# American Institute of Mathematical Sciences

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

## Time decay of solutions to the compressible Euler equations with damping

 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.
Citation: Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic and 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. [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. [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. [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. [5] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific Publishing Co., River Edge, NJ, 1997. [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. [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. [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. [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. [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. [11] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740. [12] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations Of Magnetohydrodynamics, Ph.D thesis, Kyoto University, Kyoto, 1983. [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. [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. [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. [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. [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. [18] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'Orsay, (1978), 46-53. [19] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [5] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific Publishing Co., River Edge, NJ, 1997. [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. [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. [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. [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. [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. [11] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740. [12] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations Of Magnetohydrodynamics, Ph.D thesis, Kyoto University, Kyoto, 1983. [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. [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. [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. [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. [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. [18] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'Orsay, (1978), 46-53. [19] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. [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. [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. [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. [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.
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