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

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Abstract

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 $\|u\|_{L_{T}^{\infty}\left(B_{2 \infty}^{\frac{1}{2}}\right)}$ and the quantity $\sup _{I \subset[0, T[ } \frac{1}{\sqrt{|I|}} \int_{I}\|\nabla u(t)\|_{L^{2}}^{2}$ 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.

Keywords

incompressible Navier-Stokes / Besov spaces

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