# 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 & 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, (1981), 187.   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.  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.  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.  doi: 10.1080/03605302.2012.696296.  Google Scholar [5] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms,, World Scientific Publishing Co., (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.  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.  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.  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.  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.  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.  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, (1983).   Google Scholar [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).   Google Scholar [14] A. Majda, Compressible Fluid Flow and Conservation laws in Several Space Variables,, Springer-Verlag, (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.  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.   Google Scholar [17] T. Nishida, Global solutions for an initial-boundary value problem of a quasilinear hyperbolic systems,, Proc. Japan Acad., 44 (1968), 642.  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.   Google Scholar [19] L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115.   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.  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.  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.  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.  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, (1981), 187.   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.  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.  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.  doi: 10.1080/03605302.2012.696296.  Google Scholar [5] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms,, World Scientific Publishing Co., (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.  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.  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.  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.  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.  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.  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, (1983).   Google Scholar [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).   Google Scholar [14] A. Majda, Compressible Fluid Flow and Conservation laws in Several Space Variables,, Springer-Verlag, (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.  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.   Google Scholar [17] T. Nishida, Global solutions for an initial-boundary value problem of a quasilinear hyperbolic systems,, Proc. Japan Acad., 44 (1968), 642.  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.   Google Scholar [19] L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115.   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.  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.  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.  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.  doi: 10.1360/03ys9057.  Google Scholar
 [1] Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 [2] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [3] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [4] 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 [5] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456 [6] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 [7] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [8] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [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] 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 [11] Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 [12] José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271 [13] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268 [14] Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258 [15] Sergio Conti, Georg Dolzmann. Optimal laminates in single-slip elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 1-16. doi: 10.3934/dcdss.2020302 [16] Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158 [17] Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078 [18] Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044 [19] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [20] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

2019 Impact Factor: 1.311