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Energy Equality of the 3D Navier–Stokes Equations and Generalized Newtonian Equations

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Abstract

In this paper, we establish an energy conservation criterion via a combination of the velocity and the gradient of velocity for both the Cauchy and Dirichlet problems of 3D incompressible Navier–Stokes equations, which covers the classical result of Lions (Rend Semin Mat Univ Padova 30:16–23, 1960) and Shinbrot (SIAM J Math Anal 5:948–954, 1974) and recent results in Berselli and Chiodaroli (Nonlinear Anal 192:111704, 2020) and Zhang (J Math Anal Appl 480:9, 2019). The parallel result also holds for the weak solutions of the generalized Newtonian equations, which immediately entails the latest corresponding progress in Beirao da Veiga and Yang (Nonlinear Anal 185:388–402, 2019), Yang (Appl Math Lett 88:216–221, 2019) and Berselli and Chiodaroli (2020) and particularly derives several new sufficient conditions kee** energy equality.

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References

  1. Astarita, G., Marrucci, G.: Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, London (1974)

    MATH  Google Scholar 

  2. Bahouri, H., Chemin, J., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der athematischen Wissenschaften, vol. 343. Springer, Heidelberg (2011)

    Book  Google Scholar 

  3. Beirao da Veiga, H., Yang, J.: On the Shinbrot’s criteria for energy equality to Newtonian fluids: a simplified proof, and an extension of the range of application. Nonlinear Anal. 196, 111809 (2020)

    Article  MathSciNet  Google Scholar 

  4. Beirao da Veiga, H., Yang, J.: On the energy equality for solutions to Newtonian and non-Newtonian fluids. Nonlinear Anal. 185, 388–402 (2019)

    Article  MathSciNet  Google Scholar 

  5. Berselli, L.. C.., Chiodaroli, E.: On the energy equality for the 3D Navier–Stokes equations. Nonlinear Anal. 192, 111704 (2020)

    Article  MathSciNet  Google Scholar 

  6. Cheskidov, A., Constantin, P., Friedlander, S., Shvydkoy, R.: Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21, 1233–52 (2008)

    Article  MathSciNet  Google Scholar 

  7. Cheskidov, A., Luo, X.: Energy equality for the Navier–Stokes equations in weak-in-time Onsager spaces. Nonlinearity 33, 1388–1403 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  8. Constantin, P., E, W., Titi, E.. S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165, 207–209 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  9. Farwig, R., Taniuchi, Y.: On the energy equality of Navier–Stokes equations in general unbounded domains. Arch. Math. 95, 447–456 (2010)

    Article  MathSciNet  Google Scholar 

  10. Galdi, G.P.: An introduction to the Navier-Stokes initial-boundary value problem. In: Fundamental directions in mathematical fluid mechanics, Adv. Math. Fluid Mech., pp. 1–70. Birkhäuser, Basel, 2000

  11. Galdi, G.P.: Mathematical problems in classical and non-Newtonian fluid mechanics. In: Flows, Hemodynamical (ed.) Modeling, Analysis and Simulation, Oberwolfach Seminars, vol. 37, pp. 121–273. Birkhäuser, Basel (2008)

    Google Scholar 

  12. Galdi, G.P.: On the energy equality for distributional solutions to Navier–Stokes equations. Proc. Am. Math. Soc. 147, 785–792 (2019)

    Article  MathSciNet  Google Scholar 

  13. Hille, E., Philips, R.S.: Functional Aanalysis and Semigroups, vol. 31. American Mathematical Society Colloquium Publications, Providence (1957)

    Google Scholar 

  14. Hopf, E.: Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen. Math. Nachr. (Germ.) 4, 213–231 (1950)

    Article  Google Scholar 

  15. Ladyzhenskaya, O.A.: New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them. Proc. Steklov Inst. Math. 102, 95–18 (1967)

    Google Scholar 

  16. Ladyzhenskaia, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gorden and Breach, New York (1969)

    Google Scholar 

  17. Leray, J.: Sur le mouvement déun liquide visqueux emplissant léspace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  18. Lions, J.L.: Sur la régularité et l’unicité des solutions turbulentes des équations de Navier Stokes. Rend. Semin. Mat. Univ. Padova 30, 16–23 (1960)

    MATH  Google Scholar 

  19. Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs, vol. 13. CRC Press, Boca Raton (1996)

    Book  Google Scholar 

  20. Málek, J., Nečas, J., Růžička, M.: On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case \(p \ge 2\). Adv. Differ. Equ. 6, 257–302 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Nguyen, Q., Nguyen, P., Tang, B.: Energy equalities for compressible Navier–Stokes equations. Nonlinearity 32, 4206–4231 (2019)

    Article  MathSciNet  ADS  Google Scholar 

  22. Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13, 116–162 (1955)

    MathSciNet  MATH  Google Scholar 

  23. Shinbrot, M.: The energy equation for the Navier–Stokes system. SIAM J. Math. Anal. 5, 948–954 (1974)

    Article  MathSciNet  ADS  Google Scholar 

  24. Shvydkoy, R.: A geometric condition implying an energy equality for solutions of the 3D Navier–Stokes equation. J. Dyn. Differ. Equ. 21, 117–125 (2009)

    Article  MathSciNet  Google Scholar 

  25. Wang, Y., Ye, Y.: Energy conservation via a combination of velocity and its gradient in the Navier–Stokes system. ar**v:2106.01233

  26. Wei, W., Wang, Y., Ye, Y.: Gagliardo–Nirenberg inequalities in Lorentz type spaces and energy equality for the Navier–Stokes system. ar**v:2106.11212

  27. Wolf, J.: Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9, 104–138 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  28. Yang, J.: The energy equality for weak solutions to the equations of non-Newtonian fluids. Appl. Math. Lett. 88, 216–221 (2019)

    Article  MathSciNet  Google Scholar 

  29. Zhang, Z.: Remarks on the energy equality for the non-Newtonian fluids. J. Math. Anal. Appl. 480, 9 (2019)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express their deepest gratitude to the two anonymous referees and the editors for carefully reading our manuscript whose invaluable comments and suggestions helped to improve the paper greatly. The authors would like to express their sincere gratitude to Dr. **aoxin Zheng at Beihang University for the discussion of Theorem 1.2. Wang was partially supported by the National Natural Science Foundation of China under Grant (Nos. 11971446, 12071113 and 11601492).

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  1. College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, Henan, People’s Republic of China

    Yanqing Wang, Xue Mei & Yike Huang

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  1. Yanqing Wang
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  2. Xue Mei
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Correspondence to Yanqing Wang.

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Communicated by G. P. Galdi.

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Wang, Y., Mei, X. & Huang, Y. Energy Equality of the 3D Navier–Stokes Equations and Generalized Newtonian Equations. J. Math. Fluid Mech. 24, 65 (2022). https://doi.org/10.1007/s00021-022-00687-2

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