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

Abstract

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, $$\dot{B}^0_{\infty, 1}(\mathbb{R}^3)$$ . For the Navier–Stokes equations the convergence of the velocity to the self-similar singularity is in L q (B(z,r)) for some $$q\in [2, \infty)$$ , 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 $$q\in [3, \infty)$$ we present a simple alternative proof of the similar result in Hou and Li in arXiv-preprint, math.AP/0603126.