Non linear equivalence of some scaling invariant norms for solutions of incompressible Navier-Stokes equations

Motivated by the study of the possible breakdown of regularity for solutions of the incompressible Navier-Stokes system in the whole three-dimensional space ${\mathbb{R}}^3$, we prove that the norm $\Vert u{\Vert }_{L_T^{\mathrm{\infty }}(B_{2\mathrm{\infty }}^{\frac{1}{2}})}$ and the quantity $\underset{I\subset [0,T[}{sup}\frac{1}{\sqrt{|I|}}{\int }_I\Vert \mathrm{\nabla }u(t){\Vert }_{L^2}^{2}dt$ are non linearly equivalent. The proofs rely on the Duhamel formula, dyadic localization in the Fourier space, and for one of them, on the energy inequality.