Forward discretely self-similar solutions of the Navier-Stokes equations II

TL;DR

This paper constructs forward discretely self-similar solutions to the 3D Navier-Stokes equations for large initial data in the weak $L^3$ space, without requiring continuity or local boundedness, extending the class of known solutions.

Contribution

It introduces a method to construct discretely self-similar solutions for large initial data in $L^3_w$, without additional regularity assumptions.

Findings

Existence of discretely self-similar solutions for large initial data.

Solutions are constructed in the sense of Lemarié-Rieusset.

Method applies to any $-1$-homogeneous initial data in $L^3_w$.

Abstract

For any discretely self-similar, incompressible initial data $v_0$ which satisfies $\|v_0 \|_{L^3_w(\mathbb R^3)}\leq c_0$ where $c_0$ is allowed to be large, we construct a forward discretely self-similar local Leray solution in the sense of Lemari\'e-Rieusset to the 3D Navier-Stokes equations in the whole space. No further assumptions are imposed on the initial data; in particular, the data is not required to be continuous or locally bounded on $\mathbb R^3\setminus \{0\}$. The same method is used to construct self-similar solutions for any $-1$-homogeneous initial data in $L^3_w$.