Glasnik Matematicki, Vol. 61, No. 1 (2026), 161-174. \( \)

FINITE-TIME BLOW-UP OF A CLASSICAL SOLUTION TO THE TWO-FLUID MODEL WITH DENSITY-DEPENDENT VISCOSITY

Kaiyue Wang and Tong Tang

School of Mathematical Science, Yangzhou University, 225002 Yangzhou, China
e-mail:wky17861193382@163.com

School of Mathematical Science, Yangzhou University, 225002 Yangzhou, P. R. China
e-mail:tt0507010156@126.com

Abstract.   This paper concerns the initial-boundary value problem for a compressible two-fluid model with density-dependent viscosities (possibly degenerating in vacuum), subject to Dirichlet boundary conditions. We prove that the two-fluid system with non-monotone pressure will blow up in finite time under the assumption that the initial densities include an isolated mass group.

2020 Mathematics Subject Classification.   76T10, 35Q30, 35M33

Key words and phrases.   Two-fluid model, blow-up, non-monotone pressure

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.61.1.07

References:

  1. C. E. Brennen, Fundamentals of multiphase flow, Cambridge University Press, Cambridge, 2005.
    CrossRef

  2. D. Bresch and P. E. Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. of Math. (2) 188 (2018), 577–684.
    MathSciNet    CrossRef

  3. D. Bresch, P. B. Mucha and E. Zatorska, Finite-energy solutions for compressible two-fluid Stokes system, Arch. Ration. Mech. Anal. 232 (2019), 987–1029.
    MathSciNet    CrossRef

  4. S. M. Chen and C. J. Zhu, The global classical solution to a 1D two-fluid model with density-dependent viscosity and vacuum, Sci. China Math. 65 (2022), 2563–2582.
    MathSciNet    CrossRef

  5. B. Duan, Z. Luo and W. Yan, Finite-time blow-up of classical solutions to the rotating shallow water system with degenerate viscosity, Z. Angew. Math. Phys. 70 (2019), Paper No. 54.
    MathSciNet    CrossRef

  6. S. Evje, A compressible two-phase model with pressure-dependent well-reservoir interaction, SIAM J. Math. Anal. 45 (2013), 518–546.
    MathSciNet    CrossRef

  7. E. Feireisl, Compressible Navier-Stokes equations with a non-monotone pressure law, J. Differential Equations 184 (2002), 97–108.
    MathSciNet    CrossRef

  8. E. Feireisl, Dynamics of viscous compressible fluids, Oxford University Press, Oxford, 2004.
    MathSciNet

  9. F. F. Hao, H. L. Li, L. Y. Shang and S. Zhao, Recent progress on outflow/inflow problem for viscous multi-phase flow, Commun. Appl. Math. Comput. 5 (2023), 987–1014.
    MathSciNet    CrossRef

  10. M. Ishii, Thermo-fluid dynamic theory of two-fluid flow, Eyrolles, Paris, 1975.

  11. M. Ishii, One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-fluid flow regimes, Argonne National Laboratory Report, USA, 1977, ANL-77-47.

  12. B. J. Jin, Y. S. Kwon, Š. Nečasová and A. Novotný, Existence and stability of dissipative turbulent solutions to a simple bi-fluid model of compressible fluids, J. Elliptic Parabol. Equ. 7 (2021), 537–570.
    MathSciNet    CrossRef

  13. Q. S. Jiu, Y. X. Wang and Z. P. Xin, Remarks on blow-up of smooth solutions to the compressible fluid with constant and degenerate viscosities, J. Differential Equations 259 (2015), 2981–3003.
    MathSciNet    CrossRef

  14. M. Kalousek, S. Mitra and Š. Nečasová, The existence of a weak solution for a compressible multicomponent fluid structure interaction problem, J. Math. Pures Appl. (9) 184 (2024), 118–189.
    MathSciNet    CrossRef

  15. M. Kalousek and Š. Nečasová, On existence of weak solutions to a Baer-Nunziato type system, J. Differential Equations 452 (2026), Paper No. 113804.
    MathSciNet    CrossRef

  16. A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech. 41 (1977), 273–282.
    MathSciNet    CrossRef

  17. S. Kracmar, Y. S. Kwon, Š. Nečasová and A. Novotný, Weak solutions for a bifluid model for a mixture of two compressible noninteracting fluids with general boundary data, SIAM J. Math. Anal. 54 (2022), 818–871.
    MathSciNet    CrossRef

  18. H. L. Li, Y. X. Wang and Z. P. Xin, Non-existence of classical solutions with finite energy to the Cauchy problem of the compressible Navier-Stokes equations, Arch. Ration. Mech. Anal. 232 (2019), 557–590.
    MathSciNet    CrossRef

  19. M. L. Li, Z. A. Yao and R. F. Yu, Non-existence of global classical solutions to barotropic compressible Navier-Stokes equations with degenerate viscosity and vacuum, J. Differential Equations 306 (2022), 280–295.
    MathSciNet    CrossRef

  20. H. L. Li and L. Y. Shou, Global existence of weak solutions to the drift-flux system for general pressure laws, Sci. China Math. 66 (2023), 251–284.
    MathSciNet    CrossRef

  21. P. L. Lions, Mathematical topics in fluid dynamics. Vol. 2, Oxford University Press, New York, 1998.
    MathSciNet

  22. A. J. Majda and A. L. Bertozzi, Vorticity and incompressible flow, Cambridge University Press, England, 2002.
    MathSciNet

  23. 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), 67–104.
    MathSciNet    CrossRef

  24. A. Novotný and M. Pokorný, Weak solutions for some compressible multicomponent fluid models, Arch. Ration. Mech. Anal. 235 (2020), 355–403.
    MathSciNet    CrossRef

  25. M. Pokorný, A. Wróblewska-Kamińska and E. Zatorska, Two-phase compressible/incompressible Navier-Stokes system with inflow-outflow boundary conditions, J. Math. Fluid Mech. 24 (2022), Paper No. 87.
    MathSciNet    CrossRef

  26. O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, J. Differential Equations 245 (2008), 1762–1774.
    MathSciNet    CrossRef

  27. A. Vasseur, H. Y. Wen and C. Yu, Global weak solution to the viscous two-fluid model with finite energy, J. Math. Pures Appl. 125 (2019), 247–282.
    MathSciNet    CrossRef

  28. G. B. Wallis, One-dimensional two-phase flow, McGraw-Hill, New York, 1969.

  29. H. Y. Wen, On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions, Calc. Var. Partial Differential Equations 60 (2021), Paper No. 158.
    MathSciNet    CrossRef

  30. Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equations with compact density, Comm. Pure Appl. Math. 51 (1988), 229–240.
    MathSciNet    CrossRef

  31. Z. P. Xin and W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations, Comm. Math. Phys. 321 (2013), 529–541.
    MathSciNet    CrossRef

  32. L. Yao, T. Zhang and C. J. Zhu, Existence of asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model, SIAM J. Math. Anal. 42 (2010), 1874–1897.
    MathSciNet    CrossRef

  33. N. Zuber, On the dispersed two-phase flow in the laminar flow regime, Chem. Eng. Sci. 19 (1964), 897–917.
    CrossRef
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