American Institute of Mathematical Sciences

Advanced
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

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 & 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

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, (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

[1]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[2]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[3]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[4]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304
[5]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318
[6]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[7]

Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293
[8]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[9]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345
[10]

Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020356
[11]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465
[12]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299
[13]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218
[14]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453
[15]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353
[16]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[17]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[18]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
[19]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112
[20]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences