The key finding in this mathematical preprint is conditional: under the paper’s assumptions, a forcing term f with the required local integrability gives a weak solution a spatial gradient Du with higher local integrability. In plain language, the equation’s input can be controlled in one local sense while the solution’s change across space can be controlled in a stronger one—but only within the theorem’s stated exponent range.
The result is framed as a Calderón–Zygmund estimate, a bound linking the integrability of an equation’s data to that of its gradient. The paper studies nonlinear parabolic systems whose growth is governed by a power p and whose data enter in non-divergence form.
What the theorem says
The core model is the nonlinear system ∂t u − div A(x,t,Du)=f on a bounded space-time cylinder. The objects being analyzed are local weak solutions, coefficient fields and forcing terms, not participants or laboratory samples.
In its main formulation, the theorem turns the specified local integrability of f into local integrability of Du at exponent s. It also gives a quantitative estimate on every parabolic cylinder Q_R compactly contained in Ω_T, with R in (0,1]. The bound uses lower-order gradient and scaled-datum terms, while its constant depends on n, N, p, ν, L, s and the coefficient modulus ω(·).
An equivalent formulation uses q as the integrability exponent for f: the theorem states that local L^q integrability of |f| gives local integrability of |Du| at exponent q[n(p−1)+p]/(n+2−q). That formulation remains limited to the paper’s stated q range.
How the estimate is built
The proof is built around comparisons. It contrasts the original solution with homogeneous problems whose vector field is frozen in the spatial variable. For the coefficient cases labeled VMO, the argument uses two successive comparisons: first with a homogeneous problem and then with a frozen problem.
Other ingredients include intrinsic parabolic cylinders—space-time regions scaled to the equation—and a Vitali covering step that produces a countable pairwise disjoint subfamily. Higher-integrability lemmas for homogeneous comparison systems provide the gradient control needed in the stated range of powers starting at p.
A conditional, local result
The authors present the framework as a unified treatment of the singular p<2 and degenerate p≥2 regimes. They also state that the estimates do not require differentiability of the coefficients with respect to the gradient variable.
The conclusions are conditional on local weak-solution status and on structural condition (2.2) together with either (2.3) or conditions (2.4)–(2.6). The estimates are local, applying on cylinders compactly contained in the domain, and the admissible exponent ranges are restricted by the theorem. The supplied analysis does not establish global or boundary estimates.
The supplied document is an arXiv version 1 preprint dated 20 August 2026. Because the work contains no empirical participant or biological sample, it is a mathematical result rather than evidence about human disease, treatment or prognosis.
Paper data and sources
Original title: Calderón-Zygmund estimates for parabolic systems with $p$-growth and non-divergence data
Authors: Pêdra Andrade, Kristian Moring
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text