Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations

In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as $t$ goes to the possible time of singularity $T$. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, ${\dot{B}}_{1,\mathrm{\infty }}^0({\mathbb{R}}^3)$. For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in $L^q(B(z,r))$ for some $q\in [2,\mathrm{\infty })$, where the ball of radius $r$ is shrinking toward a possible singularity point $z$ at the order of $\sqrt{T-t}$ as $t$ approaches to $T$. In the $L^q({\mathbb{R}}^3)$ convergence case with $q\in [3,\mathrm{\infty })$ we present a simple alternative proof of the similar result in \cite{hou}.