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 η(ρ) = ρα. The existence of unique global H2m-solutions (m ∈ ℕ) to the free boundary problem is proven for when \(0 < \alpha < {1 \over 4}\). Furthermore, we obtain the global C∞-solutions if the initial data is smooth.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Blesgen T. A generalizaion of the Navier-Stokes equations to two-phase flow. J Phys D Appl Phys, 1999, 32: 1119–1123
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
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
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
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
Dai W, Ding S, Li Y. Global strong solutions of the compressible Navier-Stokes/Allen-Cahn system with density-dependent viscosity. preprint
Ding S, Huang J, Liu X, Wen H. Global C∞-solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity. Math Methods Appl Sci, 2011, 34(12): 1499–1511
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
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
Freistühler 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
Fang D, Zhang T. Compressible Navier-Stokes equations with vacuum state in one dimension. Commun Pure Appl Anal, 2004, 3(4): 675–694
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
Grad H. Asymptotic theory of the Boltzmann equation II//Laurmann J, ed. Rarefied Gas Dynamics, Vol 1. New York: Academic Press, 1963: 26–59
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
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
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
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
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
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
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
Liu J. Local existence of solution to free boundary value problem for compressible Navier-Stokes equations. Acta Math Sci, 2012, 32B(4): 1298–1320
Liu T, Xin Z, Yang T. Vacuum states of compressible flow. Discrete Conti Dyn Syst, 1998, 4: 1–32
Okada M, Matuš-Nečasová Š, 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
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
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
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
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
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
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
Yang T, Zhu C. Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Comm Math Phys, 2002, 230(2): 329–363
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
Zhang J. Regularity of solutions to 1D compressible Navier-Stokes-Allen-Cahn system. Appl Anal, 2021, 100(9): 1827–1842
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Additional information
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).
Rights and permissions
About this article
Cite this article
Ding, S., Li, Y. & Wang, Y. Global solutions to 1D compressible Navier-Stokes/Allen-Cahn system with density-dependent viscosity and free-boundary. Acta Math Sci 44, 195–214 (2024). https://doi.org/10.1007/s10473-024-0111-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-024-0111-5