Estimates on fractional higher derivatives of weak solutions for the Navier–Stokes equations

  • Kyudong Choi

    Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI 53706, USA
  • Alexis F. Vasseur

    Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA

Abstract

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 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 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.

Cite this article

Kyudong Choi, Alexis F. Vasseur, Estimates on fractional higher derivatives of weak solutions for the Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 5, pp. 899–945

DOI 10.1016/J.ANIHPC.2013.08.001