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

TL;DR

This paper constructs discretely self-similar solutions to the Navier-Stokes equations with local energy inequality, providing explicit pressure formulas and new local energy estimates for data in $L^2_{loc}$, advancing understanding of weak solutions.

Contribution

It introduces a method to construct discretely self-similar suitable weak solutions satisfying the classical local energy inequality for $L^2_{loc}$ data, with explicit pressure formulas and local energy estimates.

Findings

Constructed discretely self-similar solutions satisfying classical local energy inequality.

Derived explicit pressure formulas in terms of velocity.

Developed new local energy estimates for discretely self-similar solutions.

Abstract

Chae and Wolf recently constructed discretely self-similar solutions to the Navier-Stokes equations for any discretely self similar data in $L^2_{\mathrm{loc}}$. 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 $L^2_{\mathrm{loc}}$ and an approximation of divergence free, discretely self-similar vector fields in $L^2_{\mathrm{loc}}$ by divergence free, discretely self-similar elements of $L^3_w$.