[Paper Review] Magnetic helicity and subsolutions in ideal MHD

Daniel Faraco, Sauli Lindberg|arXiv (Cornell University)|Jan 15, 2018

Navier-Stokes equation solutions32 references3 citations

TL;DR

This paper establishes that ideal 2D magnetohydrodynamics (MHD) admits no weak or subsolutions with compactly supported time dependence and non-trivial magnetic fields, due to the structure of the Λ-convex hull of the constraint set. It further proves that mean-square magnetic potential is conserved in 2D for subsolutions and weak limits in the energy space $L^∞_t L^2_x$, while magnetic helicity is conserved in 3D under $L^3$-integrability, with the Λ-convex hull being large enough to allow nontrivial smooth, compactly supported strict subsolutions in 3D.

ABSTRACT

We show that ideal 2D MHD does not possess weak solutions (or even subsolutions) with compact support in time and non-trivial magnetic field. We also show that the $Λ$-convex hull of ideal MHD has empty interior in both 2D and 3D; this is seen by finding suitable $Λ$-convex functions. As a consequence we show that mean-square magnetic potential is conserved in 2D by subsolutions and weak limits of solutions in the physically natural energy space $L^\infty_t L^2_x$, and in 3D we show the conservation of magnetic helicity by $L^3$-integrable subsolutions and weak limits of solutions. However, in 3D the $Λ$-convex hull is shown to be large enough that nontrivial smooth, compactly supported strict subsolutions exist.

Motivation & Objective

• To investigate the existence of weak and subsolutions in ideal MHD with compact support in time.
• To analyze the structure of the Λ-convex hull of the constraint set in ideal MHD and its implications for conservation laws.
• To determine whether integral invariants such as magnetic helicity and mean-square magnetic potential are preserved under weak limits and subsolutions.
• To clarify the distinction between 2D and 3D behavior in terms of subsolution existence and conservation laws.

Proposed method

• Adopt the Tartar framework to decouple ideal MHD into linear PDEs and pointwise constraints on the state variables.
• Define the wave cone $\Lambda$ and the $\Lambda$-convex hull $K^{\Lambda}$ to analyze the set of possible subsolutions.
• Construct explicit $\Lambda$-convex functions to show that the $\Lambda$-convex hull has empty interior in 2D, but is large enough in 3D to allow nontrivial compactly supported subsolutions.
• Use the convex integration method to analyze conservation of physical quantities under weak convergence.
• Apply the theory of compensated compactness and weak limits to prove conservation of mean-square magnetic potential in 2D and magnetic helicity in 3D.
• Leverage symmetry reductions and known results from convex integration in Euler equations to construct 3D MHD solutions with compact support in time.

Experimental results

Research questions

• RQ1 Can ideal 2D MHD admit weak solutions or subsolutions with compact support in time and non-trivial magnetic field?
• RQ2 Is the $\Lambda$-convex hull of the ideal MHD constraint set $K$ empty in the interior in 2D and 3D?
• RQ3 Does the mean-square magnetic potential remain conserved in 2D for subsolutions and weak limits of solutions in the energy space $L^\infty_t L^2_x$?
• RQ4 Is magnetic helicity conserved in 3D for $L^3$-integrable subsolutions and weak limits of solutions?
• RQ5 Can nontrivial smooth, compactly supported strict subsolutions exist in 3D ideal MHD?

Key findings

• In 2D, ideal MHD admits no weak solutions or subsolutions with compact support in time and non-trivial magnetic field, due to the $\Lambda$-convex hull of the constraint set having empty interior.
• The mean-square magnetic potential is conserved in 2D for subsolutions and weak limits of solutions in the energy space $L^\infty_t L^2_x$, as a consequence of the $\Lambda$-convex hull structure.
• In 3D, magnetic helicity is conserved for $L^3$-integrable subsolutions and weak limits of solutions, due to the existence of appropriate $\Lambda$-convex functions.
• The $\Lambda$-convex hull in 3D is large enough to allow the existence of nontrivial smooth, compactly supported strict subsolutions.
• The construction of compactly supported solutions in 3D via symmetry reduction (e.g., $u = (u_1, u_2, 0)$, $b = (0, 0, b_3)$) relies on orthogonal decomposition of velocity and magnetic fields, which prevents cross helicity conservation in such solutions.
• The cross helicity $\int u \cdot b\,dx$ vanishes identically in the symmetric solutions of Bronzi et al., leaving open the question of whether it is conserved in general weak solutions of 3D ideal MHD.