#### 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 ${L}_{loc}^{2}$. 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