Research article

Regularity of weak solution of the compressible Navier-Stokes equations with self-consistent Poisson equation by Moser iteration

  • Received: 13 April 2023 Revised: 08 July 2023 Accepted: 13 July 2023 Published: 19 July 2023
  • MSC : 35L65, 35Q35, 76N10

  • In this paper, we consider the regularity of the weak solution to the compressible Navier-Stokes-Poisson equations in period domain $ \Omega \subseteq \mathbb{R}^3 $ provided that the density $ \rho(t, x) $ with integrability on the space $ L^{\infty}(0, T;L^{q_0}(\Omega)) $ where $ q_0 $ satisfies a certain condition and $ T > 0 $, by which we could present that $ \sup_{t, x}\rho(t, x) < \infty $ and $ \inf_{t, x}\rho(t, x) > 0 $. Furthermore, we develop the estimate for the velocity $ \Vert u\Vert_{L^{\infty}} $ by the Moser iteration method and Gronwall inequality.

    Citation: Cuiman Jia, Feng Tian. Regularity of weak solution of the compressible Navier-Stokes equations with self-consistent Poisson equation by Moser iteration[J]. AIMS Mathematics, 2023, 8(10): 22944-22962. doi: 10.3934/math.20231167

    Related Papers:

  • In this paper, we consider the regularity of the weak solution to the compressible Navier-Stokes-Poisson equations in period domain $ \Omega \subseteq \mathbb{R}^3 $ provided that the density $ \rho(t, x) $ with integrability on the space $ L^{\infty}(0, T;L^{q_0}(\Omega)) $ where $ q_0 $ satisfies a certain condition and $ T > 0 $, by which we could present that $ \sup_{t, x}\rho(t, x) < \infty $ and $ \inf_{t, x}\rho(t, x) > 0 $. Furthermore, we develop the estimate for the velocity $ \Vert u\Vert_{L^{\infty}} $ by the Moser iteration method and Gronwall inequality.



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    [1] H. Cai, Z. Tan, Existence and stability of stationary solutions to the compressible Navier-Stokes-Poisson equations, Nonlinear Anal.-Real, 32 (2016), 260–293. https://doi.org/10.1016/j.nonrwa.2016.04.010 doi: 10.1016/j.nonrwa.2016.04.010
    [2] Y. Cho, H. Choe, H. Kim, Unique solvability of the boundary value problems for compressible viscous fluids, J. Math. Pure. Appl., 83 (2004), 243–275. https://doi.org/10.1016/j.matpur.2003.11.004 doi: 10.1016/j.matpur.2003.11.004
    [3] H. Choe, H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differ. Equations, 190 (2003), 504–523. https://doi.org/10.1016/S0022-0396(03)00015-9 doi: 10.1016/S0022-0396(03)00015-9
    [4] H. Choe, B. Jin, Regularity of weak solutions of the compressible Navier-Stokes equations, J. Korean Math. Soc., 40 (2003), 1031–1050. https://doi.org/10.4134/JKMS.2003.40.6.1031 doi: 10.4134/JKMS.2003.40.6.1031
    [5] B. Desjardins, Regularity of weak solutions of the compressible isentropic navier-stokes equations, Commun. Part. Diff. Eq., 22 (1997), 977–1008. https://doi.org/10.1080/03605309708821291 doi: 10.1080/03605309708821291
    [6] E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358–392. https://doi.org/10.1007/PL00000976 doi: 10.1007/PL00000976
    [7] G. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, New York: Springer, 2011. https://doi.org/10.1007/978-0-387-09620-9
    [8] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Berlin: Springer, 2001. https://doi.org/10.1007/978-3-642-61798-0
    [9] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differ. Equations, 120 (1995), 215–254. https://doi.org/10.1006/jdeq.1995.1111 doi: 10.1006/jdeq.1995.1111
    [10] C. Jia, Z. Tan, Regularity for the weak solutions to certain parabolic systems under certain growth condition, J. Math. Anal. Appl., 468 (2018), 324–343. https://doi.org/10.1016/j.jmaa.2018.08.014 doi: 10.1016/j.jmaa.2018.08.014
    [11] T. Kobayashi, T. Suzuki, Weak solutions to the Navier-Stokes-Poisson equation, Adv. Math. Sci. Appl., 18 (2008), 141–168.
    [12] P. Lions, Mathematical topics in fluid mechanics: volume 2: compressible models, Oxford: Oxford University Press, 1998.
    [13] O. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, 2 Eds., New York: Gordon and Breach Science Publishers, 1964.
    [14] Š. Matušů-Nečasová, M. Okada, T. Makino, Free boundary problem for the equation of spherically symmetric motion of viscous gas (II), Japan J. Indust. Appl. Math., 12 (1995), 195–203. https://doi.org/10.1007/BF03167288 doi: 10.1007/BF03167288
    [15] Š. Matušů-Nečasová, M. Okada, T. Makino, Free boundary problem for the equation of spherically symmetric motion of viscous gas (III), Japan J. Indust. Appl. Math., 14 (1997), 199–213. https://doi.org/10.1007/BF03167264 doi: 10.1007/BF03167264
    [16] G. Mingione, Nonlinear aspects of Calderón-Zygmund theory, Jahresber. Dtsch. Math. Ver., 112 (2010), 159–191. https://doi.org/10.1365/s13291-010-0004-5 doi: 10.1365/s13291-010-0004-5
    [17] M. Okada, Š. Matušů-Nečasová, T. Makino, Free boundary problem for the equation of one dimensional motion of compressible gas with density-dependent viscosity, Ann. Univ. Ferrara, 48 (2002), 1–20. https://doi.org/10.1007/BF02824736 doi: 10.1007/BF02824736
    [18] M. Okada, T. Makino, Free boundary problem for the equation of spherically symmetric motion of viscous gas, Japan J. Industr. Appl. Math., 10 (1993), 219. https://doi.org/10.1007/BF03167573 doi: 10.1007/BF03167573
    [19] Y. Sun, C. Wang, Z. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Rational Mech. Anal., 201 (2011), 727–742. https://doi.org/10.1007/s00205-011-0407-1 doi: 10.1007/s00205-011-0407-1
    [20] Z. Tan, Y. Wang, Propagation of density-oscillations in solutions to the compressible Navier-Stokes-Poisson system, Chin. Ann. Math. Ser. B, 29 (2008), 501–520. https://doi.org/10.1007/s11401-007-0380-z doi: 10.1007/s11401-007-0380-z
    [21] Z. Tan, T. Yang, H. Zhao, Q. Zou, Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data, SIAM J. Math. Anal., 45 (2013), 547–571. https://doi.org/10.1137/120876174 doi: 10.1137/120876174
    [22] Z. Tan, Y. Zhang, Strong solutions of the coupled Navier-Stokes-Poisson equations for isentropic compressible fluids, Acta Math. Sci., 30 (2010), 1280–1290. https://doi.org/10.1016/S0252-9602(10)60124-5 doi: 10.1016/S0252-9602(10)60124-5
    [23] R. Temam, Navier-Stokes equations and nonlinear functional analysis, Philadelphia: Society of Industrial and Applied Mathematics, 1995. https://doi.org/10.1007/978-0-387-09620-9
    [24] A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm.-Sci., 10 (1983), 607–647.
    [25] H. Wen, C. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. Math., 248 (2013), 534–572. https://doi.org/10.1016/j.aim.2013.07.018
    [26] J. Yin, Z. Tan, Global existence of the radially symmetric strong solution to Navier-Stokes-Poisson equations for isentropic compressible fluids, Acta Math. Sin.-English Ser., 25 (2009), 1703–1720. https://doi.org/10.1007/s10114-009-6595-z doi: 10.1007/s10114-009-6595-z
    [27] Y. Zhang, Z. Tan, On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow, Math. Method. Appl. Sci., 30 (2007), 305–329. https://doi.org/10.1002/mma.786 doi: 10.1002/mma.786
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