Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping

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November 2014, 13(6): 2211-2228. doi: 10.3934/cpaa.2014.13.2211

Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping

Shifeng Geng ^1, and Lina Zhang ^2,

1.	School of Mathematics and Computational Science, Xiangtan University, Hunan 411105
2.	School of Mathematical Science and Computing Technology, Central South University, Changsha 410075, China

Received April 2013 Revised December 2013 Published July 2014

Abstract
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This paper is concerned with large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. For the nonlinear damping case, i.e. $\beta \neq 0,$ results for the linear damping case are extended to the case of nonlinear damping. Compared with the results obtained by Marcati and Pan, better decay estimates are obtained in this paper.

Keywords: system of compressible adiabatic flow through porous media, Large-time behavior, convergence rates., nonlinear damping.

Mathematics Subject Classification: 35L45, 35L60, 35L65, 76R5.

Citation: Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211

References:

[1]	S. Geng and Z. Wang, Convergence rates to nonlinear diffusion waves for solutions to the system of compressible adiabatic flow through porous media, Comm. Partial Differential Equations, 36 (2011), 850-872. doi: 10.1080/03605302.2010.520052.
[2]	L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys., 143 (1992), 599-605. doi: 10.1007/BF02099268.
[3]	L. Hsiao and T.-P. Liu, Nonlinear diffusion phenomena of nonlinear hyperbolic system, Chin. Ann. Math. Ser. B, 14 (1993), 465-480.
[4]	L. Hsiao and T. Luo, Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media, J. Differential Equations, 125 (1996), 329-365. doi: 10.1006/jdeq.1996.0034.
[5]	L. Hsiao and D. Serre, Large-time behavior of solutions for the system of compressible adiabatic flow through porous media, Chin. Ann. Math. Ser. B, 16 (1995), 431-444.
[6]	L. Hsiao and D. Serre, Global existence of solutions for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal., 27 (1996), 70-77. doi: 10.1137/S0036141094267078.
[7]	M. Jiang and C. Zhu, Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant, Discrete Contin. Dyn. Syst. Ser. A, 23 (2009), 887-918. doi: 10.3934/dcds.2009.23.887.
[8]	H. Ma and M. Mei, Best asymptotic profile for linear damped p-system with boundary effect, J. Differential Equations, 249 (2010), 446-484. doi: 10.1016/j.jde.2010.04.008.
[9]	P. Marcati and M. Mei, B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech., 7 (2005), S224-S240. doi: 10.1007/s00021-005-0155-9.
[10]	P. Marcati and K. Nishihara, The $L^p-L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3.
[11]	P. Marcati and R. Pan, On the diffusive profiles for the system of compressible adiabatice flow through porous media, SIAM J. Math. Anal., 33 (2001), 790-826. doi: 10.1137/S0036141099364401.
[12]	M. Mei, Nonlinear diffusion waves for hyperbolic $p$-system with nonlinear damping, J. Differential Equations, 247 (2009), 1275-1296. doi: 10.1016/j.jde.2009.04.004.
[13]	M. Mei, Best asymptotic profile for hyperbolic p-system with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.
[14]	K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations, 131 (1996), 171-188. doi: 10.1006/jdeq.1996.0159.
[15]	K. Nishihara, Asymptotic toward the diffusion wave for a one-dimensional compressible flow through porous media, Proceedings of the Royal Society of Edinburgh, 133A (2003), 177-196. doi: 10.1017/S0308210500002341.
[16]	K. Nishihara and M. Nishikawa, Asymptotic behavior of solutions to the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal., 33 (2001), 216-239. doi: 10.1137/S003614109936467X.
[17]	K. Nishihara, W. Wang and T. Yang, $L_p$ -convergence rate to nonlinear diffusion waves for p-system with damping, J. Differential Equations, 161 (1999), 191-218. doi: 10.1006/jdeq.1999.3703.
[18]	M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107-132.
[19]	R. Pan, Darcy's law as long-time limit of adiabatic porous media flow, J. Differential Equations, 220 (2006), 121-146. doi: 10.1016/j.jde.2004.10.013.
[20]	H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, J. Differential Equations, 174 (2001), 200-236. doi: 10.1006/jdeq.2000.3936.
[21]	C. Zhu, Convergence rates to nonlinear diffusion waves for weak solutions to $p$-system with damping, Sci. Chin. Ser. A, 46 (2003), 562-575. doi: 10.1360/03ys9057.
[22]	C. Zhu and M. Jiang, $L^p$-decay rates to nonlinear diffusion waves for $p$-system with nonlinear damping, Sciences in China, Series A, 49 (2006), 721-739. doi: 10.1007/s11425-006-0721-5.

show all references

References:

[1]	S. Geng and Z. Wang, Convergence rates to nonlinear diffusion waves for solutions to the system of compressible adiabatic flow through porous media, Comm. Partial Differential Equations, 36 (2011), 850-872. doi: 10.1080/03605302.2010.520052.
[2]	L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys., 143 (1992), 599-605. doi: 10.1007/BF02099268.
[3]	L. Hsiao and T.-P. Liu, Nonlinear diffusion phenomena of nonlinear hyperbolic system, Chin. Ann. Math. Ser. B, 14 (1993), 465-480.
[4]	L. Hsiao and T. Luo, Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media, J. Differential Equations, 125 (1996), 329-365. doi: 10.1006/jdeq.1996.0034.
[5]	L. Hsiao and D. Serre, Large-time behavior of solutions for the system of compressible adiabatic flow through porous media, Chin. Ann. Math. Ser. B, 16 (1995), 431-444.
[6]	L. Hsiao and D. Serre, Global existence of solutions for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal., 27 (1996), 70-77. doi: 10.1137/S0036141094267078.
[7]	M. Jiang and C. Zhu, Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant, Discrete Contin. Dyn. Syst. Ser. A, 23 (2009), 887-918. doi: 10.3934/dcds.2009.23.887.
[8]	H. Ma and M. Mei, Best asymptotic profile for linear damped p-system with boundary effect, J. Differential Equations, 249 (2010), 446-484. doi: 10.1016/j.jde.2010.04.008.
[9]	P. Marcati and M. Mei, B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech., 7 (2005), S224-S240. doi: 10.1007/s00021-005-0155-9.
[10]	P. Marcati and K. Nishihara, The $L^p-L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3.
[11]	P. Marcati and R. Pan, On the diffusive profiles for the system of compressible adiabatice flow through porous media, SIAM J. Math. Anal., 33 (2001), 790-826. doi: 10.1137/S0036141099364401.
[12]	M. Mei, Nonlinear diffusion waves for hyperbolic $p$-system with nonlinear damping, J. Differential Equations, 247 (2009), 1275-1296. doi: 10.1016/j.jde.2009.04.004.
[13]	M. Mei, Best asymptotic profile for hyperbolic p-system with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.
[14]	K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations, 131 (1996), 171-188. doi: 10.1006/jdeq.1996.0159.
[15]	K. Nishihara, Asymptotic toward the diffusion wave for a one-dimensional compressible flow through porous media, Proceedings of the Royal Society of Edinburgh, 133A (2003), 177-196. doi: 10.1017/S0308210500002341.
[16]	K. Nishihara and M. Nishikawa, Asymptotic behavior of solutions to the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal., 33 (2001), 216-239. doi: 10.1137/S003614109936467X.
[17]	K. Nishihara, W. Wang and T. Yang, $L_p$ -convergence rate to nonlinear diffusion waves for p-system with damping, J. Differential Equations, 161 (1999), 191-218. doi: 10.1006/jdeq.1999.3703.
[18]	M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107-132.
[19]	R. Pan, Darcy's law as long-time limit of adiabatic porous media flow, J. Differential Equations, 220 (2006), 121-146. doi: 10.1016/j.jde.2004.10.013.
[20]	H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, J. Differential Equations, 174 (2001), 200-236. doi: 10.1006/jdeq.2000.3936.
[21]	C. Zhu, Convergence rates to nonlinear diffusion waves for weak solutions to $p$-system with damping, Sci. Chin. Ser. A, 46 (2003), 562-575. doi: 10.1360/03ys9057.
[22]	C. Zhu and M. Jiang, $L^p$-decay rates to nonlinear diffusion waves for $p$-system with nonlinear damping, Sciences in China, Series A, 49 (2006), 721-739. doi: 10.1007/s11425-006-0721-5.

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