If a mild solution to 3D incompressible Navier-Stokes with v0 in Ḣ^{1/2} and Ω0 in L^{r0} (r0∈(1,2)) blows up at T*, then for any 2<p<∞ and unit vector e the integral ∫_0^{T*} ||(v(t)|e)||_{Ḃ^{1/2+2/p}_{2,∞}}^p dt diverges at T*.
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On the Critical One Components Regularity for the $3-D$ Navier-Stokes System in $L^p_T(\dot{B}^{\frac 1 2+\frac 2 p}_{2,\infty})$ spaces
If a mild solution to 3D incompressible Navier-Stokes with v0 in Ḣ^{1/2} and Ω0 in L^{r0} (r0∈(1,2)) blows up at T*, then for any 2<p<∞ and unit vector e the integral ∫_0^{T*} ||(v(t)|e)||_{Ḃ^{1/2+2/p}_{2,∞}}^p dt diverges at T*.