Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant

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March 2012, 11(2): 475-500. doi: 10.3934/cpaa.2012.11.475

Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant

Shifeng Geng ^1, and Zhen Wang ^2,

1.	School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
2.	Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, China

Received August 2010 Revised January 2011 Published October 2011

Abstract
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We realize the best asymptotic profile for the solutions to the nonisentropic $p$-system with damping on quadrant is a particular solution of the IBVP for the corresponding nonlinear parabolic equation with special initial data, and we further show the convergence rates to this particular asymptotic profile. This rates are same to that for the isentropic case obtained by H. Ma and M. Mei (J. Differential Equations 249 (2010), 446--484).

Keywords: boundary effect, convergence rates., Asymptotic behavior, the system of compressible adiabatic flow.

Mathematics Subject Classification: 35B40, 35L50, 35L65, 76R5.

Citation: Shifeng Geng, Zhen Wang. Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant. Communications on Pure and Applied Analysis, 2012, 11 (2) : 475-500. doi: 10.3934/cpaa.2012.11.475

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. 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.
[4]	L. Hsiao and D. Serre, Large-time behavior of solutions for the system of compressible adiabatic flow through porous media, Chinese Ann. Math. Ser. B, 16 (1995), 431-444.
[5]	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.
[6]	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.
[7]	P. Marcati, M. Mei and 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.
[8]	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.
[9]	P. Marcati and R. Pan, On the diffusive profiles for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal., 33 (2001), 790-826. doi: 10.1137/S0036141099364401.
[10]	A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation,, Publ. Res. Inst. Math. Sci., 13 (): 349. doi: 10.2977/prims/1195189813.
[11]	M. Mei, Best asymptotic profile for hyperbolic $p$-system with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.
[12]	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.
[13]	K. Nishihara, Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 177-196. doi: 10.1017/S0308210500002341.
[14]	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.
[15]	K. Nishihara, W. Wang and T. Yang, $L^p$-convergence rate to nonlinear diffusion waves for $p$-system with damping, J. Differential Equations, 161 (2000), 191-218. doi: 10.1006/jdeq.1999.3703.
[16]	R. Pan, Boundary effects and large time behavior for the system of compressible adiabatic flow through porous media, Michigan Math. J., 49 (2001), 519-540. doi: 10.1307/mmj/1012409969.
[17]	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.
[18]	B. Said-Houari, Convergence to strong nonlinear diffusion waves for solutions to $p$-system with damping, J. Differential Equations, 247 (2009), 917-930. doi: 10.1016/j.jde.2009.04.011.
[19]	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.
[20]	C. Zhu, Convergence rates to nonlinear diffusion waves for weak entropy solutions to $p$-system with damping, Sci. China Ser. A, 46 (2003), 562-575. doi: 10.1007/BF02884028.

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. 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.
[4]	L. Hsiao and D. Serre, Large-time behavior of solutions for the system of compressible adiabatic flow through porous media, Chinese Ann. Math. Ser. B, 16 (1995), 431-444.
[5]	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.
[6]	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.
[7]	P. Marcati, M. Mei and 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.
[8]	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.
[9]	P. Marcati and R. Pan, On the diffusive profiles for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal., 33 (2001), 790-826. doi: 10.1137/S0036141099364401.
[10]	A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation,, Publ. Res. Inst. Math. Sci., 13 (): 349. doi: 10.2977/prims/1195189813.
[11]	M. Mei, Best asymptotic profile for hyperbolic $p$-system with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.
[12]	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.
[13]	K. Nishihara, Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 177-196. doi: 10.1017/S0308210500002341.
[14]	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.
[15]	K. Nishihara, W. Wang and T. Yang, $L^p$-convergence rate to nonlinear diffusion waves for $p$-system with damping, J. Differential Equations, 161 (2000), 191-218. doi: 10.1006/jdeq.1999.3703.
[16]	R. Pan, Boundary effects and large time behavior for the system of compressible adiabatic flow through porous media, Michigan Math. J., 49 (2001), 519-540. doi: 10.1307/mmj/1012409969.
[17]	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.
[18]	B. Said-Houari, Convergence to strong nonlinear diffusion waves for solutions to $p$-system with damping, J. Differential Equations, 247 (2009), 917-930. doi: 10.1016/j.jde.2009.04.011.
[19]	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.
[20]	C. Zhu, Convergence rates to nonlinear diffusion waves for weak entropy solutions to $p$-system with damping, Sci. China Ser. A, 46 (2003), 562-575. doi: 10.1007/BF02884028.

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