A compensated Piola principle for critical nondiffusive parabolic systems

Abstract

We introduce a compensated Piola graph method for Lagrangian stability estimates in critical homogeneous Besov spaces below the usual product threshold. For $d\ge2$, $2d\le p<\infty$, and $s=d/p$, we establish local Hadamard well-posedness and a continuation criterion in the scaling-critical Besov phase space $\dot B^{s-1}_{p,1}(\mathbb R^d)^d\times \dot B^s_{p,1}(\mathbb R^d)$, for a class of incompressible parabolic systems coupled to nondiffusive internal variables. The main obstruction is that the formal Piola product between the inverse deformation gradient and the Lagrangian velocity is only borderline at $p=2d$ and is not continuous in general for $p>2d$. We replace this product by a closed solenoidal Piola graph, using a compensated divergence structure that survives in the high-integrability range. The principle applies to viscous non-resistive MHD, Hookean incompressible viscoelasticity, and nondiffusive Oldroyd--B systems with affine objective terms. In particular, it closes the previously untreated high-$p$ Hadamard well-posedness range for critical non-resistive MHD and for the nondiffusive Oldroyd--B systems considered here; combined with the known low-$p$ MHD theory, it gives the finite-$p$, $q=1$ critical Besov picture for non-resistive MHD.