Prodi-Serrin condition for 3D Navier-Stokes equations via one directional derivative of velocity
Abstract
In this paper, we consider the conditional regularity of weak solution to the 3D Navier-Stokes equations. More precisely, we prove that if one directional derivative of velocity, say ∂3 u, satisfies ∂3 u ∈L p0 , 1 (0 , T ;L q0 (R3)) with 2/p0 + 3/q0 = 2 and 3/2 <q0 < + ∞, then the weak solution is regular on (0 , T ]. The proof is based on the new local energy estimates introduced by Chae-Wolf (2019) [4] and Wang-Wu-Zhang (2020) [21].
- Publication:
-
Journal of Differential Equations
- Pub Date:
- October 2021
- DOI:
- 10.1016/j.jde.2021.07.015
- arXiv:
- arXiv:2102.06497
- Bibcode:
- 2021JDE...298..500H
- Keywords:
-
- Navier-Stokes equations;
- Regularity of weak solutions;
- Serrin-Prodi condition;
- Mathematics - Analysis of PDEs