doi: 10.3934/dcdsb.2020047

Global existence and convergence rates of solutions for the compressible Euler equations with damping

1. 

School of Mathematics and Statistics, Shaoguan University, 512005, Shaoguan, China

2. 

Department of Mathematics, Sun Yat-sen University, 510275, Guangzhou, China

* Corresponding author: Yin Li

Received  January 2019 Revised  September 2019 Published  February 2020

Fund Project: We would like to express our sincere thanks to Academician Boling Guo of institute of Applied Physics and Computational Mathematics in Beijing for their fruitful help and discussions. This work is partially supported by the National Natural Science Foundation of China(Nos.11926354,11701380 and 11971496),Natural Science Foundation of Guangdong Province (Nos.2019A1515011320,2017A030307022,2016A030310019 and 2016A030307042),the Education Research Platform Project of Guangdong Province(No.2018179).

The Cauchy problem for the 3D compressible Euler equations with damping is considered. Existence of global-in-time smooth solutions is established under the condition that the initial data is small perturbations of some given constant state in the framework of Sobolev space $ H^3(\mathbb{R}^{3}) $ only, but we don't need the bound of $ L^1 $ norm. Moreover, the optimal $ L^{2} $-$ L^{2} $ convergence rates are also obtained for the solution. Our proof is based on the benefit of the low frequency and high frequency decomposition, here, we just need spectral analysis of the low frequency part of the Green function to the linearized system, so that we succeed to avoid some complicate analysis.

Citation: Ruiying Wei, Yin Li, Zheng-an Yao. Global existence and convergence rates of solutions for the compressible Euler equations with damping. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020047
References:
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[2]

Q. Chen and Z. Tan, Time decay of solutions to the compressible Euler equations with damping, Kinet. Relat. Models, 7 (2014), 605-619.  doi: 10.3934/krm.2014.7.605.  Google Scholar

[3]

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

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

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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

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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.  Google Scholar

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L. Hsiao and R. H. Pan, The damped $p$-system with boundary effects, in Nonlinear PDE's, Dynamics and Continuum Physics, Contemp. Math., 255, Amer. Math. Soc., Providence, RI, 2000, 109–123. doi: 10.1090/conm/255/03977.  Google Scholar

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F. M. HuangP. Marcati and R. H. Pan, Convergence to 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

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F. M. HuangR. 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

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

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A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

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

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T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publications Mathématiques d'Orsay, Département de Mathématique, Université de Paris-Sud, Orsay, 1978.  Google Scholar

[15]

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

[16]

R. H. Pan and K. Zhao, The 3D compressible Euler equations with damping in a bounded domain, J. Differential Equations, 246 (2009), 581-596.  doi: 10.1016/j.jde.2008.06.007.  Google Scholar

[17]

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

[18]

Z. Tan and Y. Wang, Global solution and large-time behavior of the 3D compressible Euler equations with damping, J. Differential Equations, 254 (2013), 1686-1704.  doi: 10.1016/j.jde.2012.10.026.  Google Scholar

[19]

Z. TanY. J. Wang and Y. Wang, Stability of steady states of the Navier-Stokes-Poisson equations with non-flat doping profile, SIAM J. Math. Anal., 47 (2015), 179-209.  doi: 10.1137/130950069.  Google Scholar

[20]

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

[21]

D. H. Wang, Global solutions and relaxation limits of Euler-Poisson equations, Z. Angew. Math. Phys., 52 (2001), 620-630.  doi: 10.1007/s00033-001-8135-2.  Google Scholar

[22]

W. K. 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]

W. J. Wang and W. K. Wang, Large time behavior for the system of a viscous liquid-gas two-phase flow model in $\mathbb{R}^{3}$, J. Differential Equations, 261 (2016), 5561-5589.  doi: 10.1016/j.jde.2016.08.013.  Google Scholar

[24]

Y. WangC. Liu and Z. Tan, Well-posedness on a new hydrodynamic model of the fluid with the dilute charged particles, J. Differential Equations, 262 (2017), 68-115.  doi: 10.1016/j.jde.2016.09.026.  Google Scholar

[25]

Y. N. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279.  doi: 10.1007/s002050050188.  Google Scholar

[26]

H. J. 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.  Google Scholar

show all references

References:
[1] R. Adams, Sobolev Spaes, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975.   Google Scholar
[2]

Q. Chen and Z. Tan, Time decay of solutions to the compressible Euler equations with damping, Kinet. Relat. Models, 7 (2014), 605-619.  doi: 10.3934/krm.2014.7.605.  Google Scholar

[3]

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

[4]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 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., Inc., River Edge, NJ, 1997. doi: 10.1142/3538.  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]

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.  Google Scholar

[8]

L. Hsiao and R. H. Pan, The damped $p$-system with boundary effects, in Nonlinear PDE's, Dynamics and Continuum Physics, Contemp. Math., 255, Amer. Math. Soc., Providence, RI, 2000, 109–123. doi: 10.1090/conm/255/03977.  Google Scholar

[9]

F. M. HuangP. Marcati and R. H. Pan, Convergence to 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

[10]

F. M. HuangR. 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

[11]

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

[12]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[13]

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

[14]

T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publications Mathématiques d'Orsay, Département de Mathématique, Université de Paris-Sud, Orsay, 1978.  Google Scholar

[15]

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

[16]

R. H. Pan and K. Zhao, The 3D compressible Euler equations with damping in a bounded domain, J. Differential Equations, 246 (2009), 581-596.  doi: 10.1016/j.jde.2008.06.007.  Google Scholar

[17]

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

[18]

Z. Tan and Y. Wang, Global solution and large-time behavior of the 3D compressible Euler equations with damping, J. Differential Equations, 254 (2013), 1686-1704.  doi: 10.1016/j.jde.2012.10.026.  Google Scholar

[19]

Z. TanY. J. Wang and Y. Wang, Stability of steady states of the Navier-Stokes-Poisson equations with non-flat doping profile, SIAM J. Math. Anal., 47 (2015), 179-209.  doi: 10.1137/130950069.  Google Scholar

[20]

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

[21]

D. H. Wang, Global solutions and relaxation limits of Euler-Poisson equations, Z. Angew. Math. Phys., 52 (2001), 620-630.  doi: 10.1007/s00033-001-8135-2.  Google Scholar

[22]

W. K. 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]

W. J. Wang and W. K. Wang, Large time behavior for the system of a viscous liquid-gas two-phase flow model in $\mathbb{R}^{3}$, J. Differential Equations, 261 (2016), 5561-5589.  doi: 10.1016/j.jde.2016.08.013.  Google Scholar

[24]

Y. WangC. Liu and Z. Tan, Well-posedness on a new hydrodynamic model of the fluid with the dilute charged particles, J. Differential Equations, 262 (2017), 68-115.  doi: 10.1016/j.jde.2016.09.026.  Google Scholar

[25]

Y. N. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279.  doi: 10.1007/s002050050188.  Google Scholar

[26]

H. J. 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.  Google Scholar

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