We study weak solutions of the 3D Navier–Stokes equations with initial data. We prove that is locally integrable in space–time for any real α such that . Up to now, only the second derivative was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-. These estimates depend only on the -norm of the initial data and on the domain of integration. Moreover, they are valid even for 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.
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{\textendash}Stokes} equations}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {899--945}, publisher = {Elsevier}, volume = {31}, number = {5}, year = {2014}, doi = {10.1016/j.anihpc.2013.08.001}, mrnumber = {3258360}, zbl = {1297.76047}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.001/} }
TY - JOUR AU - Choi, Kyudong AU - Vasseur, Alexis F. TI - Estimates on fractional higher derivatives of weak solutions for the Navier–Stokes equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 899 EP - 945 VL - 31 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.001/ DO - 10.1016/j.anihpc.2013.08.001 LA - en ID - AIHPC_2014__31_5_899_0 ER -
%0 Journal Article %A Choi, Kyudong %A Vasseur, Alexis F. %T Estimates on fractional higher derivatives of weak solutions for the Navier–Stokes equations %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 899-945 %V 31 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.08.001/ %R 10.1016/j.anihpc.2013.08.001 %G en %F 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/articles/10.1016/j.anihpc.2013.08.001/
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