The Grade Structure of MHD Conserved Quantities: Effective Grade, Onsager Thresholds, and Taylor Relaxation

We introduce the effective grade of a conserved quantity: \(\mathrm{grade}_{\mathrm{eff}}(Q) = \mathrm{grade}_{\mathrm{nom}}(Q) - \delta_{\mathrm{constr}}(Q)\), where \(\mathrm{grade}_{\mathrm{nom}}\) is the polynomial degree in the dynamical fields and \(\delta_{\mathrm{constr}}\) is the derivative order gained from structural constraints (gauge freedom, differential form closedness, potential structure). For the three conserved quantities of ideal incompressible MHD on \(\mathbb{T}^3\):

| Quantity | Expression | \(\mathrm{grade}_{\mathrm{nom}}\) | \(\delta_{\mathrm{constr}}\) | \(\mathrm{grade}_{\mathrm{eff}}\) | Predicted threshold | Proved threshold | |:---|:---|:---:|:---:|:---:|:---|:---| | Total energy | \(\frac{1}{2}\int (\lvert u\rvert^2 + \lvert B\rvert^2)\) | 2 | 0 | 2 | \(C^{0,1/3}\) | \(C^{0,1/3}\) (CET, Kang-Lee) | | Cross-helicity | \(\int u \cdot B\) | 2 | 0 | 2 | \(C^{0,1/3}\) | \(C^{0,1/3}\) (CET, Kang-Lee) | | Magnetic helicity | \(\int A \cdot B\) | 2 | 1 | 1 | \(L^3\) | \(L^3\) (Kang-Lee) |

The effective grade formula recovers all known Onsager-type thresholds for MHD as structural predictions. We prove that the Tartar-Murat div-curl compensation lemma — the mechanism behind helicity's robustness — is a manifestation of grade-1 stability under weak convergence. We reinterpret Taylor-Woltjer relaxation as grade minimization: the magnetic field evolves toward the configuration of minimal total grade (force-free/Beltrami fields) subject to the grade-1 topological invariant (helicity). We show that the Elsässer variables \(z^\pm = u \pm B\) diagonalize the grade-2 energy and provide the natural basis for the grade decomposition of MHD. We discuss how the nonlinear simpleness constraint \(\omega \wedge \omega = 0\) on the Faraday 2-form — the obstruction to extending Székelyhidi's bounded solutions to Hölder continuous ones — manifests as a grade constraint that is not removable by the standard convex integration framework.