# American Institute of Mathematical Sciences

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