Advertisement

Mathematical Theory of Incompressible Flows: Local Existence, Uniqueness, and Blow-Up of Solutions in Sobolev–Gevrey Spaces

  • Wilberclay G. Melo
  • Natã Firmino Rocha
  • Ezequiel Barbosa
Chapter
First Online:

Abstract

This work establishes the local existence and uniqueness as well as the blow-up criteria for solutions of the Navier–Stokes equations in Sobolev–Gevrey spaces. More precisely, if the maximal time of existence of solutions for these equations is finite, we demonstrate the explosion, near this instant, of some limits superior and integrals involving a specific usual Lebesgue spaces and, as a consequence, we prove the lower bounds related to Sobolev–Gevrey spaces.

Keywords

Navier–Stokes equations Local existence and uniqueness of solutions Blow-up criteria Sobolev–Gevrey spaces 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The author Natã Firmino Rocha was partially supported by CAPES grant 1579575. The author Ezequiel Barbosa was partially supported by CNPq.

References

  1. 1.
    J. Benameur, On the blow-up criterion of 3D Navier-Stokes equations. J. Math. Anal. Appl. 371, 719–727 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Benameur, On the blow-up criterion of the periodic incompressible fluids. Math. Methods Appl. Sci. 36, 143–153 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Benameur, On the exponential type explosion of Navier-Stokes equations. Nonlinear Anal. 103, 87–97 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. Benameur, L. Jlali, On the blow-up criterion of 3D-NSE in Sobolev–Gevrey spaces. J. Math. Fluid Mech. 18, 805–822 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Biswas, Local existence and Gevrey regularity of 3-D Navier-Stokes equations with initial data. J. Differ. Equ. 215, 429–447 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Braz e Silva, W.G. Melo, P.R. Zingano, Some remarks on the paper “On the blow up criterion of 3D Navier-Stokes equations” by J. Benameur. C. R. Acad. Sci. Paris Ser. I 352, 913–915 (2014)Google Scholar
  7. 7.
    P. Braz e Silva, W.G. Melo, P.R. Zingano, Lower bounds on blow-up of solutions for magneto-micropolar fluid systems in homogeneous Sobolev spaces. Acta Appl. Math. 147, 1–17 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics, vol. 3 (Elsevier, North Holland, 2004), pp. 161–244zbMATHGoogle Scholar
  9. 9.
    J.-Y. Chemin, About Navier-Stokes Equations. Publication du Laboratoire Jacques-Louis Lions, Universite de Paris VI, R96023 (1996)Google Scholar
  10. 10.
    J. Lorenz, P.R. Zingano, Properties at potential blow-up times for the incompressible Navier-Stokes equations. Bol. Soc. Paran. Mat. 35, 127–158 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    W.G. Melo, The magneto-micropolar equations with periodic boundary conditions: solution properties at potential blow-up times. J. Math. Anal. Appl. 435, 1194–1209 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Wilberclay G. Melo
    • 1
  • Natã Firmino Rocha
    • 2
  • Ezequiel Barbosa
    • 2
  1. 1.Departamento de MatemáticaUniversidade Federal de SergipeSão CristóvãoBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

Personalised recommendations

Cite chapter