Global existence of uniformly locally energy solutions for the incompressible fractional Navier-Stokes equations

Abstract

In this paper, we introduce the concept of local Leray solutions starting from a locally square-integrable initial data to the fractional Navier-Stokes equations with $s\in [3/4,1)$. Furthermore, we prove its local in time existence when $s\in (3/4,1)$. In particular, if the locally square-integrable initial data vanishs at infinity, we show that the fractional Navier-Stokes equations admit a global-in-time local Leray solution when $s\in [5/6,1)$. For such local Leray solutions starting from locally square-integrable initial data vanishing at infinity, the singularity only occurs in $B_R(0)$ for some $R$.

26 Pages,

References in corpus (3)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations
• On higher order extensions for the fractional Laplacian
• The generalized Caffarelli-Kohn-Nirenberg Theorem for the hyperdissipative Navier-Stokes system