Motivated by the study of the possible breakdown of regularity for solutions of the incompressible Navier-Stokes system in the whole three-dimensional space $ {{\mathop{\mathbb R\kern 0pt}\nolimits}}^3 $, we prove that the norm $ \| u\|_{L^\infty_T(B^{\frac 12}_{2\infty})} $ 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.
Citation: Jean-Yves Chemin. Non linear equivalence of some scaling invariant norms for solutions of incompressible Navier-Stokes equations[J]. Communications in Analysis and Mechanics, 2025, 17(4): 944-954. doi: 10.3934/cam.2025038
Motivated by the study of the possible breakdown of regularity for solutions of the incompressible Navier-Stokes system in the whole three-dimensional space $ {{\mathop{\mathbb R\kern 0pt}\nolimits}}^3 $, we prove that the norm $ \| u\|_{L^\infty_T(B^{\frac 12}_{2\infty})} $ 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.
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Jean-Yves Chemin. Non linear equivalence of some scaling invariant norms for solutions of incompressible Navier-Stokes equations[J]. Communications in Analysis and Mechanics, 2025, 17(4): 944-954. doi: 10.3934/cam.2025038
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