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2019 Discretely self-similar solutions to the Navier–Stokes equations with data in $L^2_{\mathrm{loc}}$ satisfying the local energy inequality
Zachary Bradshaw, Tai-Peng Tsai
Anal. PDE 12(8): 1943-1962 (2019). 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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Zachary Bradshaw. Tai-Peng Tsai. "Discretely self-similar solutions to the Navier–Stokes equations with data in $L^2_{\mathrm{loc}}$ satisfying the local energy inequality." Anal. PDE 12 (8) 1943 - 1962, 2019. https://doi.org/10.2140/apde.2019.12.1943

Information

Received: 24 January 2018; Revised: 16 October 2018; Accepted: 30 November 2018; Published: 2019
First available in Project Euclid: 14 December 2019

zbMATH: 07143407
MathSciNet: MR4023972
Digital Object Identifier: 10.2140/apde.2019.12.1943

Subjects:
Primary: 35Q30, 76D05

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 8 • 2019
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