Remarks on the asymptotically discretely self-similar solutions of the Navier–Stokes and the Euler equations

Abstract

We study scenarios of self-similar type blow-up for the incompressible NavierStokes and the Euler equations. The previous notions of the discretely (backward) self-similar solution and the asymptotically self-similar solution are generalized to the locally asymptotically discretely self-similar solution. We prove that there exists no such locally asymptotically discretely self-similar blow-up for the 3D Navier-Stokes equations if the blow-up profile is a time periodic function belonging to C 1 (R; L 3 (R 3 ) ∩ C 2 (R 3 )). For the 3D Euler equations we show that the scenario of asymptotically discretely self-similar blow-up is excluded if the blow-up profile satisfies suitable integrability conditions. Mathematics Subject Classification(2000): 76B03, 35Q311 The main theorems Navier-Stokes equationsWe consider the Cauchy problem of the 3D Navier-stokes equations.where v(x, t) = (v 1 (x, t), v 2 (x, t), v 3 (x, t)) is the velocity, p = p(x, t) is the pressure, and v 0 (x) is the initial data satisfying divv 0 = 0. We study the possibility of finite time blow-up of smooth solution of (NS). By translation of time we may assume that the solution is smooth for t < 0, and the blow-up happens at t = 0. We say a solution 1

Removing discretely self-similar singularities for the 3D Navier–Stokes equations

“…Although the proof is similar to the proof of Theorem 1.2 in [2], we write it in detail for reader's convenience.…”

“…By Q we denote the space time cylinder Ω × (0, T ). We denote V 1,2 σ (Q) the space of all vector functions L ∞ (0, T ; L 2 (Ω))∩L 2 (0, T ; W 1, 2 (Ω)) fulfilling ∇•u = 0 a.e. in Q.…”

Existence of Vortex Rings in Beltrami Flows

“…Chae [Cha07], [Cha15a], [Cha15c], [Cha15b], Chae et al [CKL09], Chae and Shvydkoy [CS13], and Chae and Tsai [CT14] investigated the nonexistence of backward self-similar solutions to the Euler equations for α −1:…”