GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY*

  • Shijin Ding ,
  • Yinghua Li ,
  • Yu Wang
Expand
  • School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Shijin Ding, E-mail: dingsj@scnu.edu.cn; Yu Wang, E-mail: yuwang@m.scnu.edu.cn

Received date: 2022-09-23

  Revised date: 2023-06-16

  Online published: 2024-02-27

Supported by

Ding's research was supported by the Key Project of the NSFC (12131010), the NSFC (11771155, 12271032) and the NSF of Guangdong Province (2021A1515010249, 2021A1515010303). Li's research was supported by the NSFC (11971179, 12371205).

Abstract

This paper is concerned with the Navier-Stokes/Allen-Cahn system, which is used to model the dynamics of immiscible two-phase flows. We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form of $\eta(\rho)=\rho^\alpha$. The existence of unique global $H^{2m}$-solutions $(m\in\mathbb N)$ to the free boundary problem is proven for when $0<\alpha<\frac14$. Furthermore, we obtain the global $C^\infty$-solutions if the initial data is smooth.

Cite this article

Shijin Ding , Yinghua Li , Yu Wang . GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY*[J]. Acta mathematica scientia, Series B, 2024 , 44(1) : 195 -214 . DOI: 10.1007/s10473-024-0111-5

References

[1] Blesgen T. A generalizaion of the Navier-Stokes equations to two-phase flow. J Phys D Appl Phys, 1999, 32: 1119-1123
[2] Chen M X, Guo X. Global large solutions for a coupled compressible Navier-Stokes/Allen-Cahn system with initial vacuum. Nonlinear Anal Real World Appl, 2017, 37: 350-373
[3] Chen S, Zhu C. Blow-up criterion and the global existence of strong/classical solutions to Navier-Stokes/Allen-Cahn system. Z Angew Math Phys, 2021, 72(1): Art 14
[4] Chen Y, He Q, Huang B, Shi X. Global strong solution to a thermodynamic compressible diffuse interface model with temperature dependent heat-conductivity in 1-D. Math Methods Appl Sci, 2021, 44: 12945-12962
[5] Chen Y, He Q, Huang B, Shi X.The Cauchy problem for non-isentropic compressible Navier-Stokes/Allen-Cahn system with degenerate heat-conductivity. arXiv:2005.11205
[6] Dai W, Ding S, Li Y.Global strong solutions of the compressible Navier-Stokes/Allen-Cahn system with density-dependent viscosity. preprint
[7] Ding S, Huang J, Liu X, Wen H. Global C$^\infty$-solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity. Math Methods Appl Sci, 2011, 34(12): 1499-1511
[8] Ding S, Li Y, Luo W. Global solutions for a coupled compressible Navier-Stokes/Allen-Cahn system in 1D. J Math Fluid Mech, 2013, 15(2): 335-360
[9] Ding S, Li Y, Tang Y. Strong solutions to 1D compressible Navier-Stokes/Allen-Cahn system with free boundary. Math Methods Appl Sci, 2019, 42(14): 4780-4794
[10] Freist$\rm\ddot{u}$hler H, Kotschote M. Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids. Arch Ration Mech Anal, 2017, 224(1): 1-20
[11] Fang D, Zhang T. Compressible Navier-Stokes equations with vacuum state in one dimension. Commun Pure Appl Anal, 2004, 3(4): 675-694
[12] Fang D, Zhang T. Compressible Navier-Stokes equations with vacuum state in the case of general pressure law. Math Methods Appl Sci, 2006, 29(10): 1081-1106
[13] Grad H.Asymptotic theory of the Boltzmann equation II//Laurmann J, ed. Rarefied Gas Dynamics, Vol 1. New York: Academic Press, 1963: 26-59
[14] Guo Z, Jiang S, Xie F. Global existence and asymptotic behavior of weak solutions to the 1D compressible Navier-Stokes equations with degenerate viscosity coefficient. Asymptot Anal, 2008, 60(1/2): 101-123
[15] He Q, Shi X. Energy stable discontinuous Galerkin method for compressible Navier-Stokes-Allen-Cahn system. Commun Nonlinear Sci Numer Simul, 2021, 98: Art 105771
[16] Heida M, Malek J, Rajagopal K R. On the development and generalizations of Allen-Cahn and Stefan equations within a thermodynamic framework. Z Angew Math Phys, 2012, 63: 759-776
[17] Jiang S. Global smooth solutions of the equations of a viscous, heat-conducting, one-dimensional gas with density-dependent viscosity. Math Nachr, 1998, 190: 169-183
[18] Jiang S, Xin Z, Zhang P. Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. Methods Appl Anal, 2005, 12(3): 239-251
[19] Kong H, Li, H, Zhang X. A blow-up criterion of spherically symmetric strong solutions to 3d compressible Navier-Stokes equations with free boundary. Acta Math Sci, 2016, 36B(4): 1153-1166
[20] Li Y, Yan Y, Ding S, Chen G. Global weak solutions fro 1D compressible Navier-Stokes/Allen-Cahn system with vacuum. Z Angew Math Phys, 2023, 74(1): Art 2
[21] Liu J. Local existence of solution to free boundary value problem for compressible Navier-Stokes equations. Acta Math Sci, 2012, 32B(4): 1298-1320
[22] Liu T, Xin Z, Yang T. Vacuum states of compressible flow. Discrete Conti Dyn Syst, 1998, 4: 1-32
[23] Okada M, Matu$\rm\breve{s}\dot{u}$-Ne$\rm\breve{c}$asov$\rm\acute{a}$ $\rm\breve{S}$, Makino T. Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity. Ann Univ Ferrara Sez VII (NS), 2002, 48: 1-20
[24] Qin Y, Huang L, Yao Z. Regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. J Differ Equ, 2008, 245: 3956-3973
[25] Su M. On global classical solutions to one-dimensional compressible Navier-Stokes/Allen-Cahn system with density-dependent viscosity and vacuum. Bound Value Probl, 2021, 2021: Art 92
[26] Vong S W, Yang T, Zhu C, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum (II). J Differ Equ, 2003, 192(2): 475-501
[27] Yan Y, Ding S, Li Y. Strong solutions for 1D compressible Navier-Stokes/Allen-Cahn system with phase variable dependent viscosity. J Differ Equ, 2022, 326(25): 1-48
[28] Yang X. A novel fully decoupled scheme with second-order time accuracy and unconditional energy stability for the Navier-Stokes equations coupled with mass-conserved Allen-Cahn phase-field model of two-phase incompressible flow. Internat J Numer Methods Engrg, 2021, 122(5): 1283-1306
[29] Yang T, Yao Z, Zhu C. Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Commun Partial Differ Equ, 2001, 26(5/6): 965-981
[30] Yang T, Zhu C. Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Comm Math Phys, 2002, 230(2): 329-363
[31] Yang T, Zhao H. A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. J Differ Equ, 2002, 184(1): 163-184
[32] Zhang J. Regularity of solutions to 1D compressible Navier-Stokes-Allen-Cahn system. Appl Anal, 2021, 100(9): 1827-1842
Options
Outlines

/