Remarks on Sharp Interface Limit for an Incompressible Navier-Stokes and Allen-Cahn Coupled System

Abstract

The authors are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper. When the thickness of the diffuse interfacial zone, which is parameterized by ε, goes to zero, they prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the L ^∞(L ²) ∩ L ²(H ¹) sense on a uniform time interval independent of the small parameter ε. The proof consists of two parts: One is the construction of a suitable approximate solution and another is the estimate of the error functions in Sobolev spaces. Besides the careful energy estimates, a spectral estimate of the linearized operator for the incompressible Navier-Stokes and Allen-Cahn coupled system around the approximate solution is essentially used to derive the uniform estimates of the error functions. The convergence of the velocity is well expected due to the fact that the layer of the velocity across the diffuse interfacial zone is relatively weak.

Keywords

Sharp interface limit / Incompressible Navier-Stokes equations / Allen-Cahn equation / Spectral estimate / Energy estimates

References

[1] Abels H. On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal, 2009, 194(2): 463-506

[2] Abels H, Depner D, Garcke H. Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities. J. Math. Fluid Mech., 2013, 15(3): 453-480

[3] Abels, H. and Fei, M., Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System with Different Viscosities, 2022, arXiv:2201.09343.

[4] Abels H, Liu Y. Sharp interface limit for a Stokes/Allen-Cahn system. Arch. Ration. Mech. Anal., 2018, 229(1): 417-502

[5] Abels H, Marquardt A. Sharp interface limit of a Stokes/Cahn-Hilliard system, part I: Convergence result. Interfaces Free Bound, 2021, 23(3): 353-402

[6] Abels H, Röger M. Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2009, 26(6): 2403-2424

[7] Abels H, Wilke M. Well-posedness and qualitative behaviour of solutions for a two-phase Navier-Stokes-Mullins-Sekerka system. Interfaces Free Bound, 2013, 15(1): 39-75

[8] Alikakos N D, Bates P W, Chen X. Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Rational Mech. Anal, 1994, 128(2): 165-205

[9] Boyer F. Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal., 1999, 20(2): 175-212

[10] Chen X. Generation and propagation of interfaces for reaction-diffusion equations. J. Differential Equations, 1992, 96(1): 116-141

[11] Chen X. Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differential Equations, 1994, 19(7–8): 1371-1395

[12] Chen X, Hilhorst D, Logak E. Mass conserving Allen-Cahn equation and volume preserving mean curvature flow. Interfaces Free Bound, 2010, 12(4): 527-549

[13] De Mottoni P, Schatzman M. Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc., 1995, 347(5): 1533-1589

[14] Elliott C M, Zheng S. On the Cahn-Hilliard equation. Arch. Rational Mech. Anal, 1986, 96(4): 339-357

[15] Evans L O, Soner H M, Souganidis P E. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math., 1992, 45(9): 1097-1123

[16] Fusco G. A geometric approach to the dynamics of u _t = ε ² u _xx + f(u) for small ε. Problems involving change of type (Stuttgart, 1988), 1990, Berlin: Springer-Verlag 53-73

[17] Gal C G, Grasselli M. Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2010, 27(1): 401-436

[18] Gal C G, Grasselli M. Trajectory attractors for binary fluid mixtures in 3D. Chin. Ann. Math. Ser. B, 2010, 31(5): 655-678

[19] Gurtin M E, Polignone D, Viñals J. Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci., 1996, 6(6): 815-831

[20] Hohenberg P C, Halperin B I. Theory of dynamic critical phenomena. Rev. Mod. Phys., 1977, 49(3): 435-479

[21] Könne M, Prüss J, Wilke M. Qualitative behaviour of solutions for the two-phase Navier-Stokes equations with surface tension. Math. Ann., 2013, 356(2): 737-792

[22] Liu C, Shen J. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D, 2003, 179(3–4): 211-228

[23] Liu S, Wang F, Zhao H. Global existence and asymptotics of solutions of the Cahn-Hilliard equation. J. Differential Equations, 2007, 238(2): 426-469

[24] Liu, Y., Moving interface in a viscous incompressible flow, 2022, arXiv:2201.09423.

[25] Nirenberg L. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 1959, 13: 115-162

[26] Prüss J, Shibata Y, Shimizu S, Simonett G. On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities. Evol. Equ. Control Theory, 2012, 1(1): 171-194

[27] Prüss J, Simonett G. On the two-phase Navier-Stokes equations with surface tension. Interfaces Free Bound, 2010, 12(3): 311-345

[28] Xu X, Zhao L, Liu O. Axisymmetric solutions to coupled Navier-Stokes/Allen-Cahn equations. SIAM J. Math. Anal, 2009, 41(6): 2246-2282 10

[29] Zhao L, Guo B, Huang H. Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system. J. Math. Anal. Appl, 2011, 384(2): 232-245