Vol. 12, No. 8, 2019

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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
Abstract

Chae and Wolf recently constructed discretely self-similar solutions to the Navier–Stokes equations for any discretely self-similar data in Lloc2. 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 Lloc2 and an approximation of divergence-free, discretely self-similar vector fields in Lloc2 by divergence-free, discretely self-similar elements of Lw3.

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