Estimates on fractional higher derivatives of weak solutions for the Navier–Stokes equations
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 5, p. 899-945

We study weak solutions of the 3D Navier–Stokes equations with ${L}^{2}$ initial data. We prove that ${\nabla }^{\alpha }u$ is locally integrable in space–time for any real α such that $1<\alpha <3$. Up to now, only the second derivative ${\nabla }^{2}u$ was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-${L}_{\mathrm{𝑙𝑜𝑐}}^{4/\left(\alpha +1\right)}$. These estimates depend only on the ${L}^{2}$-norm of the initial data and on the domain of integration. Moreover, they are valid even for $\alpha ⩾3$ as long as u is smooth. The proof uses a standard approximation of Navier–Stokes from Leray and blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced.

DOI : https://doi.org/10.1016/j.anihpc.2013.08.001
Classification:  76D05,  35Q30
Keywords: Navier–Stokes equations, Fluid mechanics, Blow-up techniques, Weak solutions, Higher derivatives, Fractional derivatives
@article{AIHPC_2014__31_5_899_0,
author = {Choi, Kyudong and Vasseur, Alexis F.},
title = {Estimates on fractional higher derivatives of weak solutions for the Navier--Stokes equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {5},
year = {2014},
pages = {899-945},
doi = {10.1016/j.anihpc.2013.08.001},
zbl = {1297.76047},
mrnumber = {3258360},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_5_899_0}
}

Choi, Kyudong; Vasseur, Alexis F. Estimates on fractional higher derivatives of weak solutions for the Navier–Stokes equations. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 5, pp. 899-945. doi : 10.1016/j.anihpc.2013.08.001. http://www.numdam.org/item/AIHPC_2014__31_5_899_0/

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