-
Previous Article
Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane
- DCDS Home
- This Issue
-
Next Article
Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one
LP decay for general hyperbolic-parabolic systems of balance laws
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA |
We study time asymptotic decay of solutions for a general system of hyperbolic-parabolic balance laws in multi space dimensions. The system has physical viscosity matrices and a lower order term for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper we obtain optimal $L^p$ decay rates for $p≥2$. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively.
References:
[1] |
J. F. Clarke and M. McChesney,
Dynamics of Relaxing Gases 2nd edition, Butterworths, London, 1976. |
[2] |
K. O. Friedrichs and P. D. Lax,
Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), 1686-1688.
|
[3] |
S. Kawashima,
Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics Doctoral thesis, Kyoto University, 1983. |
[4] |
S. Kawashima and W.-A. Yong,
Decay estimates for hyperbolic balance laws, Z. Anal. Anwend., 28 (2009), 1-33.
doi: 10.4171/ZAA/1369. |
[5] |
T.-P Liu and Y Zen,
arge time behavior of solutions for gene, Mem. Amer. Math. Soc., 1525 (1997), ⅷ+120 pp.
doi: 10.1090/memo/0599. |
[6] |
T.-P. Liu and Y. Zeng,
Compressible Navier-Stokes equations with zero heat conductivity, J. Differential Equations, 153 (1999), 225-291.
doi: 10.1006/jdeq.1998.3554. |
[7] |
T.-P. Liu and Y. Zen,
Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc., 234 (2015), ⅵ+168 pp.
doi: 10.1090/memo/1105. |
[8] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
|
[9] |
Y. Shizuta and S. Kawashima,
Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.
doi: 10.14492/hokmj/1381757663. |
[10] |
W. Vincenti and C. Kruger Jr, Introduction to Physical Gas Dynamics, Krieger Publishing Company, 1986.
![]() |
[11] |
Y. Zeng,
Global existence theory for a general class of hyperbolic balance laws, Bulletin, Inst. Math. Academia Sinica, 10 (2015), 143-170.
|
[12] |
Y. Zeng,
On Cauchy problems of thermal non-equilibrium flows with small data, Bull. Braz. Math. Soc. (N.S.), 47 (2016), 799-809.
doi: 10.1007/s00574-016-0187-1. |
[13] |
Y. Zeng,
Global existence theory for general hyperbolic-parabolic balance laws with application, J. Hyperbolic Differ. Equ., 14 (2017), 359-391.
doi: 10.1142/S0219891617500126. |
[14] |
Y. Zeng and J. Chen,
Pointwise time asymptotic behavior of solutions to a general class of hyperbolic balance laws, J. Differential Equations, 260 (2016), 6745-6786.
doi: 10.1016/j.jde.2016.01.013. |
show all references
References:
[1] |
J. F. Clarke and M. McChesney,
Dynamics of Relaxing Gases 2nd edition, Butterworths, London, 1976. |
[2] |
K. O. Friedrichs and P. D. Lax,
Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), 1686-1688.
|
[3] |
S. Kawashima,
Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics Doctoral thesis, Kyoto University, 1983. |
[4] |
S. Kawashima and W.-A. Yong,
Decay estimates for hyperbolic balance laws, Z. Anal. Anwend., 28 (2009), 1-33.
doi: 10.4171/ZAA/1369. |
[5] |
T.-P Liu and Y Zen,
arge time behavior of solutions for gene, Mem. Amer. Math. Soc., 1525 (1997), ⅷ+120 pp.
doi: 10.1090/memo/0599. |
[6] |
T.-P. Liu and Y. Zeng,
Compressible Navier-Stokes equations with zero heat conductivity, J. Differential Equations, 153 (1999), 225-291.
doi: 10.1006/jdeq.1998.3554. |
[7] |
T.-P. Liu and Y. Zen,
Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc., 234 (2015), ⅵ+168 pp.
doi: 10.1090/memo/1105. |
[8] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
|
[9] |
Y. Shizuta and S. Kawashima,
Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.
doi: 10.14492/hokmj/1381757663. |
[10] |
W. Vincenti and C. Kruger Jr, Introduction to Physical Gas Dynamics, Krieger Publishing Company, 1986.
![]() |
[11] |
Y. Zeng,
Global existence theory for a general class of hyperbolic balance laws, Bulletin, Inst. Math. Academia Sinica, 10 (2015), 143-170.
|
[12] |
Y. Zeng,
On Cauchy problems of thermal non-equilibrium flows with small data, Bull. Braz. Math. Soc. (N.S.), 47 (2016), 799-809.
doi: 10.1007/s00574-016-0187-1. |
[13] |
Y. Zeng,
Global existence theory for general hyperbolic-parabolic balance laws with application, J. Hyperbolic Differ. Equ., 14 (2017), 359-391.
doi: 10.1142/S0219891617500126. |
[14] |
Y. Zeng and J. Chen,
Pointwise time asymptotic behavior of solutions to a general class of hyperbolic balance laws, J. Differential Equations, 260 (2016), 6745-6786.
doi: 10.1016/j.jde.2016.01.013. |
[1] |
Bopeng Rao, Xu Zhang. Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2789-2809. doi: 10.3934/cpaa.2021119 |
[2] |
Rinaldo M. Colombo, Graziano Guerra. Hyperbolic balance laws with a dissipative non local source. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1077-1090. doi: 10.3934/cpaa.2008.7.1077 |
[3] |
Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271 |
[4] |
Guochun Wu, Han Wang, Yinghui Zhang. Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in $ \mathbb R^3 $. Electronic Research Archive, 2021, 29 (6) : 3889-3908. doi: 10.3934/era.2021067 |
[5] |
Huashui Zhan. On a hyperbolic-parabolic mixed type equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 605-624. doi: 10.3934/dcdss.2017030 |
[6] |
Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic and Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725 |
[7] |
Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 |
[8] |
Marcello D'Abbicco, Ruy Coimbra Charão, Cleverson Roberto da Luz. Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2419-2447. doi: 10.3934/dcds.2016.36.2419 |
[9] |
Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic and Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883 |
[10] |
Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227 |
[11] |
Stephan Gerster, Michael Herty. Discretized feedback control for systems of linearized hyperbolic balance laws. Mathematical Control and Related Fields, 2019, 9 (3) : 517-539. doi: 10.3934/mcrf.2019024 |
[12] |
Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic and Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035 |
[13] |
Toyohiko Aiki, Joost Hulshof, Nobuyuki Kenmochi, Adrian Muntean. Analysis of non-equilibrium evolution problems: Selected topics in material and life sciences. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : i-iii. doi: 10.3934/dcdss.2014.7.1i |
[14] |
Andrea Bondesan, Laurent Boudin, Marc Briant, Bérénice Grec. Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2549-2573. doi: 10.3934/cpaa.2020112 |
[15] |
Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018 |
[16] |
Isabelle Choquet, Brigitte Lucquin-Desreux. Non equilibrium ionization in magnetized two-temperature thermal plasma. Kinetic and Related Models, 2011, 4 (3) : 669-700. doi: 10.3934/krm.2011.4.669 |
[17] |
Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407 |
[18] |
Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032 |
[19] |
Shuai Liu, Yuzhu Wang. Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021041 |
[20] |
Anton Arnold, Sjoerd Geevers, Ilaria Perugia, Dmitry Ponomarev. On the exponential time-decay for the one-dimensional wave equation with variable coefficients. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022105 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]