Abstract
Let us consider initial data \({v_0}\) for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to \({L^{\frac 32}\cap L^2}\). We prove that if the solution associated with \({v_0}\) blows up at a finite time \({T^\star}\), then for any p in \({]4,\infty[}\), and any unit vector e of \({\mathbb{R}^3}\), the L p norm in time with value in \({\dot{H}^{\frac 12+\frac 2 p }}\)of \({(v|e)_{\mathbb{R}^3}}\) blows up at \({T^\star}\).
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Communicated by F. Lin
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Chemin, JY., Zhang, P. & Zhang, Z. On the Critical One Component Regularity for 3-D Navier-Stokes System: General Case. Arch Rational Mech Anal 224, 871–905 (2017). https://doi.org/10.1007/s00205-017-1089-0
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DOI: https://doi.org/10.1007/s00205-017-1089-0