Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T05:09:43.691Z Has data issue: false hasContentIssue false
  • English
  • Français

Global well-posedness and decay estimates for three-dimensional compressible Navier–Stokes–Allen–Cahn systems

Part of: Equations of mathematical physics and other areas of application Compressible fluids and gas dynamics, general

Published online by Cambridge University Press:  11 October 2021

Xiaopeng Zhao
Xiaopeng Zhao*
Affiliation:
College of Sciences, Northeastern University, Shenyang 110004, China ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot {H}^{-s}$($0\leq s<\frac {3}{2}$) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

Keywords

Compressible Navier–Stokes–Allen–Cahn equationsdecay estimatesglobal well-posednesspure energy method

MSC classification

Primary: 35Q30: Navier-Stokes equations
Secondary: 76N10: Existence, uniqueness, and regularity theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 5 , October 2022 , pp. 1291 - 1322
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, D. M., McFadden, G. B. and Wheeler, A. A.. Diffuse-interface methods in fluid mechanics. Ann. Rev. Fluid Mech. 30 (1998), 139–165.CrossRefGoogle Scholar
Blesgen, T.. A generalisation of Navier–Stokes equations to two-phase-flows. J. Phys. D: Appl. Phys. 32 (1999), 1119–1123.CrossRefGoogle Scholar
Chen, M. and Guo, X.. Global large solutions for a coupled compressible Navier–Stokes/Allen–Cahn system with initial vacuum. Nonlinear Anal. Real World Appl. 37 (2017), 350–373.CrossRefGoogle Scholar
Chen, S., Wen, H. and Zhu, C.. Global existence of weak solution to compressible Navier–Stokes/Allen–Cahn system in three dimensions. J. Math. Anal. Appl. 477 (2019), 1265–1295.CrossRefGoogle Scholar
Ding, S., Li, Y. and Luo, W. Global solutions for a coupled compressible Navier–Stokes/Allen–Cahn system in 1D. J. Math. Fluid Mech. 15 (2013), 335–360.CrossRefGoogle Scholar
Ding, S., Li, Y. and Tang, Y.. Strong solutions to 1D compressible Navier–Stokes/Allen–Cahn system with free boundary. Math. Methods Appl. Sci. 42 (2019), 4780–4794.CrossRefGoogle Scholar
Fan, J., Alsaedi, A., Hayat, T., Nakamura, G. and Zhou, Y.. On strong solutions to the compressible Hall-magnetohydrodynamics system. Nonlinear Anal. Real World Appl. 22 (2015), 423–434.CrossRefGoogle Scholar
Fan, J. and Li, F.. Regularity criteria for Navier–Stokes–Allen–Cahn and related systems. Front. Math. China 14 (2019), 301–314.CrossRefGoogle Scholar
Favre, G. and Schimperna, G.. On a Navier–Stokes–Allen–Cahn model with inertial effects. J. Math. Anal. Appl. 475 (2019), 811–838.CrossRefGoogle Scholar
Feireisl, E., Petzeltova, H., Rocca, E. and Schimperna, G.. Analysis of a phase-field model for two-phase compressible fluids. Math. Models Methods Appl. Sci. 20 (2010), 1129–1160.CrossRefGoogle Scholar
Gal, C. G. and Grasselli, M.. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete Contin. Dyn. Syst. 28 (2010), 1–39.CrossRefGoogle Scholar
Gal, C. G. and Grasselli, M.. Trajectory attractors for binary fluid mixtures in 3D. Chin. Ann. Math. Ser. B 31 (2010), 655–678.CrossRefGoogle Scholar
Grafakos, L.. Classical and modern Fourier analysis (Upper Saddle River, NJ: Pearson Education Inc., Prentice-Hall, 2004).Google Scholar
Gurti, M. E., Polignone, D. and Vinals, J.. Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6 (1996), 815–831.CrossRefGoogle Scholar
Jiang, J., Li, Y. and Liu, C.. Strong solutions for an incompressible Navier–Stokes/Allen–Cahn system with different densities. Discrete Contin. Dyn. Syst. 37 (2017), 3243–3284.CrossRefGoogle Scholar
Kato, T. and Ponce, G.. Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41 (1988), 891–907.CrossRefGoogle Scholar
Kotschote, M.. Strong solutions of the Navier–Stokes equations for a compressible fluid of Allen–Cahn type. Arch. Ration. Mech. Anal. 206 (2012), 489–514.CrossRefGoogle Scholar
Li, Y. and Huang, M.. Strong solutions for an incompressible Navier–Stokes/Allen–Cahn system with different densities. Z. Angew. Math. Phys. 69 (2018), 68.CrossRefGoogle Scholar
Li, Y., Ding, S. and Huang, M.. Blow-up criterion for an incompressible Navier–Stokes/Allen–Cahn system with different densities. Discrete Contin. Dyn. Syst. Ser B. 21 (2016), 1507–1523.CrossRefGoogle Scholar
Liu, C., Shen, J. and Yang, X.. Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density. J. Sci. Comput. 62 (2015), 601–622.CrossRefGoogle Scholar
Lowengrub, J. and Truskinovsky, L.. Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. London, Ser. A 454 (1998), 2617–2654.CrossRefGoogle Scholar
Luo, T., Yin, H. and Zhu, C.. Stability of the rarefaction wave for a coupled compressible Navier–Stokes/Allen–Cahn system. Math. Methods Appl. Sci. 41 (2018), 4724–4736.CrossRefGoogle Scholar
Nirenberg, L.. On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 13 (1959), 115–162.Google Scholar
Song, C., Zhang, J. and Wang, Y.. Time-periodic solution to the compressible Navier–Stokes/Allen–Cahn system. Acta Math. Sinica, Engl. Ser. 36 (2020), 419–442.CrossRefGoogle Scholar
Stein, E. M.. Singular integrals and differentiability properties of functions (Princeton, NJ: Princeton University Press, 1970).Google Scholar
Wang, Y.. Decay of the Navier–Stokes–Poisson equations. J. Differ. Equ. 253 (2012), 273–297.CrossRefGoogle Scholar
Wei, R., Li, Y. and Yao, Z.. Decay of the compressible magnetohydrodynamic equations. Z. Angew. Math. Phys. 66 (2015), 2499–2524.CrossRefGoogle Scholar
Witterstein, G.. Sharp interface limit of phase change flows. Adv. Math. Sci. Appl. 20 (2011), 585–629.Google Scholar
Xu, X., Zhao, L. and Liu, C.. Axisymmetric solutions to coupled Navier–Stokes/Allen–Cahn equations. SIAM J. Math. Anal. 41 (2010), 2246–2282.CrossRefGoogle Scholar
Yin, H. and Zhu, C.. Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier–Stokes/Allen–Cahn system. J. Differ. Equ. 266 (2019), 7291–7326.CrossRefGoogle Scholar
Zhao, L., Guo, B. and Huang, H.. Vanishing viscosity limit for a coupled Navier–Stokes/Allen–Cahn system. J. Math. Anal. Appl. 384 (2011), 232–245.CrossRefGoogle Scholar