Lower Bounds of Potential Blow-Up Solutions of the Three-dimensional Navier-Stokes Equations in ${\dot{H}}^{\frac{3}{2}}$

We improve previous known lower bounds for Sobolev norms of potential blow-up solutions to the three-dimensional Navier-Stokes equations in ${\dot{H}}^{\frac{3}{2}}$. We also present an alternate proof for the lower bound for the ${\dot{H}}^{\frac{5}{2}}$ blow-up.