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Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials
Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping
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 |
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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