# American Institute of Mathematical Sciences

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

 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

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).
Citation: Shifeng Geng, Zhen Wang. Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant. Communications on Pure & 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, (2011), 850.  doi: 10.1080/03605302.2010.520052.  Google Scholar [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.  doi: 10.1007/BF02099268.  Google Scholar [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.  doi: 10.1006/jdeq.1996.0034.  Google Scholar [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.   Google Scholar [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.  doi: 10.1137/S0036141094267078.  Google Scholar [6] H. Ma and M. Mei, Best asymptotic profile for linear damped $p$-system with boundary effect,, J. Differential Equations, 249 (2010), 446.  doi: 10.1016/j.jde.2010.04.008.  Google Scholar [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).  doi: 10.1007/s00021-005-0155-9.  Google Scholar [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.  doi: 10.1016/S0022-0396(03)00026-3.  Google Scholar [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.  doi: 10.1137/S0036141099364401.  Google Scholar [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.  Google Scholar [11] M. Mei, Best asymptotic profile for hyperbolic $p$-system with damping,, SIAM J. Math. Anal., 42 (2010), 1.  doi: 10.1137/090756594.  Google Scholar [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.  doi: 10.1006/jdeq.1996.0159.  Google Scholar [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.  doi: 10.1017/S0308210500002341.  Google Scholar [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.  doi: 10.1137/S003614109936467X.  Google Scholar [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.  doi: 10.1006/jdeq.1999.3703.  Google Scholar [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.  doi: 10.1307/mmj/1012409969.  Google Scholar [17] R. Pan, Darcy's law as long-time limit of adiabatic porous media flow,, J. Differential Equations, 220 (2006), 121.  doi: 10.1016/j.jde.2004.10.013.  Google Scholar [18] B. Said-Houari, Convergence to strong nonlinear diffusion waves for solutions to $p$-system with damping,, J. Differential Equations, 247 (2009), 917.  doi: 10.1016/j.jde.2009.04.011.  Google Scholar [19] H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of $p$-system with damping,, J. Differential Equations, 174 (2001), 200.  doi: 10.1006/jdeq.2000.3936.  Google Scholar [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.  doi: 10.1007/BF02884028.  Google Scholar

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##### 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, (2011), 850.  doi: 10.1080/03605302.2010.520052.  Google Scholar [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.  doi: 10.1007/BF02099268.  Google Scholar [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.  doi: 10.1006/jdeq.1996.0034.  Google Scholar [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.   Google Scholar [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.  doi: 10.1137/S0036141094267078.  Google Scholar [6] H. Ma and M. Mei, Best asymptotic profile for linear damped $p$-system with boundary effect,, J. Differential Equations, 249 (2010), 446.  doi: 10.1016/j.jde.2010.04.008.  Google Scholar [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).  doi: 10.1007/s00021-005-0155-9.  Google Scholar [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.  doi: 10.1016/S0022-0396(03)00026-3.  Google Scholar [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.  doi: 10.1137/S0036141099364401.  Google Scholar [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.  Google Scholar [11] M. Mei, Best asymptotic profile for hyperbolic $p$-system with damping,, SIAM J. Math. Anal., 42 (2010), 1.  doi: 10.1137/090756594.  Google Scholar [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.  doi: 10.1006/jdeq.1996.0159.  Google Scholar [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.  doi: 10.1017/S0308210500002341.  Google Scholar [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.  doi: 10.1137/S003614109936467X.  Google Scholar [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.  doi: 10.1006/jdeq.1999.3703.  Google Scholar [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.  doi: 10.1307/mmj/1012409969.  Google Scholar [17] R. Pan, Darcy's law as long-time limit of adiabatic porous media flow,, J. Differential Equations, 220 (2006), 121.  doi: 10.1016/j.jde.2004.10.013.  Google Scholar [18] B. Said-Houari, Convergence to strong nonlinear diffusion waves for solutions to $p$-system with damping,, J. Differential Equations, 247 (2009), 917.  doi: 10.1016/j.jde.2009.04.011.  Google Scholar [19] H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of $p$-system with damping,, J. Differential Equations, 174 (2001), 200.  doi: 10.1006/jdeq.2000.3936.  Google Scholar [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.  doi: 10.1007/BF02884028.  Google Scholar
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