In this article, we are concerned with analytical solutions for a model of inviscid liquid-gas two-phase flow. On the basis of Yuen's works [25, 27-29] on self-similar solutions for compressible Euler equations, we present some special self-similar solutions for a model of inviscid liquid-gas two-phase flow in radial symmetry with and without rotation, and in elliptic symmetry without rotation. Some blowup phenomena and the global existence of the solutions obtained are classified.
Jianwei DONG
,
Manwai YUEN
. SOME SPECIAL SELF-SIMILAR SOLUTIONS FOR A MODEL OF INVISCID LIQUID-GAS TWO-PHASE FLOW[J]. Acta mathematica scientia, Series B, 2021
, 41(1)
: 114
-126
.
DOI: 10.1007/s10473-021-0107-3
[1] An H L, Yuen M W. Drifting solutions with elliptic symmetry for the compressible Navier-Stokes equations with density-dependent viscosity. J Math Phys, 2014, 55(5):053506
[2] Chen Y S, Zhang M. A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions. Kinet Relat Model, 2016, 9(3):429-441
[3] Cui H B, Wen H Y, Yin H Y. Global classical solutions of viscous liquid-gas two-phase flow model. Math Meth Appl Sci, 2013, 36(5):567-583
[4] Deng Y B, Xiang J L, Yang T. Blowup phenomena of solutions to Euler-Poisson equations. J Math Anal Appl, 2003, 286(1):295-306
[5] Evje S. Global weak solutions for a compressible gas-liquid model with well-deformation interaction. J Differential Equations, 2011, 251(8):2352-2386
[6] Evje S. Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells. SIAM J Math Anal, 2011, 43(4):1887-1922
[7] Evje S, Karlsen K H. Global existence of weak solutions for a viscous two-phase model. J Differential Equations, 2008, 245(9):2660-2703
[8] Evje S, Karlsen K H. Global weak solutions for a viscous liquid-gas model with singular pressure law. Commun Pure Appl Anal, 2009, 8(6):1867-1894
[9] Friis H A, Evje S. Global weak solutions for a gas-liquid model with external forces general pressure law. SIAM J Appl Math, 2011, 71(2):409-442
[10] Friis H A, Evje S. Well-posedness of a compressible gas-liquid model with a friction term important for well control operations. SIAM J Appl Math, 2011, 71(6):2014-2047
[11] Hao C C, Li H L. Well-posedness for a multidimensional viscous liquid-gas two-phase flow model. SIAM J Math Anal, 2012, 44(3):1304-1332
[12] Huang F M, Wang D H, Yuan D F. Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Disc Conti Dyn Sys, 2019, 39(6):3535-3575
[13] Kwong M K, Yuen M W. Periodic solutions of 2D isothermal Euler-Poisson equations with possible applications to spiral and disk-like galaxies. J Math Anal Appl, 2014, 420(2):1854-1863
[14] Li T H. Some special solutions of the multidimensional Euler equations in RN. Commun Pure Appl Anal, 2005, 4(4):757-762
[15] Li T H, Wang D H. Blowup phenomena of solutions to the Euler equations for compressible fluid flow. J Differential Equations, 2006, 221(1):91-101
[16] Makino T. Exact solutions for the compressible Eluer equation. Journal of Osaka Sangyo University Natural Sciences, 1993, 95(1):21-35
[17] Ruan L Z, Wang D H, Weng S K, Zhu C J. Rectilinear vortex sheets of inviscid liquid-gas two-phase flow:linear stability. Commun Math Sci, 2016, 14(3):735-776
[18] Wang W J, Wang W K. Large time behavior for the system of a viscous liquid-gas two-phase flow model in R3. J Differential Equations, 2016, 261(10):5561-5589
[19] Wen H Y, Yao L, Zhu C J. A blow-up criterion of strong solution to a 3D viscous liquid-gas two-phase flow model with vacuum. J Math Pures Appl, 2012, 97(3):204-229
[20] Yao L, Zhu C J. Existence and uniqueness of global weak solution to a two-phase flow model with vacuum. Math Ann, 2011, 349(4):903-928
[21] Yao L, Zhu C J. Free boundary value problem for a viscous two-phase model with mass-dependent viscosity. J Differential Equations, 2009, 247(10):2705-2739
[22] Yao L, Zhang T, Zhu C J. A blow-up criterion for a 2D viscous liquid-gas two-phase flow model. J Differential Equations, 2011, 250(8):3362-3378
[23] Yao L, Zhang T, Zhu C J. Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model. SIAM J Math Anal, 2010, 42(4):1874-1897
[24] Yuen M W. Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars. J Math Anal Appl, 2008, 341(1):445-456
[25] Yuen M W. Analytical solutions to the Navier-Stokes equations. J Math Phys, 2008, 49(11):113102
[26] Yuen M W. Blowup solutions for a class of fluid dynamical equations in RN. J Math Anal Appl, 2007, 329(2):1064-1079
[27] Yuen M W. Rotational and self-similar solutions for the compressible Euler equations in R3. Commun Nonlinear Sci Numer Simul, 2015, 20(3):634-640
[28] Yuen M W. Self-similar solutions with elliptic symmetry for the compressible Euler and Navier-Stokes equations in RN. Commun Nonlinear Sci Numer Simul, 2012, 17(12):4524-4528
[29] Yuen M W. Vortical and self-similar flows of 2D compressible Euler equations. Commun Nonlinear Sci Numer Simul, 2014, 19(7):2172-2180
[30] Zhang T, Zheng Y X. Exact spiral solutions of the two-dimensional Euler equations. Discrete Contin Dynam Systems, 1997, 3(1):117-133
