Existence of infinite-energy and discretely self-similar global weak solutions for 3D MHD equations

TL;DR

This paper proves the existence of global weak solutions and discretely self-similar solutions for 3D MHD equations with specific initial data, extending understanding of solution behaviors in weighted spaces.

Contribution

It establishes the existence of infinite-energy and discretely self-similar solutions for 3D MHD equations, inspired by recent Navier-Stokes research.

Findings

Existence of global weak solutions in weighted spaces.

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

Extension of methods from Navier-Stokes to MHD equations.

Abstract

This paper deals with the existence of global weak solutions for 3D MHD equations when the initial data belong to the weighted spaces $L^2_{w_\gamma}$, with $w_\gamma(x)=(1+| x|)^{-\gamma}$ and $0 \leq \gamma \leq 2$. Moreover, we prove the existence of discretely self-similar solutions for 3D MHD equations for discretely self-similar initial data which are locally square integrable. Our methods are inspired of a recent work of P. Fern\'andez-Dalgo and P.G. Lemari\'e-Riseusset for the 3D Navier-Stokes equations.