SOME SPECIAL SELF-SIMILAR SOLUTIONS FOR A MODEL OF INVISCID LIQUID-GAS TWO-PHASE FLOW

doi:10.1007/s10473-021-0107-3

Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (1): 114-126.doi: 10.1007/s10473-021-0107-3

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SOME SPECIAL SELF-SIMILAR SOLUTIONS FOR A MODEL OF INVISCID LIQUID-GAS TWO-PHASE FLOW

Jianwei DONG¹, Manwai YUEN²

1. School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450015, China;
2. Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong, China

Received:2019-11-05 Revised:2020-08-27 Online:2021-02-25 Published:2021-04-06
Contact: Manwai YUEN E-mail:nevetsyuen@hotmail.com
About author:Jianwei DONG,E-mail:dongjianweiccm@163.com
Supported by:
The research was supported by the Project of Youth Backbone Teachers of Colleges and Universities in Henan Province (2019GGJS176), the Natural Science Foundation of Henan Province Science and Technology Department (162300410077), the Outstanding Youth Foundation of Science and Technology Innovation of Henan Province (2018JQ0004), the Aeronautical Science Foundation of China (2017ZD55014), the Basic Research Projects of Key Scientific Research Projects Plan in Henan Higher Education Institutions (20zx003), and the Internal Research Grant from the Education University of Hong Kong (RG 15/2018-2019R).

Abstract

Abstract: 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.

Key words: two-phase flow, analytical solutions, self-similar, blowup, global solutions

CLC Number:

35L60

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.

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References

[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

SOME SPECIAL SELF-SIMILAR SOLUTIONS FOR A MODEL OF INVISCID LIQUID-GAS TWO-PHASE FLOW

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