ORIGINAL PAPER

Global Existence of Smooth Solutions for the One-Dimensional Full Euler System for a Dusty Gas

Expand
  • Department of Mathematics, Shanghai University, Shanghai, 200444, China

Received date: 2021-05-23

  Revised date: 2021-05-23

  Online published: 2023-08-29

Supported by

This work is partially supported by the National Natural Science Foundation of China (12071278).

Abstract

We study the existence of global-in-time classical solutions for the one-dimensional nonisentropic compressible Euler system for a dusty gas with large initial data. Using the characteristic decomposition method proposed by Li et al. (Commun Math Phys 267: 1–12, 2006), we derive a group of characteristic decompositions for the system. Using these characteristic decompositions, we find a sufficient condition on the initial data to ensure the existence of global-in-time classical solutions.

Cite this article

Geng Lai, Yingchun Shi . Global Existence of Smooth Solutions for the One-Dimensional Full Euler System for a Dusty Gas[J]. Communications on Applied Mathematics and Computation, 2023 , 5(3) : 1235 -1246 . DOI: 10.1007/s42967-022-00197-y

References

[1] Chen, G.: Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations. J. Hyperbol. Differ. Equ. 8, 671–690 (2011)
[2] Chen, G., Pan, R.H., Zhu, S.G.: Singularity formation for the compressible Euler equations. SIAM J. Math. Anal. 49, 2591–2614 (2017)
[3] Chen, G., Young, R.: Shock-free solutions of the compressible Euler equations. Arch. Rational Mech. Anal. 217, 1265–1293 (2015)
[4] Grassin, M.: Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J. 47, 1397–1432 (1998)
[5] Hu, Y., Li, J.Q., Sheng, W.C.: Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations. Z. Angew. Math. Phys. 63, 1021–1046 (2012)
[6] John, F.: Formation of singularities in one-dimensional nonlinear waves propagation. Commun. Pure Appl. Math. 27, 377–405 (1974)
[7] Lai, G.: On the expansion of a wedge of van der Waals gas into a vacuum. J. Differ. Equ. 259, 1181–1202 (2015)
[8] Lai, G.: On the expansion of a wedge of van der Waals gas into a vacuum II. J. Differ. Equ. 260, 3538–3575 (2016)
[9] Lai, G.: Interaction of composite waves of the two-dimensional full Euler equations for van der Waals gases. SIAM J. Math. Anal. 50, 3535–3597 (2018)
[10] Lai, G.: Global solutions to a class of two-dimensional Riemann problems for the isentropic Euler equations with general equations of state. Indiana Univ. Math. J. 68, 1409–1464 (2019)
[11] Lai, G.: Global non-isentropic rotational supersonic flows in a semi-infinite divergent duct. SIAM J. Math. Anal. 52, 5121–5154 (2020)
[12] Lai, G., Zhao, Q.: Existence of global bounded smooth solutions for the one-dimensional nonisentropic Euler system. Math. Meth. Appl. Sci. 44, 2226–2236 (2021)
[13] Lax, P.: Development of singularities of solutions on nonlinear hyperbolic partial differential equations. J. Math. Phys. 5, 611–613 (1964)
[14] Li, J.Q., Yang, Z.C., Zheng, Y.X.: Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations. J. Differ. Equ. 250, 782–798 (2011)
[15] Li, J.Q., Zhang, T., Zheng, Y.X.: Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations. Commun. Math. Phys. 267, 1–12 (2006)
[16] Li, J.Q., Zheng, Y.X.: Interaction of rarefaction waves of the two-dimensional self-similar Euler equations. Arch. Ration. Mech. Anal. 193, 623–657 (2009)
[17] Li, J.Q., Zheng, Y.X.: Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations. Commun. Math. Phys. 296, 303–321 (2010)
[18] Li, T.T.: Global Classical Solutions for Quasilinear Hyperbolic System. Wiley, Oxford (1994)
[19] Li, T.T., Yu, W.C.: Boundary value problem for quasilinear hyperbolic systems. Duke University, Durham (1985)
[20] Lin, L.W., Vong, S.: A note on the existence and nonexistence of globally bounded classical solutions for nonisentropic gas dynamics. Acta Math. Sci. 26, 537–540 (2006)
[21] Liu, F.G.: Global smooth resolvability for one-dimensional gas dynamics systems. Nonlinear Anal. 36, 25–34 (1999)
[22] Liu, T.P.: Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations. J. Differ. Equ. 30, 92–111 (1979)
[23] Magali, L.M.: Global smooth solutions of Euler equations for Van der Waals gases. SIAM J. Math. Anal. 43, 877–903 (2011)
[24] Pan, R.H., Zhu, Y.: Singularity formation for one dimensional full Euler equations. J. Differ. Equ. 261, 7132–7144 (2016)
[25] Saffmann, P.G.: The stability of laminar flow of a dusty gas. J. Fluid Mech. 13, 120–128 (1962)
[26] Serre, D.: Solutions classiques globales des équations d'Euler pour un fluide parfait compressible. Ann. Inst. Fourier 47, 139–153 (1997)
[27] Sideris, T.C.: Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys. 101, 475–485 (1985)
[28] Steiner, H., Hirschler, T.: A self-similar solution of a shock propagation in a dusty gas. Eur. J. Mech. B Fluids 21, 371–380 (2002)
[29] Zhao, Y.C.: A class of global smooth solutions of the one dimensional gas dynamics system. IMA Series No. 545. June (1989)
[30] Zhu, C.J.: Global smooth solution of the nonisentropic gas dynamics system. Proc. R. Soc. Edinb. 126A, 768–775 (1996)
Options
Outlines

/