American Institute of Mathematical Sciences

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

LP decay for general hyperbolic-parabolic systems of balance laws

Yanni Zeng

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.

Keywords: Hyperbolic-parabolic, balance laws, time-decay rates, gas flows in thermal non-equilibrium.
Mathematics Subject Classification: Primary:35B40, 35M31;Secondary:35Q35.
Citation: Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete & Continuous Dynamical Systems, 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. Google Scholar

[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.   Google Scholar

[3]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics Doctoral thesis, Kyoto University, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar

[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.  Google Scholar

[10] W. Vincenti and C. Kruger Jr, Introduction to Physical Gas Dynamics, Krieger Publishing Company, 1986.   Google Scholar
[11]

Y. Zeng, Global existence theory for a general class of hyperbolic balance laws, Bulletin, Inst. Math. Academia Sinica, 10 (2015), 143-170.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. F. Clarke and M. McChesney, Dynamics of Relaxing Gases 2nd edition, Butterworths, London, 1976. Google Scholar

[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.   Google Scholar

[3]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics Doctoral thesis, Kyoto University, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar

[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.  Google Scholar

[10] W. Vincenti and C. Kruger Jr, Introduction to Physical Gas Dynamics, Krieger Publishing Company, 1986.   Google Scholar
[11]

Y. Zeng, Global existence theory for a general class of hyperbolic balance laws, Bulletin, Inst. Math. Academia Sinica, 10 (2015), 143-170.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021045
[2]

Charles Amorim, Miguel Loayza, Marko A. Rojas-Medar. The nonstationary flows of micropolar fluids with thermal convection: An iterative approach. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2509-2535. doi: 10.3934/dcdsb.2020193
[3]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021071
[4]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[5]

Danielle Hilhorst, Pierre Roux. A hyperbolic-elliptic-parabolic PDE model describing chemotactic E. Coli colonies. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021033
[6]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
[7]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073
[8]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024
[9]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201
[10]

De-han Chen, Daijun jiang. Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021023
[11]

Tomoyuki Tanaka, Kyouhei Wakasa. On the critical decay for the wave equation with a cubic convolution in 3D. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021048
[12]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[13]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91
[14]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021031
[15]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026
[16]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
[17]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068
[18]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017
[19]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[20]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (91)
  • HTML views (77)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences