# On approximate solutions of the equations of incompressible magnetohydrodynamics

## TL;DR

This paper develops a framework for analyzing approximate solutions of incompressible magnetohydrodynamics (MHD) equations, enabling the determination of solution existence time and error bounds, with applications including global existence results for specific flows.

## Contribution

It introduces a novel a posteriori analysis method for MHD equations, extending previous approaches for Navier-Stokes, and applies it to complex three-dimensional flows.

## Key findings

- Lower bounds on solution existence time can be computed from approximate solutions.
- Global existence of MHD solutions is established under certain conditions.
- The method successfully analyzes a Galerkin approximation of the ABC flow in 3D.

## Abstract

Inspired by an approach proposed previously for the incompressible Navier-Stokes (NS) equations, we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics (MHD) on a torus of arbitrary dimension d; this setting involves a Sobolev space of infinite order, made of C^infinity vector fields (with vanishing divergence and mean) on the torus. Given any approximate solution of the MHD Cauchy problem, its a posteriori analysis with the method of the present work allows to infer a lower bound on the time of existence of the exact solution, and to bound from above the Sobolev distance of any order between the exact and the approximate solution. In certain cases the above mentioned lower bound on the time of existence is found to be infinite, so one infers the global existence of the exact MHD solution. We present some applications of this general scheme; the most sophisticated one lives in dimension d=3, with the ABC flow (perturbed magnetically) as an initial datum, and uses for the Cauchy problem a Galerkin approximate solution in 124 Fourier modes. We illustrate the conclusions arising in this case from the a posteriori analysis of the Galerkin approximant; these include the derivation of global existence of the exact MHD solution with the ABC datum, when the dimensionless viscosity and resistivity are equal and stay above an explicitly given threshold value.