# Discretely self-similar solutions to the Navier-Stokes equations with Besov space data

## TL;DR

This paper constructs both self-similar and discretely self-similar solutions to the 3D Navier-Stokes equations with initial data in certain Besov spaces, extending previous results and providing concrete examples.

## Contribution

It introduces new methods to construct self-similar solutions for larger initial data in Besov spaces, expanding the class of initial conditions for Navier-Stokes solutions.

## Key findings

- Constructed self-similar solutions for large Besov space data.
- Extended previous results from $L^3_w$ to Besov spaces.
- Provided explicit examples of initial vector fields in the relevant spaces.

## Abstract

We construct self-similar solutions to the three dimensional Navier-Stokes equations for divergence free, self-similar initial data that can be large in the critical Besov space $\dot B^{-1+3/p}_{p,\infty}$ where $3< p< 6$. We also construct discretely self-similar solutions for divergence free initial data in $\dot B^{-1+3/p}_{p,\infty}$ for $3<p<6$ that is discretely self-similar for some scaling factor $\lambda>1$. These results extend those of \cite{BT1} which dealt with initial data in $L^3_w$ since $L^3_w\subsetneq \dot B^{-1+3/p}_{p,\infty}$ for $p>3$. We also provide several concrete examples of vector fields in the relevant function spaces.