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Diffusion and kinetic transport with very weak confinement
Global existence and long time behavior of the Ellipsoidal-Statistical-Fokker-Planck model for diatomic gases
1. | School of Mathematical Sciences, Capital Normal University, Beijing 100048, China |
2. | Institute of Applied Physics and Computational Mathematics, Beijing 100088, China |
We are concerned with the global existence and long time behavior of the solutions to the ES-FP model for diatomic gases proposed in [
References:
[1] |
M. A. AI-Gwaiz, Sturm-Liouville Theory and its Applications, Springer-Verlag, London, 2008. |
[2] |
P. Andriès, P. Le Tallec, J.-F. Perlat and B. Perthame,
The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B Fluids, 19 (2000), 813-830.
doi: 10.1016/S0997-7546(00)01103-1. |
[3] |
J. Bang and S.-B. Yun,
Stationary solutions for the ellipsoidal BGK model in a slab, J. Differential Equations, 261 (2016), 5803-5828.
doi: 10.1016/j.jde.2016.08.022. |
[4] |
W. Beckner,
A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc., 105 (1989), 397-400.
doi: 10.2307/2046956. |
[5] |
J. A. Carrillo, R. J. Duan and A. Moussa,
Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system, Kinet. Relat. Models, 4 (2011), 227-258.
doi: 10.3934/krm.2011.4.227. |
[6] |
C. Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press, New York, 1990.
doi: 10.1007/978-1-4899-7291-0.![]() ![]() ![]() |
[7] |
C. Cercignani, The Boltzmann Equation and its Application, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[8] |
P. Degond and B. Lucquin-Desreux,
The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci., 2 (1992), 167-182.
doi: 10.1142/S0218202592000119. |
[9] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996.
doi: 10.1137/1.9781611971477. |
[10] |
H. Grad,
On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[11] |
L. Gross,
Logarithmic Sobolev inequality, Amer. J. Math., 97 (1975), 1061-1083.
doi: 10.2307/2373688. |
[12] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[13] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[14] |
Y. Guo,
The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[15] |
L. H. Holway, Kinetic theory of shock structure using an ellipsoidal distribution function, in Rarefied Gas Dynamics Vol. I, Academic Press, New York, 1966,193–215. |
[16] |
P. Jenny, M. Torrihon and S. Heinz,
A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion, J. Comput. Phys., 229 (2010), 1077-1098.
doi: 10.1016/j.jcp.2009.10.008. |
[17] |
S. Kawashima,
The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.
doi: 10.1007/BF03167846. |
[18] |
H. L. Li, J. W. Sun, T. Yang and M. Y. Zhong, Large time behavior of solutions to Vlasov-Poisson-Landau (Fokker-Planck) equations, Sci Sin Math(in Chinese), 46 (2016), 981–1004. Available from: https://doi.org/10.1360/N012015-00230. |
[19] |
D. Liberzon and R. W. Brockett,
Spectral analysis of Fokker-Planck and related operators arising from linear stochastic differential equations, SIAM J. Control Optim., 38 (2000), 1453-1467.
doi: 10.1137/S0363012998338193. |
[20] |
L. Luo and H. J. Yu,
Spectrum analysis of the linear Fokker-Planck equation, Anal. Appl. (Singap.), 15 (2017), 313-331.
doi: 10.1142/S0219530515500219. |
[21] |
J. Mathiaud and L. Mieussens,
A Fokker-Planck Mode of the Boltzmann equation with correct Prandtl number, J. Stat. Phys., 162 (2016), 397-414.
doi: 10.1007/s10955-015-1404-9. |
[22] |
J. Mathiaud and L. Mieussens,
A Fokker-Planck model of the Boltzmann equation with correct Prandtl number for polyatomic gases, J. Stat. Phys., 168 (2017), 1031-1055.
doi: 10.1007/s10955-017-1837-4. |
[23] |
J. Nash,
Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.
doi: 10.2307/2372841. |
[24] |
S. J. Park and S.-B. Yun, Cauchy problem for the ellipsoidal-BGK model of the Boltzmann equation, J. Math. Phys., 57 (2016), 081512, 19 pp.
doi: 10.1063/1.4960745. |
[25] |
S. J. Park and S.-B. Yun,
Cauchy problem for the ellipsoidal BGK model for polyatomic particles, J. Differential Equations, 266 (2019), 7678-7708.
doi: 10.1016/j.jde.2018.12.013. |
[26] |
J. W. Sun and L. Jing,
Global existence and long time behavior of the ellipsoidal-Fokker-Planck equation, Appl. Anal., 98 (2019), 1605-1625.
doi: 10.1080/00036811.2018.1434154. |
[27] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, 1 (2002), 71–305. |
[28] |
N. Wiener, The Fourier Integral and Certain of its Applications, Reprint of the 1933 edition. With a foreword by Jean-Pierre Kahane. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988.
doi: 10.1017/CBO9780511662492. |
[29] |
T. Yang and H. J. Yu,
Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560.
doi: 10.1016/j.jde.2009.11.027. |
[30] |
T. Yang and H. J. Yu,
Optimal convergence rates of classical solutions for Vlasov-Poisson-Blltzmann system, Comm. Math. Phys., 301 (2011), 319-355.
doi: 10.1007/s00220-010-1142-4. |
[31] |
T. Yang and H. J. Yu,
Optimal convergence rates of Landau equation with external forcing in the whole space, Acta Math. Sci. Ser. B, 29 (2009), 1035-1062.
doi: 10.1016/S0252-9602(09)60085-0. |
[32] |
T. Yang and H. J. Yu,
Global solutions to the relativistic Landau-Maxwell system in the whole space, J. Math. Pures Appl., 97 (2012), 602-634.
doi: 10.1016/j.matpur.2011.09.006. |
[33] |
T. Yang and H. J. Yu,
Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM J. Math. Anal., 42 (2010), 459-488.
doi: 10.1137/090755796. |
[34] |
S.-B. Yun,
Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354.
doi: 10.1137/130932399. |
[35] |
S.-B. Yun,
Classical solutions for the ellipsoidal BGK model with fixed collision frequency, J. Differential Equations, 259 (2015), 6009-6037.
doi: 10.1016/j.jde.2015.07.016. |
[36] |
S.-B. Yun,
Ellipsoidal BGK model for polyatomic molecules near Maxwellians: A dichotomy in the dissipation estimate, J. Differential Equations, 266 (2019), 5566-5614.
doi: 10.1016/j.jde.2018.10.036. |
show all references
References:
[1] |
M. A. AI-Gwaiz, Sturm-Liouville Theory and its Applications, Springer-Verlag, London, 2008. |
[2] |
P. Andriès, P. Le Tallec, J.-F. Perlat and B. Perthame,
The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B Fluids, 19 (2000), 813-830.
doi: 10.1016/S0997-7546(00)01103-1. |
[3] |
J. Bang and S.-B. Yun,
Stationary solutions for the ellipsoidal BGK model in a slab, J. Differential Equations, 261 (2016), 5803-5828.
doi: 10.1016/j.jde.2016.08.022. |
[4] |
W. Beckner,
A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc., 105 (1989), 397-400.
doi: 10.2307/2046956. |
[5] |
J. A. Carrillo, R. J. Duan and A. Moussa,
Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system, Kinet. Relat. Models, 4 (2011), 227-258.
doi: 10.3934/krm.2011.4.227. |
[6] |
C. Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press, New York, 1990.
doi: 10.1007/978-1-4899-7291-0.![]() ![]() ![]() |
[7] |
C. Cercignani, The Boltzmann Equation and its Application, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[8] |
P. Degond and B. Lucquin-Desreux,
The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci., 2 (1992), 167-182.
doi: 10.1142/S0218202592000119. |
[9] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996.
doi: 10.1137/1.9781611971477. |
[10] |
H. Grad,
On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[11] |
L. Gross,
Logarithmic Sobolev inequality, Amer. J. Math., 97 (1975), 1061-1083.
doi: 10.2307/2373688. |
[12] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[13] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[14] |
Y. Guo,
The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[15] |
L. H. Holway, Kinetic theory of shock structure using an ellipsoidal distribution function, in Rarefied Gas Dynamics Vol. I, Academic Press, New York, 1966,193–215. |
[16] |
P. Jenny, M. Torrihon and S. Heinz,
A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion, J. Comput. Phys., 229 (2010), 1077-1098.
doi: 10.1016/j.jcp.2009.10.008. |
[17] |
S. Kawashima,
The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.
doi: 10.1007/BF03167846. |
[18] |
H. L. Li, J. W. Sun, T. Yang and M. Y. Zhong, Large time behavior of solutions to Vlasov-Poisson-Landau (Fokker-Planck) equations, Sci Sin Math(in Chinese), 46 (2016), 981–1004. Available from: https://doi.org/10.1360/N012015-00230. |
[19] |
D. Liberzon and R. W. Brockett,
Spectral analysis of Fokker-Planck and related operators arising from linear stochastic differential equations, SIAM J. Control Optim., 38 (2000), 1453-1467.
doi: 10.1137/S0363012998338193. |
[20] |
L. Luo and H. J. Yu,
Spectrum analysis of the linear Fokker-Planck equation, Anal. Appl. (Singap.), 15 (2017), 313-331.
doi: 10.1142/S0219530515500219. |
[21] |
J. Mathiaud and L. Mieussens,
A Fokker-Planck Mode of the Boltzmann equation with correct Prandtl number, J. Stat. Phys., 162 (2016), 397-414.
doi: 10.1007/s10955-015-1404-9. |
[22] |
J. Mathiaud and L. Mieussens,
A Fokker-Planck model of the Boltzmann equation with correct Prandtl number for polyatomic gases, J. Stat. Phys., 168 (2017), 1031-1055.
doi: 10.1007/s10955-017-1837-4. |
[23] |
J. Nash,
Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.
doi: 10.2307/2372841. |
[24] |
S. J. Park and S.-B. Yun, Cauchy problem for the ellipsoidal-BGK model of the Boltzmann equation, J. Math. Phys., 57 (2016), 081512, 19 pp.
doi: 10.1063/1.4960745. |
[25] |
S. J. Park and S.-B. Yun,
Cauchy problem for the ellipsoidal BGK model for polyatomic particles, J. Differential Equations, 266 (2019), 7678-7708.
doi: 10.1016/j.jde.2018.12.013. |
[26] |
J. W. Sun and L. Jing,
Global existence and long time behavior of the ellipsoidal-Fokker-Planck equation, Appl. Anal., 98 (2019), 1605-1625.
doi: 10.1080/00036811.2018.1434154. |
[27] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, 1 (2002), 71–305. |
[28] |
N. Wiener, The Fourier Integral and Certain of its Applications, Reprint of the 1933 edition. With a foreword by Jean-Pierre Kahane. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988.
doi: 10.1017/CBO9780511662492. |
[29] |
T. Yang and H. J. Yu,
Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560.
doi: 10.1016/j.jde.2009.11.027. |
[30] |
T. Yang and H. J. Yu,
Optimal convergence rates of classical solutions for Vlasov-Poisson-Blltzmann system, Comm. Math. Phys., 301 (2011), 319-355.
doi: 10.1007/s00220-010-1142-4. |
[31] |
T. Yang and H. J. Yu,
Optimal convergence rates of Landau equation with external forcing in the whole space, Acta Math. Sci. Ser. B, 29 (2009), 1035-1062.
doi: 10.1016/S0252-9602(09)60085-0. |
[32] |
T. Yang and H. J. Yu,
Global solutions to the relativistic Landau-Maxwell system in the whole space, J. Math. Pures Appl., 97 (2012), 602-634.
doi: 10.1016/j.matpur.2011.09.006. |
[33] |
T. Yang and H. J. Yu,
Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM J. Math. Anal., 42 (2010), 459-488.
doi: 10.1137/090755796. |
[34] |
S.-B. Yun,
Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354.
doi: 10.1137/130932399. |
[35] |
S.-B. Yun,
Classical solutions for the ellipsoidal BGK model with fixed collision frequency, J. Differential Equations, 259 (2015), 6009-6037.
doi: 10.1016/j.jde.2015.07.016. |
[36] |
S.-B. Yun,
Ellipsoidal BGK model for polyatomic molecules near Maxwellians: A dichotomy in the dissipation estimate, J. Differential Equations, 266 (2019), 5566-5614.
doi: 10.1016/j.jde.2018.10.036. |
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