January  2018, 38(1): 363-396. doi: 10.3934/dcds.2018018

LP decay for general hyperbolic-parabolic systems of balance laws

Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA

Received  May 2017 Revised  August 2017 Published  September 2017

Fund Project: This work was partially supported by a grant from the Simons Foundation (#244905 to Yanni Zeng)

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.

Citation: Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018
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]

Rinaldo M. Colombo, Graziano Guerra. Hyperbolic balance laws with a dissipative non local source. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1077-1090. doi: 10.3934/cpaa.2008.7.1077

[2]

Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271

[3]

Huashui Zhan. On a hyperbolic-parabolic mixed type equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 605-624. doi: 10.3934/dcdss.2017030

[4]

Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic & Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725

[5]

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 & Continuous Dynamical Systems - A, 2016, 36 (5) : 2419-2447. doi: 10.3934/dcds.2016.36.2419

[6]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883

[7]

Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227

[8]

Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032

[9]

Stephan Gerster, Michael Herty. Discretized feedback control for systems of linearized hyperbolic balance laws. Mathematical Control & Related Fields, 2019, 9 (3) : 517-539. doi: 10.3934/mcrf.2019024

[10]

Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035

[11]

Toyohiko Aiki, Joost Hulshof, Nobuyuki Kenmochi, Adrian Muntean. Analysis of non-equilibrium evolution problems: Selected topics in material and life sciences. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : i-iii. doi: 10.3934/dcdss.2014.7.1i

[12]

Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018

[13]

Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407

[14]

Isabelle Choquet, Brigitte Lucquin-Desreux. Non equilibrium ionization in magnetized two-temperature thermal plasma. Kinetic & Related Models, 2011, 4 (3) : 669-700. doi: 10.3934/krm.2011.4.669

[15]

Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031

[16]

Elena Bonetti, Pierluigi Colli, Mauro Fabrizio, Gianni Gilardi. Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1001-1026. doi: 10.3934/dcdsb.2006.6.1001

[17]

Fengbai Li, Feng Rong. Decay of solutions to fractal parabolic conservation laws with large initial data. Communications on Pure & Applied Analysis, 2013, 12 (2) : 973-984. doi: 10.3934/cpaa.2013.12.973

[18]

Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control & Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002

[19]

Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 203-222. doi: 10.3934/nhm.2016.11.203

[20]

Graziano Crasta, Benedetto Piccoli. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 477-502. doi: 10.3934/dcds.1997.3.477

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (45)
  • HTML views (24)
  • Cited by (0)

Other articles
by authors

[Back to Top]