EXISTENCE OF WEAK SOLUTIONS TO THE BGK EQUATION AND AN APPROXIMATE CONSERVATION LAWS WITH LARGE INITIAL DATA

Yu Dong; Zaihong Jiang; Li Li

doi:10.11948/20230129

2024 Volume 14 Issue 1

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Yu Dong, Zaihong Jiang, Li Li. EXISTENCE OF WEAK SOLUTIONS TO THE BGK EQUATION AND AN APPROXIMATE CONSERVATION LAWS WITH LARGE INITIAL DATA[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 162-181. doi: 10.11948/20230129

Citation:

Yu Dong, Zaihong Jiang, Li Li. EXISTENCE OF WEAK SOLUTIONS TO THE BGK EQUATION AND AN APPROXIMATE CONSERVATION LAWS WITH LARGE INITIAL DATA[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 162-181. doi: 10.11948/20230129

EXISTENCE OF WEAK SOLUTIONS TO THE BGK EQUATION AND AN APPROXIMATE CONSERVATION LAWS WITH LARGE INITIAL DATA

1.
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China
2.
School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China

Author Bio: Email: 2011071009@nbu.edu.cn(Y. Dong); Email: lili2@nbu.edu.cn(L. Li)
Corresponding author: Email: jzhong@zjnu.cn(Z. Jiang)
Fund Project: The second named author was partially supported by National Natural Science Foundation of China (No. 12071439) and Zhejiang Provincial Natural Science Foundation of China (No. LY19A010016). The third named author was partially supported by National Natural Science Foundation of China (No. 12271276)

Abstract

Abstract

This paper studies the Cauchy problem of a BGK model and the corresponding nonlinear hyperbolic conservation laws. Given bounded initial data for the kinetic equation, the existence of weak solutions to the BGK model is obtained by the time-splitting method. Moreover, weak solutions to the limiting hyperbolic system are obtained by passing the relaxation parameter to zero in a modified BGK model.
- BGK equation /
- conservation laws /
- existence /
- time splitting method
MSC: 35A01, 35D30, 35E15, 35L65, 35Q20

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References

[1]	R. A. Adams and J. F. Fournier, Sobolev Spaces. Second Edition, Pure and Applied Mathematics 140. Amsterdam: Academic Press, 2003, xiv+305pp. Google Scholar
[2]	R. Bianchini, Relative entropy in diffusive relaxation for a class of discrete velocities BGK models, Commun. Math. Sci., 2021, 19(1), 39–54. doi: 10.4310/CMS.2021.v19.n1.a2 CrossRef Google Scholar
[3]	S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math., 2005, 161(1), 223–342. doi: 10.4007/annals.2005.161.223 CrossRef Google Scholar
[4]	F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000, x+162pp. Google Scholar
[5]	A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, Oxford, 2000, xii+250pp. Google Scholar
[6]	P. Buttá, M. Hauray and M. Pulvirenti, Particle approximation of the BGK equation, Arch. Ration. Mech. Anal., 2021, 240(2), 785–808. doi: 10.1007/s00205-021-01621-y CrossRef Google Scholar
[7]	G. Q. Chen and Y. Lu, The study on application way of the compensated compactness theory, Chinese Sci. Bull., 1989, 34, 15–19. doi: 10.1360/csb1989-34-1-15 CrossRef Google Scholar
[8]	J. F. Coulombel and T. Goudon, Entropy-based moment closure for kinetic equations, Riemann problem and invariant regions, J. Hyperbolic Differ. Equ., 2006, 3(4), 649–671. doi: 10.1142/S0219891606000951 CrossRef Google Scholar
[9]	R. E. Edwards, Functional Analysis. Theory and Applications, New York-Toronto-London: Holt, Rinehart and Winston, 1965, xiii+781pp. Google Scholar
[10]	J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 1965, 18, 697–715. doi: 10.1002/cpa.3160180408 CrossRef Google Scholar
[11]	H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Expo. Math., 2010, 28(4), 385–394. doi: 10.1016/j.exmath.2010.03.001 CrossRef Google Scholar
[12]	J. Hua, Z. Jiang and T. Yang, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, Arch. Ration. Mech. Anal., 2010, 196(2), 433–454. doi: 10.1007/s00205-009-0266-1 CrossRef Google Scholar
[13]	B. H. Hwang, T. Ruggeri and S. B. Yun, On a relativistic BGK model for polyatomic gases near equilibrium, SIAM J. Math. Anal., 2022, 54(3), 2906–2947. doi: 10.1137/21M1404946 CrossRef Google Scholar
[14]	P. D. Lax, Hyperbolic systems of conservation laws. II, Commun. Pure Appl. Math., 1957, 10, 537–566. doi: 10.1002/cpa.3160100406 CrossRef Google Scholar
[15]	C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 1996, 83(5–6), 1021–1065. doi: 10.1007/BF02179552 CrossRef Google Scholar
[16]	T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys., 1977, 57, 135–148. doi: 10.1007/BF01625772 CrossRef Google Scholar
[17]	T. P. Liu and T. Yang, Weak solutions of general systems of hyperbolic conservation laws, Comm. Math. Phys., 2002, 230(2), 289–327. doi: 10.1007/s00220-002-0705-4 CrossRef Google Scholar
[18]	Y. Lu, Hyperbolic Conservation Laws and the Compensated Compactness Method, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 2003, xii+241pp. Google Scholar
[19]	Y. Lu, Existence of global entropy solutions of a nonstrictly hyperbolic system, Arch. Ration. Mech. Anal., 2005, 178(2), 287–299. doi: 10.1007/s00205-005-0379-0 CrossRef Google Scholar
[20]	M. Perepelitsa, A kinetic model for approximately isentropic solutions of the Euler equations, J. Differential Equations, 2016, 260(11), 8229–8241. doi: 10.1016/j.jde.2016.02.022 CrossRef Google Scholar
[21]	Q. Sun, Y. Lu and C. Klingenberg, Christian Global weak solutions for a nonlinear hyperbolic system, Acta Math. Sci., 2020, 40, 1185–1194. doi: 10.1007/s10473-020-0502-1 CrossRef Google Scholar
[22]	L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., Pitman, Boston, Mass. -London, 1979, 39, 136–212. Google Scholar
[23]	A. Vasseur, Convergence of a semi-discrete kinetic scheme for the system of isentropic gas dynamics with $\gamma=3$, Indiana Univ. Math. J., 1999, 48(1), 347–364. $\gamma=3$" target="_blank">Google Scholar
[24]	G. Wang, J. Liu and L. Zhao, The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws, Bound. Value Probl., 2017, 107, 16 pp. Google Scholar
[25]	T. Yang, C. Zhu and H. Zhao, Compactness framework of L^p approximate solutions for scalar conservation laws, J. Math. Anal. Appl., 1998, 220(1), 164–186. doi: 10.1006/jmaa.1997.5845 CrossRef Google Scholar
[26]	Q. Zhang and Y. Hu, Self-similar solutions to the spherically-symmetric Euler equations with a two-constant equation of state, Indian J. Pure Appl. Math., 2019, 50(1), 35–49. doi: 10.1007/s13226-019-0305-z CrossRef Google Scholar