A Note on the Regularity Time of Leray Solutions to the Navier–Stokes Equations

Abstract

In this note we slightly improve a well known result about the regularity time of Leray solutions $${{\varvec{u}}}(\cdot ,t)$$ to the Navier–Stokes equations in $$ \mathbb {R}^{n} \!\,\!$$ ( $$n \le 4$$ ). A related result on the eventual monotonicity of $$ \,\! \Vert \;\!D^{m}{{\varvec{u}}}(\cdot ,t) \,\!\Vert _{L^{2}(\mathbb {R}^{n})} \!\;\!$$ for arbitrary $$ m \,\!$$ is also discussed.