Depletion of nonlinearity in space-analytic space-periodic solutions to equations of diffusive magnetohydrodynamics

TL;DR

This paper reveals a novel mechanism for the depletion of nonlinearity in space-analytic solutions to magnetohydrodynamics equations, using a two-phase iterative approach to bound the time of solution analyticity.

Contribution

It introduces a new bound on the space analyticity of solutions and a two-phase iterative method to analyze the nonlinear structure in magnetohydrodynamics equations.

Findings

Effective nonlinear depletion due to special structure of solutions

Expanded bounds for the time of space analyticity

Generalization to magnetohydrodynamics equations

Abstract

We consider solenoidal space-periodic space-analytic solutions to the equations of magnetohydrodynamics. An elementary bound shows that due to the special structure of the nonlinear terms in the equations for modified solutions, effectively they lack a half of the spatial gradient, which appears to be a novel mechanism for depletion of nonlinearity. We present a two-phase iterative procedure yielding an expanded bound for the guaranteed time of the space analyticity of the hydrodynamic solutions. Each iteration involves two regimes: In phase 1, the enstrophy of the modified solution and the bound for the radius of the analyticity of the original solution simultaneously increase (the bound is proportional to the elapsed time since the beginning of phase 1). In phase 2, the enstrophy and bound simultaneously decrease. It is straightforward to generalize this construction for the equations of magnetohydrodynamics.