Turbulent Dynamos on Bounded Domains and Their Generalization to the Geometric Transport Equation

TL;DR

This paper constructs divergence-free velocity and magnetic fields on bounded domains that demonstrate arbitrarily fast magnetic energy growth, using convex integration and introducing explicit potentials for localization.

Contribution

It introduces a novel convex integration scheme with explicit potentials for bounded domains, enabling the construction of solutions with rapid magnetic energy growth and a unified approach to the geometric transport equation.

Findings

Constructed solutions with arbitrarily fast magnetic energy growth.

Developed a convex integration scheme with explicit potentials for localization.

Unified the treatment of transport and Maxwell equations within the geometric transport framework.

Abstract

For any smooth bounded domain $\Omega \subset \mathbb{R}^3$, we construct a divergence-free velocity field $u \in L_t^1 W^{1,p}(\Omega)$ for all $p < \infty$, and magnetic fields $B^\epsilon \in L_t^p C^{m}(\Omega)$ for all $p < \infty$ and $m\in \mathbb{N}$, that solve the kinematic dynamo equation and exhibit arbitrarily fast growth of any magnetic energy mode, uniformly in the vanishing-diffusivity limit $\epsilon \to 0$. The construction is based on the convex integration scheme of Modena-Sz\'ekelyhidi and Cheskidov-Luo. The main novelty lies in the introduction of explicit potentials, which allow the solutions to be localized and avoid the need to work with the anti-curl operator. In addition, we present a unified scheme for the geometric transport equation (GTE), which encompasses both the transport and Maxwell equations.