Regularity criteria for Navier-Stokes-Allen-Cahn and related systems

Abstract

We prove a regularity criterion for the 3D Navier-Stokes-Allen-Cahn system in a bounded smooth domain which improves the result obtained by Y. Li, S. Ding, and M. Huang [Discrete Contin. Dyn. Syst. Ser. B, 2016, 21(5): 1507–1523]. We also present a similar result to the 3D Navier-Stokes-Cahn-Hilliard system.

Keywords

Regularity criterion / Navier-Stokes-Allen-Cahn system / Navier-Stokes-Cahn-Hilliard system

References

[1] Abels H. On a diffuse interface model for two-phase ows of viscous, incompressible uids with matched densities. Arch Ration Mech Anal, 2009, 194: 463–506

[2] Adams R A, Fournier J J F. Sobolev Spaces. 2nd ed. Amsterdam: Elsevier/Academic Press, 2003

[3] Beirão da Veiga H, Crispo F. Sharp inviscid limit results under Navier type boundary conditions. An L^p theory. J Math Fluid Mech, 2010, 12: 397–411

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

[5] Cho Y, Kim H. Unique solvability for the density-dependent Navier-Stokes equations. Nonlinear Anal, 2004, 59: 465–489

[6] 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: 401–436

[7] Kotschote M, Zacher R. Strong solutions in the dynamical theory of compressible uid mixtures. Math Models Methods Appl Sci, 2015, 25: 1217–1256

[8] Li Y, Ding S, Huang M. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete Contin Dyn Syst Ser B, 2016, 21: 1507–1523

[9] Li Y, Huang M. Strong solutions for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Z Angew Math Phys, 2018, 69: Art 68 (18pp)

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

[11] Lunardi A. Interpolation Theory. 2nd ed. Pisa: Edizioni della Normale, 2009

[12] Starovoitov V N. On the motion of a two-component uid in the presence of capillary forces. Mat Zametki, 1997, 62: 293–305

[13] Xu X, Zhao L, Liu C. Axisymmetric solutions to coupled Navier-Stokes/Allen-Cahn equations. SIAM J Math Anal, 2010, 41: 2246–2282

[14] Yang X, Feng J J, Liu C, Shen J. Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method. J Comput Phys, 2006, 218: 417–428

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

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