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

Dongho Chae

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arXiv · 1306.0305

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

Dongho Chae

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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 $C^1(\Bbb R ; L^3(\Bbb R^3)\cap C^2 (\Bbb R^3))$. 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.

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Dongho Chae. 2013-06-03. Remarks on the asymptotically discretely self-similar solutions of the Navier-Stokes and the Euler equations. https://arxiv.org/abs/1306.0305

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