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


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.


Navier–Stokes equations Local existence and uniqueness of solutions Blow-up criteria Sobolev–Gevrey spaces 



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


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© 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

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