Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 78, pp. 1-12.

Title: Solutions for the Navier-Stokes equations with critical and subcritical 
       fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaces
       
Authors: Wilberclay G. Melo (Univ. Federal de Sergipe, Sao Cristovao, SE, Brazil)
         Nata F. Rocha (Univ. Estadual do Piaui, Teresina, PI, Brazil)
         Natielle dos Santos Costa (Univ. Federal de Sergipe, Sao Cristovao, SE, Brazil)

Abstract:
In this article, we prove the existence of a unique global solution for the
critical case of the generalized Navier-Stokes equations in Lei-Lin and
Lei-Lin-Gevrey spaces, by assuming that the initial data is small enough.
Moreover, we obtain a unique local solution for the subcritical case of this
system, for any initial data, in these same spaces. It is important to point
out that our main result is obtained by discussing some properties of the
solutions for the heat equation with fractional dissipation.

Submitted April 18, 2023. Published November 10, 2023.
Math Subject Classifications: 35A01, 35Q35, 42B37.
Key Words: Navier-Stokes equations; global and local solutions; Lei-Lin-Gevrey spaces.