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 Navier–Stokes 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 C1(R;L3(R3)∩C2(R3)). 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.