Research article

Space-time decay rate of high-order spatial derivative of solution for 3D compressible Euler equations with damping

  • Received: 03 November 2022 Revised: 12 January 2023 Accepted: 28 January 2023 Published: 12 May 2023
  • We are concerned with the space-time decay rate of high-order spatial derivatives of solutions for 3D compressible Euler equations with damping. For any integer $ \ell\geq3 $, Kim (2022) showed the space-time decay rate of the $ k(0\leq k\leq \ell-2) $th-order spatial derivative of the solution. By making full use of the structure of the system, and employing different weighted energy methods for $ 0\leq k \leq \ell-2, k = \ell-1, k = \ell $, it is shown that the space-time decay rate of the $ (\ell-1) $th-order and $ \ell $th-order spatial derivative of the strong solution in weighted Lebesgue space $ L_\sigma^2 $ are $ t^{-\frac{3}{4}-\frac{\ell-1}{2}+\frac{\sigma}{2}} $ and $ t^{-\frac{3}{4}-\frac{\ell}{2}+\frac{\sigma}{2}} $ respectively, which are totally new as compared to that of Kim (2022) [1].

    Citation: Qin Ye. Space-time decay rate of high-order spatial derivative of solution for 3D compressible Euler equations with damping[J]. Electronic Research Archive, 2023, 31(7): 3879-3894. doi: 10.3934/era.2023197

    Related Papers:

  • We are concerned with the space-time decay rate of high-order spatial derivatives of solutions for 3D compressible Euler equations with damping. For any integer $ \ell\geq3 $, Kim (2022) showed the space-time decay rate of the $ k(0\leq k\leq \ell-2) $th-order spatial derivative of the solution. By making full use of the structure of the system, and employing different weighted energy methods for $ 0\leq k \leq \ell-2, k = \ell-1, k = \ell $, it is shown that the space-time decay rate of the $ (\ell-1) $th-order and $ \ell $th-order spatial derivative of the strong solution in weighted Lebesgue space $ L_\sigma^2 $ are $ t^{-\frac{3}{4}-\frac{\ell-1}{2}+\frac{\sigma}{2}} $ and $ t^{-\frac{3}{4}-\frac{\ell}{2}+\frac{\sigma}{2}} $ respectively, which are totally new as compared to that of Kim (2022) [1].



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    [1] J. Kim, Space-time decay rate for 3D compressible Euler equations with damping, J. Evol. Equations, 22 (2022), 1424–3199. https://doi.org/10.1007/s00028-022-00830-6 doi: 10.1007/s00028-022-00830-6
    [2] L. Hsiao, T. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. Math. Phys., 143 (1992), 599–605. http://projecteuclid.org/euclid.cmp/1104249084
    [3] L. Hsiao, T. Liu, Nonlinear diffusive phenomena of nonlinear hyperbolic systems, Chin. Ann. Math. Ser. B, 14 (1993), 465–480.
    [4] L. Hsiao, D. Serre, Global existence of solutions for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal., 27 (1996), 70–77. https://doi.org/10.1137/S0036141094267078 doi: 10.1137/S0036141094267078
    [5] K. Nishihara, W. Wang, T. Yang, $L_p$-convergence rate to nonlinear diffusion waves for $p$-system with damping, J. Differ. Equations, 161 (2000), 191–21. https://doi.org/10.1006/jdeq.1999.3703 doi: 10.1006/jdeq.1999.3703
    [6] H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of $p$-system with damping, J. Differ. Equations, 174 (2001), 200–236. https://doi.org/10.1006/jdeq.2000.3936 doi: 10.1006/jdeq.2000.3936
    [7] Q. Chen, Z. Tan, Time decay of solutions to the compressible Euler equations with damping, Kinet. Rel. Models, 166 (2003), 359–376. https://doi.org/10.1007/s00205-002-0234-5 doi: 10.1007/s00205-002-0234-5
    [8] F. Huang, P. Marcati, R. Pan, Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 176 (2005), 1–24. https://doi.org/10.1007/s00205-004-0349-y doi: 10.1007/s00205-004-0349-y
    [9] F. Huang, R. Pan, Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differ. Equations, 220 (2006), 207–233. https://doi.org/10.1016/j.jde.2005.03.0125 doi: 10.1016/j.jde.2005.03.0125
    [10] C. Zhu, Convergence rates to nonlinear diffusion waves for weak entropy solutions to $p$-system with damping, Sci. China Ser. A, 46 (2003), 562–575. https://doi.org/10.1360/03ys9057 doi: 10.1360/03ys9057
    [11] J. Jang, N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Commun. Pure Appl. Math., 62 (2009), 1327–1385. https://doi.org/10.1002/cpa.20285 doi: 10.1002/cpa.20285
    [12] R. Duan, M. Jiang, Y. Zhang, Boundary effect on asymptotic behavior of solutions to the $p$-system with time-dependent damping, Adv. Math. Phys., (2020), 3060867. https://doi.org/10.1155/2020/3060867 doi: 10.1155/2020/3060867
    [13] L. Hsiao, R. Pan, The damped $p$-system with boundary effects, in Nonlinear PDE's, Dynamics and Continuum Physics, 255 (2000), 109–123. https://doi.org/10.1090/conm/255/03977
    [14] J. Jang, N. Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Commun. Pure Appl. Math., 68 (2015), 61–111. https://doi.org/10.1002/cpa.21517 doi: 10.1002/cpa.21517
    [15] W. Wang, T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differ. Equations, 173 (2001), 410–450. https://doi.org/10.1006/jdeq.2000.3937 doi: 10.1006/jdeq.2000.3937
    [16] T. C. Sideris, B. Thomases, D. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Commun. Partial Differ. Equations, 28 (2003), 795–816. https://doi.org/10.1081/PDE-120020497 doi: 10.1081/PDE-120020497
    [17] J. Liao, W. Wang, T, Yang, $L^p$ convergence rates of planar waves for multi-dimensional Euler equations with damping, J. Differ. Equations, 247 (2009), 303–329. https://doi.org/10.1016/j.jde.2009.03.011 doi: 10.1016/j.jde.2009.03.011
    [18] D. Fang, J. Xu, Existence and asymptotic behavior of $C^1$ solutions to the multi-dimensional compressible Euler equations with damping, Nonlinear Anal. Theory Methods Appl., 70 (2009), 244–261. https://doi.org/10.1016/j.na.2007.11.049 doi: 10.1016/j.na.2007.11.049
    [19] Z. Tan, Y. Wang, On hyperbolic-dissipative systems of composite type, J. Differ. Equations, 260 (2016), 1091–1125. https://doi.org/10.1016/j.jde.2015.09.025 doi: 10.1016/j.jde.2015.09.025
    [20] Z. Tan, Y. Wang, Global solution and large-time behavior of the $3D$ compressible Euler equations with damping, J. Differ. Equations, 254 (2013), 1686–1704. https://doi.org/10.1016/j.jde.2012.10.026 doi: 10.1016/j.jde.2012.10.026
    [21] M. Jiang, Y. Zhang, Existence and asymptotic behavior of global smooth solution for $p$-system with nonlinear damping and fixed boundary effect, Math. Methods Appl. Sci., 37 (2014), 2585–2596. https://doi.org/10.1002/mma.2998 doi: 10.1002/mma.2998
    [22] R. Pan, K. Zhao, The 3D compressible Euler equations with damping in a bounded domain, J. Differ. Equations, 246 (2009), 581–596. https://doi.org/10.1016/j.jde.2008.06.007 doi: 10.1016/j.jde.2008.06.007
    [23] Y. Zhang, Z. Tan, Existence and asymptotic behavior of global smooth solution for $p$-system with damping and boundary effect, Nonlinear Anal. Theory Methods Appl., 72 (2010), 605–619. https://doi.org/10.1016/j.na.2009.10.046 doi: 10.1016/j.na.2009.10.046
    [24] Q. Chen, Z. Tan, Time decay of solutions to the compressible Euler equations with damping, Kinet. Rel. Models, 7 (2014), 605–619. https://doi.org/10.3934/krm.2014.7.605 doi: 10.3934/krm.2014.7.605
    [25] Z. Tan, G. Wu, Large time behavior of solutions for compressible Euler equations with damping in $\Bbb R^3$, J. Diffe. Equations, 252 (2012), 1546–156. https://doi.org/10.1016/j.jde.2011.09.003 doi: 10.1016/j.jde.2011.09.003
    [26] L. Nirenberg, On elliptic partial differential equations, in Il principio di minimo e sue applicazioni alle equazioni funzionali, 13 (1959), 115–162.
    [27] Z. Tan, Y. Wang, F. Xu, Large-time behavior of the full compressible Euler-Poisson system without the temperature damping, Discrete Contin. Dyn. Syst., 36 (2016), 1583–1601. https://doi.org/10.3934/dcds.2016.36.1583 doi: 10.3934/dcds.2016.36.1583
    [28] S. Weng, Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 270 (2016), 2168–218. https://doi.org/10.1016/j.jfa.2016.01.021 doi: 10.1016/j.jfa.2016.01.021
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