. |
[39] |
S. Ukai, T. Yang and H. Zhao, Convergence rate to stationary solutions for Boltzmann equation with external force, Chinese Ann. Math. Ser. B., 27 (2006), 363-378.
doi: 10.1007/s11401-005-0199-4. |
[40] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, I (2002), 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
[41] |
L. Xiong, T. Wang and L. Wang, Global existence and decay of solutions to the fokker-planck-boltzmann equation, Kinet. Relat. Models., 7 (2014), 169-194.
doi: 10.3934/krm.2014.7.169. |
show all references
References:
[1] |
R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.
doi: 10.1007/s002050000083. |
[2] |
R. Alexandre R. 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. Rational Mech. Anal., 202 (2011), 599-661.
doi: 10.1007/s00205-011-0432-0. |
[3] |
R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010.
doi: 10.1016/j.jfa.2011.10.007. |
[4] |
R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, Local existence with mild regularity for the Boltzmann equation, Kinet. Relat. Models., 6 (2013), 1011-1041.
doi: 10.3934/krm.2013.6.1011. |
[5] |
D. Arsénio and N. Masmoudi, A new approach to velocity averaging lemmas in Besov spaces, J. Math.Pures Appl., 101 (2014), 495-551.
doi: 10.1016/j.matpur.2013.06.012. |
[6] |
H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[7] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[8] |
J. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998. |
[9] |
J. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Diff. Equs., 121 (1995), 314-328.
doi: 10.1006/jdeq.1995.1131. |
[10] |
R. Duan, The Boltzmann equation near equilibrium states in $R^n$, Methods Appl. Anal., 14 (2007), 227-249.
doi: 10.4310/MAA.2007.v14.n3.a2. |
[11] |
R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189.
doi: 10.1088/0951-7715/24/8/003. |
[12] |
R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: global existence and uniform stability in $L^2(H_x^N)$, J. Diff. Equs., 244 (2008), 3204-3234.
doi: 10.1016/j.jde.2007.11.006. |
[13] |
R. Duan, S. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann equation, arXiv:1310.2727. |
[14] |
R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[15] |
R. Duan and T. Yang, Stability of the one species Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal., 41 (2010), 2353-2387.
doi: 10.1137/090745775. |
[16] |
N. Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, Ann. Appl. Probab., 25 (2015), 860-897.
doi: 10.1214/14-AAP1012. |
[17] |
R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[18] |
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. |
[19] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[20] |
Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.
doi: 10.1007/s00205-003-0262-9. |
[21] |
Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[22] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[23] |
T. Kato, Quasi-linear equations of evolution with applications to partial differential equations, in: Spectral Theory and Differential Equations, in: Lecture Notes in Math., Springer-Verlag, Berlin, 448 (1975), 25-70. |
[24] |
T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Phys. D., 188 (2004), 178-192.
doi: 10.1016/j.physd.2003.07.011. |
[25] |
T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.
doi: 10.1007/s00220-003-1030-2. |
[26] |
H. J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley in Chichester, New York, 1987. |
[27] |
V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbbR_x^n$, Adv. in Math., 261 (2014), 274-332.
doi: 10.1016/j.aim.2014.04.012. |
[28] |
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. |
[29] |
T. Tao, Global well-posedness of the Benjamin-ono equation in $H^1(\mathbbR)$, J. Hyperbolic Diff. Equ., 1 (2004), 27-49.
doi: 10.1142/S0219891604000032. |
[30] |
H. Tang and Z. Liu, Continuous properties of the solution map for the Euler equations, J. Math. Phys., 55 (2014), 031504, 10pp.
doi: 10.1063/1.4867622. |
[31] |
H. Tang and Z. Liu, Well-posedness of the modfied Camassa-Holm equation in Besov spaces, Z. Angew. Math. Phys., 66 (2015), 1559-1580.
doi: 10.1007/s00033-014-0483-9. |
[32] |
H. Tang, Y. Zhao and Z. Liu, A note on the solution map for the periodic Camassa-Holm equation, Appl. Anal., 93 (2014), 1745-1760.
doi: 10.1080/00036811.2013.847923. |
[33] |
H. Tang, S. Shi and Z. Liu, The dependences on initial data for the b-family equation in critical Besov space, Monatsh. Math., 177 (2015), 471-492.
doi: 10.1007/s00605-014-0673-8. |
[34] |
H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[35] |
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. |
[36] |
S. Ukai, Solutions of the Boltzmann equation, Patterns and waves, Stud. Math. Appl., North-Holland, Amsterdam, 18 (1986), 37-96.
doi: 10.1016/S0168-2024(08)70128-0. |
[37] |
S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Anal. Appl. (Singap.), 4 (2006), 263-310.
doi: 10.1142/S0219530506000784. |
[38] |
S. Ukai, T. Yang and H. Zhao, Global solutions to the Boltzmann equation with external forces, Anal. Appl., 3 (2005), 157-193.
doi: 10.1142/S0219530505000522. |
[39] |
S. Ukai, T. Yang and H. Zhao, Convergence rate to stationary solutions for Boltzmann equation with external force, Chinese Ann. Math. Ser. B., 27 (2006), 363-378.
doi: 10.1007/s11401-005-0199-4. |
[40] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, I (2002), 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
[41] |
L. Xiong, T. Wang and L. Wang, Global existence and decay of solutions to the fokker-planck-boltzmann equation, Kinet. Relat. Models., 7 (2014), 169-194.
doi: 10.3934/krm.2014.7.169. |
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