STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM

doi:10.1007/s10473-024-0417-3

Abstract

Abstract: This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space. For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation, and where the strength of the initial phase field could be arbitrarily large, we prove that the solution of the Cauchy problem exists for all time, and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero. The proof is mainly based on a scaling argument and a basic energy method.

Key words: compressible Navier-Stokes equations, Allen-Cahn equation, rarefaction wave, sharp interface limit, stability

CLC Number:

35M10

Yunkun Chen, Bin Huang, Xiaoding Shi. STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM[J].Acta mathematica scientia,Series B, 2024, 44(4): 1507-1523.

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References

[1] Abels H, Fei M. Sharp interface limit for a Navier-Stokes/Allen-Cahn system with different viscosities. arXiv:2201.09343v2
[2] Abels H, Garcke H, Grün G. Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math Models Methods Appl Sci, 2012, 22(3): 1150013
[3] Abels H, Liu Y. Sharp interface limit for a Stokes/Allen-Cahn system. Arch Ration Mech Anal, 2018, 229(1): 417-502
[4] Anderson D, Mcfadden G, Wheeler A. Diffuse-interface methods in fluid mechanics. Annu Rev Fluid Mech, 1998, 30: 139-165
[5] Blesgen T. A generalization of the Navier-Stokes equations to two-phase flows. J Phys D: Appl Phys, 1999, 32(10): 1119-1123
[6] Chen M, 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
[7] Chen S, Wen H, Zhu C. Global existence of weak solution to compressible Navier-Stokes/Allen-Cahn system in three dimensions. J Math Anal Appl, 2019, 477(2): 1265-1295
[8] 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): 14-24
[9] Chen Y, He Q, Huang B, Shi X. Global strong solution to a thermodynamic compressible diffuse interface model with temperature-dependent heat conductivity in 1D. Math Methods Appl Sci, 2021, 44: 12945-12962
[10] 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
[11] Chen Y, He Q, Shi X, Wang X.Sharp interface limit for compressible non-isentropic phase-field model. arXiv: 2102.00705
[12] Chen Y, Hong H, Shi X. Stability of the phase separation state for compressible Navier-Stokes/Allen-Cahn system. Acta Mathematicae Applicatae Sinica, 2023, 40: 45-74
[13] Chen Y, Li H, Tang H. Optimal decay rate of the compressible Navier-Stokes/Allen-Cahn system in $\mathbb{R}^3$. J Differential Equations, 2022, 334: 157-193
[14] 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
[15] 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
[16] Feireisl E, Petzeltová H, Rocca E, Schimperna G.Analysis of a phase-field model for two-phase compressible fluids. Math Models Methods Appl Sci, 20(7): 1129-1160
[17] Freistühler H. Phase transitions and traveling waves in compressible fluids. Arch Rational Mech Anal, 2014, 211: 189-204
[18] Hensel S, Liu Y. The sharp interface limit of a Navier-Stokes/Allen-Cahn system with constant mobility: Convergence rates by a relative energy approach. arXiv:2201.09423v2
[19] Huang F, Li M, Wang Y. Zero dissipation limit to rarefaction wave with vacuum for one-dimensional compressible Navier-Stokes equations. SIAM J Math Anal, 2012, 44(3): 1742-1759
[20] Jiang S, Ni G, Sun W. Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids. SIAM J Math Anal, 2006, 38(2): 368-384
[21] Jiang S, Su X, Xie F. Remarks on sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system. arXiv:2205.01301v1
[22] Kotschote M. Strong solutions of the Navier-Stokes equations for a compressible fluid of Allen-Cahn type. Arch Ration Mech Anal, 2012, 206(2): 489-514
[23] Kotschote M. Spectral analysis for travelling waves in compressible two-phase fluids of Navier-Stokes-Allen-Cahn type. J Evol Equ, 2017, 17(1): 359-385
[24] Liu T, Xin Z. Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations. Commun Math Phys, 1988, 118(3): 451-465
[25] Lowengrub J, Truskinovsky L. Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc Royal Soc A: Math Phys Eng Sci, 1998, 454: 2617-2654
[26] Luo T, Yin H, Zhu C. Stability of the rarefaction wave for a coupled compressible Navier-Stokes/Allen-Cahn system. Math Methods Appl Sci, 2018, 41(12): 4724-4736
[27] Luo T, Yin H, Zhu C. Stability of the composite wave for compressible Navier-Stokes/Allen-Cahn system. Math Models Methods Appl Sci, 2020, 30(2): 343-385
[28] Matsumura A, Nishihara K. Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas. Jpn J Ind Appl Math, 1986, 3(1): 1-13
[29] Matsumura A, Nishihara K. Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Commun Math Phys, 1992, 144(2): 325-335
[30] Shi X, Yong Y, Zhang Y. Vanishing viscosity for non-isentropic gas dynamics with interacting shocks. Acta Math Sci, 2016, 36B(6): 1699-1720
[31] Wang X, Wang Y. The sharp interface limit of a phase field model for moving contact line problem. Methods and Applications of Analysis, 2010, 14(3): 285-292
[32] Witterstein G, Sharp interface limit of phase change flows. Adv Math Sci Appl, 2010, 20(2): 585-629
[33] Xin Z. Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases. Commun Pure Appl Math, 1993, 46(5): 621-665
[34] Xu X, Di Y, Yu H. Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines. J Fluid Mech, 2018, 849: 805-833
[35] Yan Y, Ding S, Li Y. Strong solutions for 1D compressible Navier-Stokes/Allen-Cahn system with phase variable dependent viscosity. J Differential Equations, 2022, 326: 1-48
[36] Yin H, Zhu C. Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier-Stokes/Allen-Cahn system. J Differential Equations, 2019, 266(11): 7291-7326
[37] Zhao X. Global well-posedness and decay estimates for three-dimensional compressible Navier-Stokes-Allen-Cahn system. Proc Roy Soc Edinb A: Mathematics, 2022, 152(5): 1291-1322