Lower bound on the blow-up rate of the 3D Navier-Stokes equations in H^{5/2}

Abstract

Under assumption that $T^{\ast }$ is the maximal time of existence of smooth solution of the 3D Navier-Stokes equations in the Sobolev space $H^s$, we establish lower bounds for the blow-up rate of the type${\left(T^{\ast }-t\right)}^{-\phi \left(n\right)}$, where $n$ is a natural number independent of $s$ and $\phi$ is a linear function. Using this new type in the 3D Navier-Stokes equations in the $H^{5/2}$, both on the whole space and in the periodic case, we give an answer to a question left open by James et al (2012, J. Math. Phys.). We also prove optimal lower bounds for the blow-up rate in ${\dot{H}}^{3/2}$ and in $H^1$.