Discretely self-similar solutions to the Navier–Stokes equations with data in Lloc2 satisfying the local energy inequality

Bradshaw, Zachary; Tsai, Tai-Peng

doi:10.2140/apde.2019.12.1943

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Discretely self-similar solutions to the Navier–Stokes equations with data in $L^2_{\mathrm{loc}}$ satisfying the local energy inequality

Zachary Bradshaw and Tai-Peng Tsai

Vol. 12 (2019), No. 8, 1943–1962

DOI: 10.2140/apde.2019.12.1943

Abstract

Chae and Wolf recently constructed discretely self-similar solutions to the Navier–Stokes equations for any discretely self-similar data in $L_{loc}^{2}$ . Their solutions are in the class of local Leray solutions with projected pressure and satisfy the “local energy inequality with projected pressure”. In this note, for the same class of initial data, we construct discretely self-similar suitable weak solutions to the Navier–Stokes equations that satisfy the classical local energy inequality of Scheffer and Caffarelli–Kohn–Nirenberg. We also obtain an explicit formula for the pressure in terms of the velocity. Our argument involves a new purely local energy estimate for discretely self-similar solutions with data in $L_{loc}^{2}$ and an approximation of divergence-free, discretely self-similar vector fields in $L_{loc}^{2}$ by divergence-free, discretely self-similar elements of $L_{w}^{3}$ .

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Keywords

Navier–Stokes equations, self-similar solution, weak solution

Mathematical Subject Classification 2010

Primary: 35Q30, 76D05

Milestones

Received: 24 January 2018

Revised: 16 October 2018

Accepted: 30 November 2018

Published: 28 October 2019

Authors

Zachary Bradshaw
Department of Mathematics University of Arkansas Fayetteville, AR United States

Tai-Peng Tsai
Department of Mathematics University of British Columbia Vancouver, BC Canada

Vol. 12, No. 8, 2019