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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:
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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. |
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S. Kawashima and W.-A. Yong,
Decay estimates for hyperbolic balance laws, Z. Anal. Anwend., 28 (2009), 1-33.
doi: 10.4171/ZAA/1369. |
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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. |
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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.
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| [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. |
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