Grade Regularity: A Universal Criterion for Strong Solutions of Nonlinear PDEs

We prove a universal regularity criterion for nonlinear partial differential equations based on the grade decomposition of analytic vector fields. For any evolution PDE \(\partial_t u = F(u)\) where \(F\) is analytic with grade decomposition \(F = \sum_{k=0}^{\infty} A^{(k)}\) and analyticity radius \(\rho\), we establish:

(I) The Grade Regularity Theorem. If the analyticity radius satisfies \(\rho(t) \geq \rho_{\min} > 0\) on \([0, T)\), the solution is real-analytic on \([0, T) \times \Omega\) and satisfies \(\|D^k u(t)\| \leq C \cdot k! / \rho(t)^k\) for all \(k\). In particular, the solution is a classical strong solution in \(C^\infty\).

(II) The Perturbation Stability Theorem. Consider a perturbed PDE \(\partial_t u = F(u) + G(u, \nabla u)\) where \(G\) is analytic with analyticity radius \(\rho_G\). If the unperturbed system has grade regularity with radius \(\rho_F\), the perturbed system has grade regularity with radius \(\rho_{F+G} \geq (\rho_F^{-1} + \rho_G^{-1})^{-1}\). Adding a smooth perturbation cannot destroy regularity unless it drives \(\rho_{F+G}\) to zero.

(III) Subsumption of classical criteria. We show that the Beale–Kato–Majda criterion (Navier–Stokes), the Prodi–Serrin conditions, the Grujić–Kukavica lower bounds on analyticity radius, and the Foias–Temam Gevrey regularity results are all implied by grade regularity applied to the specific grade structure of each equation.

(IV) The Grade Perturbation Calculus. We develop explicit rules for how common PDE operations — adding a forcing term, coupling two systems, taking singular limits — affect the combined grade spectrum and \(\rho\). This provides a modular toolkit for regularity analysis of composite systems.