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arxiv: 2604.21727 · v1 · submitted 2026-04-23 · 🧮 math.AP

Calder\'on-Zygmund estimates for parabolic p-Laplacian systems with non-divergence form right-hand sides

P\^edra Andrade , Verena B\"ogelein , Frank Duzaar , Kristian Moring This is my paper

Pith reviewed 2026-05-09 20:41 UTC · model grok-4.3

classification 🧮 math.AP
keywords Calderon-Zygmund estimatesparabolic p-LaplacianVMO coefficientsnon-divergence formintegrability improvementweak solutionsnonlinear parabolic systemsgradient estimates
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The pith

If the right-hand side of a parabolic p-Laplacian system is in L to the power mu s locally, then the spatial gradient of the weak solution gains L to the power s integrability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to prove local Calderon-Zygmund estimates showing that higher integrability of the non-divergence right-hand side passes to the spatial gradient of solutions to nonlinear parabolic systems with p-growth. A sympathetic reader would care because these systems arise in models of diffusion and flow, and the result gives a precise rule for how much smoothness the solution can be expected to have from the data alone. The transfer is sharp, matches the natural scaling of the parabolic equation, and reduces to the known optimal exponents when the system becomes linear at p equals 2. The argument proceeds by combining an intrinsic scaling adapted to the parabolic structure with an iterative scheme that upgrades integrability step by step.

Core claim

For weak solutions to parabolic systems with p-growth and VMO coefficients, membership of the right-hand side in L to the power mu s locally, with the exponent mu chosen depending on p, the dimension N, and a target s greater than p, implies that the spatial gradient Du belongs to L to the power s locally. This yields a sharp integrability transfer consistent with parabolic scaling and recovers the optimal linear-case exponents at p equals 2.

What carries the argument

Intrinsic scaling techniques combined with a Calderon-Zygmund iteration scheme, which upgrade integrability from the right-hand side to the gradient under the VMO condition on the coefficients.

If this is right

  • The estimates apply directly to systems rather than only scalar equations.
  • The optimal exponents known for the linear parabolic case p equals 2 are recovered as a special case.
  • The integrability transfer respects the natural parabolic scaling relation between time and space.
  • The result is local, so it applies inside any subdomain where the assumptions hold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scaling-plus-iteration approach might adapt to other growth exponents or to systems with different structural assumptions if the VMO condition can be maintained.
  • Numerical tests on explicit radial solutions could verify whether the explicit dependence of mu on p and N is sharp in practice.
  • The estimates could serve as a starting point for proving higher regularity or partial regularity results when additional structure is present.

Load-bearing premise

The coefficients satisfy the vanishing mean oscillation condition and the system obeys standard p-growth without extra singularities.

What would settle it

A concrete weak solution in which the right-hand side lies in the predicted L to the power mu s space yet the spatial gradient fails to reach L to the power s integrability.

read the original abstract

We establish local Calder\'on-Zygmund type estimates for weak solutions to nonlinear parabolic systems with $p$-growth and VMO coefficients. In particular, we prove that if the right-hand side belongs locally to $L^{\mu s}$, where the exponent $\mu$ depends explicitly on $p$, $N$, and a prescribed target exponent $s>p$, then the spatial gradient of the solution enjoys improved integrability $Du \in L^s_{\rm{loc}}$. The result provides a sharp transfer of integrability from the data to the gradient, consistent with the natural parabolic scaling, and recovers the optimal exponents in the linear case $p=2$. The proof combines intrinsic scaling techniques with a Calder\'on-Zygmund type iteration scheme.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript establishes local Calderón-Zygmund estimates for weak solutions of parabolic p-Laplacian systems with VMO coefficients and non-divergence right-hand sides. It proves that local membership of the right-hand side in L^{μs} (with μ depending explicitly on p, N and a target s > p) implies Du ∈ L^s_loc, via intrinsic scaling combined with a Calderón-Zygmund iteration scheme; the result recovers the optimal linear exponents when p = 2.

Significance. If the estimates hold, the work supplies a sharp, scaling-consistent transfer of integrability from data to gradient for nonlinear parabolic systems, extending known linear and scalar results to the vectorial VMO setting. The explicit dependence of μ and the use of intrinsic cylinders are positive features that align with the parabolic structure.

major comments (2)
  1. [Proof of Theorem 1.1 (or equivalent main result)] The iteration scheme must be checked to ensure that the VMO modulus enters the constants in a controllable way; if the smallness of the VMO oscillation is used to absorb error terms in the CZ iteration, the dependence of the final constant on the VMO modulus should be stated explicitly (likely in the proof of the main theorem).
  2. [Section 3 or 4 (preliminaries on exponents)] The explicit formula for the exponent μ(p, N, s) is asserted to be chosen so that the iteration closes and recovers the p = 2 case; this formula and the verification that it satisfies the necessary inequalities (e.g., μ > 1 and compatibility with the intrinsic scaling) must appear before the iteration begins.
minor comments (3)
  1. [Section 2] Notation for the intrinsic cylinders (e.g., the radius scaling with |Du|^{p-2}) should be introduced once and used consistently; a short table or list of the cylinders appearing in the estimates would improve readability.
  2. [Theorem 1.1] The statement of the main theorem should include the precise dependence of the constant C on the VMO modulus, p, N, s and the local L^{μs} norm of the right-hand side.
  3. [Introduction] A brief comparison paragraph with the corresponding results for divergence-form right-hand sides or for the linear case would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Proof of Theorem 1.1 (or equivalent main result)] The iteration scheme must be checked to ensure that the VMO modulus enters the constants in a controllable way; if the smallness of the VMO oscillation is used to absorb error terms in the CZ iteration, the dependence of the final constant on the VMO modulus should be stated explicitly (likely in the proof of the main theorem).

    Authors: We agree with the referee that the dependence of the constants on the VMO modulus should be made explicit. In the revised manuscript, we have included a detailed remark in the proof of the main theorem (Theorem 1.1) explaining how the smallness condition on the VMO oscillation is chosen depending on the target integrability exponent s, and we explicitly state the dependence of the constant C on the VMO modulus function ω. revision: yes

  2. Referee: [Section 3 or 4 (preliminaries on exponents)] The explicit formula for the exponent μ(p, N, s) is asserted to be chosen so that the iteration closes and recovers the p = 2 case; this formula and the verification that it satisfies the necessary inequalities (e.g., μ > 1 and compatibility with the intrinsic scaling) must appear before the iteration begins.

    Authors: We thank the referee for this observation. The explicit formula for μ(p, N, s) and the verification of the required inequalities (including μ > 1 and compatibility with intrinsic scaling) have been added at the beginning of the preliminaries section (Section 3), prior to the start of the Calderón-Zygmund iteration scheme. This ensures the iteration closes properly and recovers the optimal linear exponents when p=2. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard techniques

full rationale

The paper derives local Calderón-Zygmund estimates for weak solutions to parabolic p-growth systems with VMO coefficients by combining intrinsic scaling with a CZ-type iteration scheme. The integrability transfer from RHS in L^{μs} (with μ chosen explicitly from p, N, s) to Du ∈ L^s_loc follows directly from these methods applied to the weak solution framework, without any reduction to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The approach recovers the linear p=2 case consistently with parabolic scaling and relies on previously validated techniques external to the present work, rendering the central claim independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard PDE assumptions without introducing new free parameters or postulated entities beyond the VMO condition on coefficients.

axioms (2)
  • standard math Weak solutions exist and satisfy the natural energy estimates for parabolic p-Laplacian systems
    Invoked when stating the class of solutions to which the estimates apply.
  • domain assumption Coefficients are VMO
    Required for the Calderón-Zygmund iteration to close; stated as part of the setup.

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