Energy Equality, Vectorial L1 Contraction, and Mixed-Norm Stability for Parabolic Systems with Rough Skew-Symmetric Potentials
Abstract
We consider Dirichlet parabolic systems with measurable, real skew-symmetric zero-order potentials on bounded Lipschitz domains in three dimensions. Although skew-symmetry formally removes the potential from the energy balance, singular coefficients may leave the product CU merely integrable, so testing the equation with the solution is not justified. We overcome this obstruction through bounded-gradient entropy identities and stable approximation. For C∈ Lr(0,T;Lq(Ω;ℝm× m)) with 1≤ r≤∞, 6/5≤ q≤∞, and 2/r+3/q≤7/2, we establish existence and uniqueness in the full energy--distributional class, exact energy equality at every pair of times, Euclidean-vector L1 contraction, and strong convergence of both solutions and coefficient--solution products under strong coefficient approximation. We compare a vectorial entropy proof with a derivation by scalar entropies and spherical averaging, and explain the narrower roles of temporal regularization, convex invariance, and adjoint transposition. An application to diffusion--precession gives quantitative stability under L1 perturbations of prescribed magnetic fields for bounded initial data, including singular fields outside the main mixed-norm range.
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BibTeX
@misc{fernandez2026energy,
title = {{Energy Equality, Vectorial $L^1$ Contraction, and Mixed-Norm Stability for Parabolic Systems with Rough Skew-Symmetric Potentials}},
author = {Francisco Javier Fernández Fernández},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26333},
eprint = {2610.26333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26333},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Energy Equality, Vectorial L^1 Contraction, and Mixed-Norm Stability for Parabolic Systems with Rough Skew-Symmetric Potentials
AU - Francisco Javier Fernández Fernández
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26333
UR - https://arxiv.org/abs/2610.26333
AB - We consider Dirichlet parabolic systems with measurable, real skew-symmetric zero-order potentials on bounded Lipschitz domains in three dimensions. Although skew-symmetry formally removes the potential from the energy balance, singular coefficients may leave the product CU merely integrable, so testing the equation with the solution is not justified. We overcome this obstruction through bounded-gradient entropy identities and stable approximation. For C∈ L^r(0,T;L^q(Ω;ℝ^m× m)) with 1≤ r≤∞, 6/5≤ q≤∞, and 2/r+3/q≤7/2, we establish existence and uniqueness in the full energy--distributional class, exact energy equality at every pair of times, Euclidean-vector L^1 contraction, and strong convergence of both solutions and coefficient--solution products under strong coefficient approximation. We compare a vectorial entropy proof with a derivation by scalar entropies and spherical averaging, and explain the narrower roles of temporal regularization, convex invariance, and adjoint transposition. An application to diffusion--precession gives quantitative stability under L^1 perturbations of prescribed magnetic fields for bounded initial data, including singular fields outside the main mixed-norm range.
ER -