Well-posedness for the generalized Navier–Stokes–Landau–Lifshitz equations

Abstract

The generalized Navier–Stokes–Landau–Lifshitz equations are considered in this paper. The well-posedness for the multi-dimensional hyperviscous Navier–Stokes–Landau–Lifshitz equations is proved for \(n\ge 3\). The existence and uniqueness of the strong solutions for the generalized Navier–Stokes–Landau–Lifshitz equations are proved for \(\frac{5}{4}\le \alpha <\frac{5}{2}\) and \(\frac{5}{4}\le \beta <\frac{5}{2}\).

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Barbato, D., Morandin, F., Romito, M.: Global regularity for a slightly supercritical hyperdissipative Navier–Stokes system. Anal. PDE 7, 2009–2027 (2014)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Fan, J.S., Gao, H.J., Guo, B.L.: Regularity criteria for the Navier–Stokes–Landau–Lifshitz system. J. Math. Anal. Appl. 363, 29–37 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Gal, C.G., Guo, Y.Q.: Inertial manifolds for the hyperviscous Navier–Stokes equations. J. Differ. Equ. 265, 4335–4374 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Jiang, K.R., Liu, Z.H., Zhou, L.: Global existence and asymptotic stability of 3D generalized magnetohydrodynamic equations. J. Math. Fluid Mech. 22, 14, Art. 9 (2020)

  5. 5.

    Liu, H., Sun, C.F., Xin, J.: Well-posedness for the hyperviscous magneto-micropolar equations. Appl. Math. Lett. 107, 106403 (2020)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Pennington, N.: Low regularity global solutions for a generalized MHD-\(\alpha \) system. Nonlinear Anal. Real World Appl. 38, 171–183 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Regmi, D.: The 2D magneto-micropolar equations with partial dissipation. Math. Methods Appl. Sci. 42, 4305–4317 (2019)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Tao, T.: Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation. Anal. PDE 2, 361–366 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Wang, G.W., Guo, B.L.: Global weak solution to the quantum Navier–Stokes–Landau–Lifshitz equations with density-dependent viscosity. Discrete Contin. Dyn. Syst. Ser. B 24, 6141–6166 (2019)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Wei, R.Y., Li, Y., Yao, Z.A.: Decay rates of higher-order norms of solutions to the Navier–Stokes–Landau–Lifshitz system. Appl. Math. Mech. (Engl. Ed.) 39, 1499–1528 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Wu, J.H.: Generalized MHD equations. J. Differ. Equ. 195, 284–312 (2003)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Wu, J.H.: Regularity criteria for the generalized MHD equations. Commun. Partial Differ. Equ. 33, 285–306 (2008)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Wu, J.H.: Global regularity for a class of generalized magnetohydrodynamic equations. J. Math. Fluid Mech. 13, 295–305 (2011)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Yamazaki, K.: Global regularity of logarithmically supercritical MHD system with improved logarithmic powers. Dyn. Partial Differ. Equ. 15, 147–173 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Yang, W.R., Jiu, Q.S., Wu, J.H.: The 3D incompressible magnetohydrodynamic equations with fractional partial dissipation. J. Differ. Equ. 266, 630–652 (2019)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Zhai, X.P., Li, Y.S., Yan, W.: Global solutions to the Navier–Stokes–Landau–Lifshitz system. Math. Nachr. 289, 377–388 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Zhou, Y.: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. H. Poincare Anal. Non Linaire 24, 491–505 (2007)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The work was supported by the National Natural Science Foundation of China Nos. 11901342 and 11701269, the Natural Science Foundation of Shandong Province No. ZR2018QA002, China Postdoctoral Science Foundation No. 2019M652350, and Postdoctoral Innovation Project of Shandong Province No. 202003040

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hui Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work was supported by the National Natural Science Foundation of China Nos. 11901342 and 11701269, the Natural Science Foundation of Shandong Province No. ZR2018QA002, China Postdoctoral Science Foundation No. 2019M652350, and Postdoctoral Innovation Project of Shandong Province No. 202003040.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, H., Sun, C. & Xin, J. Well-posedness for the generalized Navier–Stokes–Landau–Lifshitz equations. Z. Angew. Math. Phys. 72, 32 (2021). https://doi.org/10.1007/s00033-020-01467-6

Download citation

Keywords

  • Navier–Stokes–Landau–Lifshitz equations
  • Strong solutions
  • A priori estimates

Mathematics Subject Classification

  • 35Q35
  • 35D35
  • 35B45