Skip to main content
SpringerLink
Log in

The Low Mach Number Limit for Isentropic Compressible Navier-Stokes Equations with a Revised Maxwell’s Law

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

We investigate the low Mach number limit for the isentropic compressible Navier-Stokes equations with a revised Maxwell’s law (with Galilean invariance) in ℝ3. By applying the uniform estimates of the error system, it is proven that the solutions of the isentropic Navier-Stokes equations with a revised Maxwell’s law converge to that of the incompressible Navier-Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Maxwell J C. On the dynamical theory of gases. Phil Trans R Soc Lond, 1867, 157: 49–88

    Google Scholar 

  2. Maisano G, Migliardo P, Aliotta F, et al. Evidence of anomalous acoustic behavior from Brillouin scattering in supercooled water. Phys Rev Lett, 1984, 52: 1025–1028

    Article  Google Scholar 

  3. Pelton M, Chakraborty D, Malachosky E, et al. Viscoelastic flows in simple liquids generated by vibrating nanostructures. Phys Rev Lett, 2013, 111: 244502

    Article  Google Scholar 

  4. Sette F, Ruocco G, Krisch M, et al. Collective dynamics in water by high energy resolution inelastic x-ray scattering. Phys Rev Lett, 1995, 75: 850–853

    Article  Google Scholar 

  5. Yong W A. Newtonian limit of Maxwell fluid flows. Arch Ration Mech Anal, 2014, 214(3): 913–922

    Article  MathSciNet  MATH  Google Scholar 

  6. Chakraborty D, Sader J E. Constitutive models for linear compressible viscoelastic flows of simple liquids at nanometer length scales. Physics of Fluids, 2015, 27: 052002

    Article  MATH  Google Scholar 

  7. Hu Y, Racke R. Compressible Navier-Stokes equations with revised Maxwell’s law. J Math Fluid Mech, 2017, 19(1): 77–90

    Article  MathSciNet  MATH  Google Scholar 

  8. Ebin D G. Motion of a slightly compressible fluid. Proc Natl Acad Sci USA, 1975, 72(2): 539–542

    Article  MathSciNet  Google Scholar 

  9. Klainerman S, Majda A. Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm Pure Appl Math, 1981, 34: 481–524

    Article  MathSciNet  MATH  Google Scholar 

  10. Klainerman S, Majda A. Compressible and incompressible fluids. Comm Pure Appl Math, 1982, 35: 629–651

    Article  MathSciNet  MATH  Google Scholar 

  11. Danchin R. Zero Mach number limit for compressible flows with periodic boundary conditions. Amer J Math, 2002, 124(6): 1153–1219

    Article  MathSciNet  MATH  Google Scholar 

  12. Desjardins B, Grenier E, Lions P L, Masmoudi N. Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J Math Pures Appl, 1999, 78(5): 461–471

    Article  MathSciNet  MATH  Google Scholar 

  13. Hoff D. The Zero-Mach limit of compressible flows. Comm Math Phys, 1998, 192(3): 543–554

    Article  MathSciNet  MATH  Google Scholar 

  14. Lions P L, Masmoudi N. Incompressible limit for a viscous compressible fluid. J Math Pures Appl, 1998, 77(6): 585–627

    Article  MathSciNet  MATH  Google Scholar 

  15. Desjardins B, Grenier E. Low Mach number limit of the viscous compressible flows in the whole space. Proc R Soc Lond Ser A Math Phys Eng Sci, 1999, 455: 2271–2279

    Article  MathSciNet  MATH  Google Scholar 

  16. Ou Y. Incompressible limit of the Navier-Stokes equations for all time. J Differ Equ, 2009, 247: 3295–3314

    Article  MathSciNet  MATH  Google Scholar 

  17. Alazard T. Low Mach number limit of the full Navier-Stokes equations. Arch Ration Mech Anal, 2006, 180(1): 1–73

    Article  MathSciNet  MATH  Google Scholar 

  18. Ou Y. Low Mach number limit for the non-isentropic Navier-Stokes equations. J Differ Equ, 2009, 246(11): 4441–4465

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu Y, Wang N. Global existence versus blow-up results for one dimensional compressible Navier-Stokes equations with Maxwell’s law. Math Nachr, 2019, 292(4): 826–840

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang N, Hu Y. Blowup of solutions for compressible Navier-Stokes equations with revised Maxwell’s law. Appl Math Lett, 2020, 103: 106221

    Article  MathSciNet  MATH  Google Scholar 

  21. Hu Y, Racke R, Wang N. Formation of singularities for one-dimensional relaxed compressible Navier-Stokes equations. J Diff Equ, 2022, 327: 145–165

    Article  MathSciNet  MATH  Google Scholar 

  22. Taylor M E. Pseudodifferential operators and nonlinear PDE. Boston: Birkhäuser, 1991

    Book  MATH  Google Scholar 

  23. Yong W A. Singular perturbations of first-order hyperbolic systems with stiff source terms. J Differ Equ, 1999, 155(1): 89–132

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang S. Low Mach number limit for the full compressible Navier-Stokes equations with Cattaneo’s heat transfer law. Nonlinear Anal, 2019, 184: 83–94

    Article  MathSciNet  MATH  Google Scholar 

  25. Racke R. Lectures on Nonlinear Evolution Equations. Weisbaden: Vieweg, 1992

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Beijing, 100083, China

    Yuxi Hu  (胡玉玺)

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Zhao Wang  (王召)

Authors
  1. Yuxi Hu  (胡玉玺)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhao Wang  (王召)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Yuxi Hu  (胡玉玺) or Zhao Wang  (王召).

Additional information

Yuxi HU was supported by the NNSFC (11701556) and the Yue Qi Young Scholar Project, China University of Mining and Technology (Beijing).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, Y., Wang, Z. The Low Mach Number Limit for Isentropic Compressible Navier-Stokes Equations with a Revised Maxwell’s Law. Acta Math Sci 43, 1239–1250 (2023). https://doi.org/10.1007/s10473-023-0314-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-023-0314-1

Key words

2010 MR Subject Classification

Search

Navigation