American Institute of Mathematical Sciences

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April  2020, 13(2): 373-400. doi: 10.3934/krm.2020013

Global existence and long time behavior of the Ellipsoidal-Statistical-Fokker-Planck model for diatomic gases

Lei Jing 1,, and  Jiawei Sun 2,

1. 

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
2. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

* Corresponding author: Lei Jing

Received  May 2019 Revised  November 2019 Published  January 2020

Fund Project: The research was supported by National Natural Science Foundation of China grants No. 11671384, 11871047, 11931010, 11671120 and by the key research project of the Academy for Multidisciplinary Studies, Capital Normal University.

We are concerned with the global existence and long time behavior of the solutions to the ES-FP model for diatomic gases proposed in [22]. The global existence of the solutions for this model near the global Maxwellian is established by nonlinear energy method based on the macro-micro decomposition. An algebraic convergence rate in time of the solutions to the equilibrium state is obtained by constructing the compensating function. Since the density distribution function $ F(t, x, v, I) $ also contains energy variable $ I $, we derive more general Poincaré inequality including variables $ v, I $ on $ \mathbb{R}^3\times \mathbb{R}^+ $ to establish the coercivity estimate of the linearized operator.

Keywords: Ellipsoidal-Statistical model, Fokker-Planck equation, diatomic gases, compensating function, generalized Poincaré inequality.
Mathematics Subject Classification: Primary: 35A01 35B40 35Q84; Secondary: 82C40.
Citation: Lei Jing, Jiawei Sun. Global existence and long time behavior of the Ellipsoidal-Statistical-Fokker-Planck model for diatomic gases. Kinetic & Related Models, 2020, 13 (2) : 373-400. doi: 10.3934/krm.2020013
References:
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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.  Google Scholar

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C. Cercignani, The Boltzmann Equation and its Application, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

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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.  Google Scholar

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H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

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J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

S.-B. Yun, Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354.  doi: 10.1137/130932399.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. A. AI-Gwaiz, Sturm-Liouville Theory and its Applications, Springer-Verlag, London, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc., 105 (1989), 397-400.  doi: 10.2307/2046956.  Google Scholar

[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.  Google Scholar

[6] C. Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press, New York, 1990.  doi: 10.1007/978-1-4899-7291-0.  Google Scholar
[7]

C. Cercignani, The Boltzmann Equation and its Application, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[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.  Google Scholar

[9]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[10]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.  Google Scholar

[11]

L. Gross, Logarithmic Sobolev inequality, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.  Google Scholar

[12]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[13]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.  doi: 10.1007/s00222-003-0301-z.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.  doi: 10.1007/BF03167846.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

S.-B. Yun, Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354.  doi: 10.1137/130932399.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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