American Institute of Mathematical Sciences

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December  2014, 7(4): 605-619. doi: 10.3934/krm.2014.7.605

Time decay of solutions to the compressible Euler equations with damping

Qing Chen 1, and  Zhong Tan 2,

1. 

School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian 361024, China
2. 

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005

Received  December 2013 Revised  July 2014 Published  November 2014

We consider the time decay rates of the solution to the Cauchy problem for the compressible Euler equations with damping. We prove the optimal decay rates of the solution as well as its higher-order spatial derivatives. The damping effect on the time decay estimates of the solution is studied in details.
Keywords: optimal decay rates, Euler equations with damping, energy method, damping effect., Compressible flow.
Mathematics Subject Classification: Primary: 35Q31, 76N10; Secondary: 35P2.
Citation: Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic & Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605
References:
[1]

C. M. Dafermos, Can dissipation prevent the breaking of waves? In Transactions of the Twenty-Sixth Conference of Army Mathematicians, ARO Rep. 81, 1, U. S. Army Res. Office, Research Triangle Park, N.C., (1981), 187-198.  Google Scholar

[2]

R. J. Duan, H. X. Liu, S. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rate for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233. doi: 10.1016/j.jde.2007.03.008.  Google Scholar

[3]

R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rate for the compressible Navier-Stokes equations with potential force, Math. Mod. Meth. Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X.  Google Scholar

[4]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. PDE., 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.  Google Scholar

[5]

L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific Publishing Co., River Edge, NJ, 1997.  Google Scholar

[6]

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.  Google Scholar

[7]

F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376. doi: 10.1007/s00205-002-0234-5.  Google Scholar

[8]

F. M. Huang, P. Marcati and R. H. Pan, Convergence to the Barenblatt Solution for the Compressible Euler Equations with Damping and Vacuum, Arch. Ration. Mech. Anal., 176 (2005), 1-24. doi: 10.1007/s00205-004-0349-y.  Google Scholar

[9]

F. M. Huang, R. H. Pan and Z. Wang, $L^1$ convergence to the Barenblatt solution for compressible Euler equations with damping, Arch. Ration. Mech. Anal., 200 (2011), 665-689. doi: 10.1007/s00205-010-0355-1.  Google Scholar

[10]

M. Jiang and C. J. Zhu, Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant, J. Differential Equations, 246 (2009), 50-77. doi: 10.1016/j.jde.2008.03.033.  Google Scholar

[11]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.  Google Scholar

[12]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations Of Magnetohydrodynamics, Ph.D thesis, Kyoto University, Kyoto, 1983. Google Scholar

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.  Google Scholar

[14]

A. Majda, Compressible Fluid Flow and Conservation laws in Several Space Variables, Springer-Verlag, Berlin/New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[15]

P. Marcati and A. Milani, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J.Differential Equations, 84 (1990), 129-147. doi: 10.1016/0022-0396(90)90130-H.  Google Scholar

[16]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J Math Kyoto Univ, 20 (1980), 67-104.  Google Scholar

[17]

T. Nishida, Global solutions for an initial-boundary value problem of a quasilinear hyperbolic systems, Proc. Japan Acad., 44 (1968), 642-646. doi: 10.3792/pja/1195521083.  Google Scholar

[18]

T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'Orsay, (1978), 46-53.  Google Scholar

[19]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  Google Scholar

[20]

T. C. Sideris, B. Thomases and D. H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. PDE., 28 (2003), 795-816. doi: 10.1081/PDE-120020497.  Google Scholar

[21]

Z. Tan and G. C. Wu, Large time behavior of solutions for compressible Euler equations with damping in $\mathbbR^{3}$, J. Differential Equations, 252 (2012), 1546-1561. doi: 10.1016/j.jde.2011.09.003.  Google Scholar

[22]

W. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differential Equations, 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937.  Google Scholar

[23]

C. J. 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: 10.1360/03ys9057.  Google Scholar

show all references

References:
[1]

C. M. Dafermos, Can dissipation prevent the breaking of waves? In Transactions of the Twenty-Sixth Conference of Army Mathematicians, ARO Rep. 81, 1, U. S. Army Res. Office, Research Triangle Park, N.C., (1981), 187-198.  Google Scholar

[2]

R. J. Duan, H. X. Liu, S. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rate for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233. doi: 10.1016/j.jde.2007.03.008.  Google Scholar

[3]

R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rate for the compressible Navier-Stokes equations with potential force, Math. Mod. Meth. Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X.  Google Scholar

[4]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. PDE., 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.  Google Scholar

[5]

L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific Publishing Co., River Edge, NJ, 1997.  Google Scholar

[6]

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.  Google Scholar

[7]

F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376. doi: 10.1007/s00205-002-0234-5.  Google Scholar

[8]

F. M. Huang, P. Marcati and R. H. Pan, Convergence to the Barenblatt Solution for the Compressible Euler Equations with Damping and Vacuum, Arch. Ration. Mech. Anal., 176 (2005), 1-24. doi: 10.1007/s00205-004-0349-y.  Google Scholar

[9]

F. M. Huang, R. H. Pan and Z. Wang, $L^1$ convergence to the Barenblatt solution for compressible Euler equations with damping, Arch. Ration. Mech. Anal., 200 (2011), 665-689. doi: 10.1007/s00205-010-0355-1.  Google Scholar

[10]

M. Jiang and C. J. Zhu, Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant, J. Differential Equations, 246 (2009), 50-77. doi: 10.1016/j.jde.2008.03.033.  Google Scholar

[11]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.  Google Scholar

[12]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations Of Magnetohydrodynamics, Ph.D thesis, Kyoto University, Kyoto, 1983. Google Scholar

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.  Google Scholar

[14]

A. Majda, Compressible Fluid Flow and Conservation laws in Several Space Variables, Springer-Verlag, Berlin/New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[15]

P. Marcati and A. Milani, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J.Differential Equations, 84 (1990), 129-147. doi: 10.1016/0022-0396(90)90130-H.  Google Scholar

[16]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J Math Kyoto Univ, 20 (1980), 67-104.  Google Scholar

[17]

T. Nishida, Global solutions for an initial-boundary value problem of a quasilinear hyperbolic systems, Proc. Japan Acad., 44 (1968), 642-646. doi: 10.3792/pja/1195521083.  Google Scholar

[18]

T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'Orsay, (1978), 46-53.  Google Scholar

[19]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  Google Scholar

[20]

T. C. Sideris, B. Thomases and D. H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. PDE., 28 (2003), 795-816. doi: 10.1081/PDE-120020497.  Google Scholar

[21]

Z. Tan and G. C. Wu, Large time behavior of solutions for compressible Euler equations with damping in $\mathbbR^{3}$, J. Differential Equations, 252 (2012), 1546-1561. doi: 10.1016/j.jde.2011.09.003.  Google Scholar

[22]

W. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differential Equations, 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937.  Google Scholar

[23]

C. J. 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: 10.1360/03ys9057.  Google Scholar

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