THE DISSIPATIVE CONDITION OF THE FIRST ORDER 3 × 3 HYPERBOLIC SYSTEM WITH CONSTANT COEFFICIENTS

Shuxin Zhang; Fangqi Chen; Zejun Wang

doi:10.11948/20240270

2025 Volume 15 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(3): 1483-1502

Next Article Previous Article

Shuxin Zhang, Fangqi Chen, Zejun Wang. THE DISSIPATIVE CONDITION OF THE FIRST ORDER 3 × 3 HYPERBOLIC SYSTEM WITH CONSTANT COEFFICIENTS[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1483-1502. doi: 10.11948/20240270

Citation:

Shuxin Zhang, Fangqi Chen, Zejun Wang. THE DISSIPATIVE CONDITION OF THE FIRST ORDER 3 × 3 HYPERBOLIC SYSTEM WITH CONSTANT COEFFICIENTS[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1483-1502. doi: 10.11948/20240270

THE DISSIPATIVE CONDITION OF THE FIRST ORDER 3 × 3 HYPERBOLIC SYSTEM WITH CONSTANT COEFFICIENTS

1.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
2.
Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing 211106, China

Author Bio: Email: zhangshuxin@nuaa.edu.cn(S. Zhang); Email: fangqichen1963@126.com(F. Chen)
Corresponding author: Email: wangzejun@gmail.com(Z. Wang)
Fund Project: The authors were supported by the National Natural Science Foundation of China (grant No. 12172166, 11671193)

Abstract

Abstract

In this paper, we study the dissipative property of the first order $3 \times 3$ hyperbolic system with constant coefficients. For the corresponding $n\times n$ system, when the coefficients matrices are symmetric, it has been studied in [16] and the well-know Kawashima-Shizuta condition is obtained. When $n=3$ and for asymmetric system, we give a sufficient condition for the system to be strictly dissipative.
- 3×3 hyperbolic system /
- dissipative property /
- Kawashima-Shizuta condition
MSC: 35L40, 35L45, 35L65

FullText(HTML)

References

[1]	L. V. Ahlfors, Complex Analysis Third Edition, New York, McGraw-Hill, 1979. Google Scholar
[2]	D. Amadori and G. Guerra, Global weak solutions for systems of balance laws, Appl. Math. Letters, 1999, 12(6), 123–127. doi: 10.1016/S0893-9659(99)00090-7 CrossRef Google Scholar
[3]	C. M. Dafermos, BV solutions for hyperbolic systems of balance laws with relaxation, J. Differ. Equ., 2013, 255(8), 2521–2533. doi: 10.1016/j.jde.2013.07.002 CrossRef Google Scholar
[4]	C. M. Dafermos and L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J., 1982, 31, 471–491. doi: 10.1512/iumj.1982.31.31039 CrossRef Google Scholar
[5]	R. Duan and H. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity, Indiana Univ. Math. J., 2008, 57(5), 2299–2319. doi: 10.1512/iumj.2008.57.3326 CrossRef Google Scholar
[6]	K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 1954, 7, 345–392. doi: 10.1002/cpa.3160070206 CrossRef Google Scholar
[7]	Z. Gao, Z. Tan and G. Wu, Global existence and convergence rates of smooth solutions for the 3-D compressible magnetohy-drodynamic equations without heat conductivity, Acta Math. Sci. Ser. B (Engl. Ed. ), 2014, 34(1), 93–106. Google Scholar
[8]	B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal., 2003, 169(2), 89–117. doi: 10.1007/s00205-003-0257-6 CrossRef Google Scholar
[9]	L. Hsiao and T. Li, Global smooth solution of Cauchy problems for a class of quasilinear hyperbolic systems, Chin. Ann. Math., Ser. B, 1983, 4(1), 107–115. Google Scholar
[10]	B. Karine and Z. Enrique, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Ration. Mech. Anal., 2011, 199(1), 177–227. doi: 10.1007/s00205-010-0321-y CrossRef Google Scholar
[11]	S. Kawashima, System of a Hyperbolic-Parabolic Composite Type with Applications to the Equations of Manetohydro-Dynamics[Ph. D. Thesis], Kyoto, Kyoto University, 1983. Google Scholar
[12]	S. Kawashima and W. Yong, Dissipative structure and entropy for hyperbolic systems of balancelaws, Arch. Ration. Mech. Anal., 2004, 174(3), 345–364. doi: 10.1007/s00205-004-0330-9 CrossRef Google Scholar
[13]	S. Kawashima and W. Yong, Decay estimates forhyperbolic balance laws, Z. Anal. Anwend., 2009, 28(1), 1–33. doi: 10.4171/zaa/1369 CrossRef Google Scholar
[14]	T. Li, Global classical solutions for quasilinear hyperbolic systems, Masson, Paris, RAM Res. Appl. Math., 1994, 135–165. Google Scholar
[15]	T. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys., 1998, 196(1), 145–173. doi: 10.1007/s002200050418 CrossRef Google Scholar
[16]	Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 1985, 14(2), 249–275. Google Scholar
[17]	T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of Electro-Magneto-Fluid dynamics, Japan J. Appl. Math., 1984, 1(2), 435–457. doi: 10.1007/BF03167068 CrossRef Google Scholar
[18]	W. Wang and Z. Wang, The pointwise estimates to solutions for 1-dimensional linear thermoelastic system with second sound, J. Math. Anal. Appl., 2007, 326(2), 1061–1075. doi: 10.1016/j.jmaa.2006.03.038 CrossRef Google Scholar
[19]	W. Wang and X. Xu, Global well-posedness for systems of hyperbolic-parabolic composite type with center manifold, J. Math. Anal. Appl., 2020, 490(2), 124320. doi: 10.1016/j.jmaa.2020.124320 CrossRef Google Scholar
[20]	W. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differential Equation, 2001, 173(2), 410–450. doi: 10.1006/jdeq.2000.3937 CrossRef Google Scholar
[21]	W. Wang and X. Yang, The pointwise estimates of solutions for Navier-Stokes equations in even space-dimensions, J. Hyperbolic Differ. Equ., 2005, 2(3), 673–695. doi: 10.1142/S0219891605000580 CrossRef Google Scholar
[22]	G. Wu, Z. Tan and J. Huang, Global existence and large time behavior for the system of compressible adiabatic flow through porous media in $\mathbb{R}^3$, J. Differ. Equ., 2013, 255(8), 865–880. $\mathbb{R}^3$" target="_blank">Google Scholar
[23]	S. Zhang, F. Chen and Z. Wang, The dissipative property of the first order $2\times 2$ hyperbolic system with constant coefficients, Comm. Pure Appl. Anal., 2023, 22(5), 1565–1584. doi: 10.3934/cpaa.2023038 CrossRef $2\times 2$ hyperbolic system with constant coefficients" target="_blank">Google Scholar
[24]	Y. Zhang and C. Zhu, Global existence and optimal convergence rates for the strong solutions in $H^2$ to the 3D viscous liquid-gas two-phase flow model, J. Differential Equation, 2015, 258(7), 2315–2338. doi: 10.1016/j.jde.2014.12.008 CrossRef $H^2$ to the 3D viscous liquid-gas two-phase flow model" target="_blank">Google Scholar