Space analyticity and bounds for derivatives of solutions to the evolutionary equations of diffusive magnetohydrodynamics

TL;DR

This paper develops new bounds for derivatives of solutions to diffusive magnetohydrodynamics equations, extending previous results from Navier-Stokes, using space analyticity and auxiliary problems to analyze solution regularity.

Contribution

It introduces an original method to derive a priori estimates for derivatives of MHD solutions, generalizing bounds known for hydrodynamics to magnetohydrodynamics.

Findings

Derived bounds for space derivatives of MHD solutions

Established analyticity regions related to Sobolev norms

Extended previous hydrodynamic results to MHD context

Abstract

In 1981, Foias, Guillop\'e and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier-Stokes equation. Such bounds are instructive in the numerical investigation of intermittency often observed in simulations, e.g., numerical study of vorticity moments by Donzis et al. (2013) revealed depletion of nonlinearity that may be responsible for smoothness of solutions to the Navier-Stokes equation. We employ an original method to derive analogous estimates for space derivatives of three-dimensional space-periodic weak solutions to the evolutionary equations of diffusive magnetohydrodynamics. Construction relies on space analyticity of the solutions at almost all times. An auxiliary problem is introduced, and a Sobolev norm of its solutions bounds from below the size in $C^3$ of the region of space analyticity of the solutions to the original problem. We recover the exponents obtained earlier for the hydrodynamic problem. The same approach is also followed here to derive and prove similar a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the weak MHD solutions. This paper is dedicated to Professor Uriel Frisch on the occasion of his 80th anniversary as a sign of appreciation of the Scientist and the Teacher.