# Weak solutions for Navier--Stokes equations with initial data in weighted $L^2$ spaces

## TL;DR

This paper proves the existence of global weak solutions to the 3D Navier-Stokes equations with initial data in weighted L2 spaces, expanding understanding of solutions with less regular initial conditions.

## Contribution

It introduces new energy control methods to establish global weak solutions with initial data in weighted L2 spaces and provides a new proof for discretely self-similar solutions.

## Key findings

- Existence of global weak solutions with initial data in weighted L2 spaces.
- New energy control techniques for Navier-Stokes equations.
- Alternative proof for discretely self-similar solutions.

## Abstract

We show the existence of global weak solutions of the 3D Navier-Stokes equations with initial velocity in the weighted spaces L 2 w$\gamma$ , where w $\gamma$ (x) = (1 + |x|) --$\gamma$ and 0 < $\gamma$ $\le$ 2, using new energy controls. As application we give a new proof of the existence of global weak discretely self-similar solutions of the 3D Navier-Stokes equations for discretely self-similar initial velocities which are locally square inte-grable.