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Gradient estimates for Leibenson's equation on Riemannian manifolds

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Pith's one-line read This paper proves explicit Li–Yau type gradient estimates for positive solutions of the doubly nonlinear Leibenson equation on Riemannian manifolds with Ricci curvature bounded below by a negative constant.

desk verdict A genuine advance in Li-Yau estimates for the doubly nonlinear equation, but the Moser iteration as written has a sign gap that needs a positive-part repair before the theorem is proved. read the letter →

arxiv 2506.07221 v1 pith:VSL4SSEV submitted 2025-06-08 math.AP

Philipp Sürig This is my paper
classification math.AP MSC 35K5558J3553C2135B05
keywords Leibensonequationdoublynonlinearparabolicp-LaplaciangradientestimatesLi-YauestimateMoseriterationRiemannianmanifoldRiccicurvature
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves Li–Yau type gradient estimates for positive solutions of the Leibenson (doubly nonlinear) equation ∂tu = Δp u^q on Riemannian manifolds whose Ricci curvature is bounded below by −K. The estimates control the quantity |∇v|^p/v − α ∂t v/v, where v is a power-like transform of u, by explicit expressions in time, the ball radius, the curvature, and the assumed bounds on |∇v|^{p−2}v. Both the slow diffusion case q(p−1) > 1 and the fast diffusion case q(p−1) < 1 are treated, with local estimates in geodesic balls and global estimates obtained by letting the radius tend to infinity. If the theorems are correct, this gives the first such estimates for the general doubly nonlinear equation on noncompact manifolds with a negative Ricci lower bound, extending earlier work that handled closed manifolds or the porous medium case p = 2.

What carries the argument

The argument is carried by a change of variables v = (q(p−1)/δ) $u^{{δ/(p−1)}}$ with δ = q(p−1) − 1 in the slow case, and v = (q(p−1)/D) $u^{{−D/(p−1)}}$ with D = 1 − q(p−1) in the fast case, which puts the equation into a form where an operator F = ∂t − δ/(p−1) v L (or with D) can be applied, with L a p-Laplacian-type elliptic operator. A nonlinear Bochner-type inequality, imported from reference [17], gives a differential inequality for F acting on Fα = |∇v|^p/v − α ∂t v/v. The proof then runs a Moser iteration scheme, defined here as a repeated application of an L^λ mean value inequality and a Caccioppoli-type estimate, using Sobolev and Faber–Krahn inequalities on geodesic balls and an auxiliary function φ(t) that absorbs the curvature term. The lower bound Λmin appears in denominators throughout this iteration, which is why the argument excludes points where ∇v = 0.

What would settle it

Compute, for the explicit Barenblatt self-similar solution of the porous medium equation ∂tu = Δu^q on R^n (so p = 2, Ric = 0), the quantity sup(|∇v|^2/v − α ∂t v/v) and compare it with the C0/t term in (1.8); if the supremum ever exceeds that bound, the global estimate is false, while matching it confirms the sharp 1/t decay in the known case.

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Extended reading notes

Core claim

On its own terms, the central claim is that a Li–Yau type gradient estimate holds for positive smooth solutions of ∂tu = Δp u^q on complete Riemannian manifolds with Ric ≥ −K, K ≥ 0, in both the slow-diffusion regime q(p−1) > 1 (Theorem 1.1) and the fast-diffusion regime 1 > q(p−1) with p − nD > 0 (Theorem 1.3). For the slow case, the estimate bounds sup_B (|∇v|^p/v − α ∂t v/v) by the explicit right-hand side (1.7), and the global analogue (1.8) is obtained by sending R → ∞. For the fast case, the analogous bound controls |∇v|^p/v + α ∂t v/v and yields the global estimate (1.16). The proofs require the two-sided pointwise bound Λmin ≤ |∇v|^{p−2}v ≤ Λmax in the relevant cylinder, and the author explicitly leaves open whether the lower bound can be relaxed to just the upper bound.

Load-bearing premise

The proof needs the uniform lower bound Λmin ≤ |∇v|^{p−2}v on the whole cylinder, because Λmin appears in denominators in the Moser iteration; at points where ∇v = 0 this condition fails, and the author leaves open whether the upper bound Λmax alone would suffice.

Editorial extensions

If this is right

  • On a geodesically complete manifold with Ric ≥ −K, every positive smooth solution satisfying the two-sided bound obeys the global estimate (1.8) in the slow-diffusion case, with the right-hand side depending explicitly on t, K, Λmin, Λmax, and α.
  • Sending R → ∞ removes the dependence on the ball radius and leaves the curvature entering through the term α^2 n K δ^2 Λmax / ((p−1)(α−1)), so the estimates are stable as K → 0.
  • In the fast-diffusion regime with p − nD > 0, the same machinery gives the global bound (1.16), which the paper notes is the first such estimate for the general doubly nonlinear equation on manifolds with Ric ≥ −K.
  • The constants are explicit in p, q, n, and α, and the geometry of the ball enters only through the Sobolev constant, so the estimates are quantitative rather than qualitative.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is whether the two-sided bound can be weakened to the upper bound alone; the author leaves this open, and the proof's only obstruction appears to be the denominator Λmin in the Moser iteration.
  • Li–Yau estimates typically imply parabolic Harnack inequalities; if the same reasoning carries through here, it would give quantitative control of ratios of positive solutions of the doubly nonlinear equation, which the paper does not state explicitly.
  • For p = 2, the estimates should reduce to the known porous-medium results of references [12] and [9]; checking this limit is a consistency test for the constants and would clarify how the general p case improves on the p = 2 machinery.
  • Because all analytic steps are local, a plausible further direction is to run the same argument for doubly nonlinear equations with lower-order terms or on weighted manifolds, though that is not attempted in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proves Li-Yau-type gradient estimates for positive smooth solutions of the doubly nonlinear equation ∂_t u = Δ_p u^q on complete Riemannian manifolds with Ricci curvature bounded below by a non-positive constant. In the slow-diffusion regime q(p-1)>1 and the fast-diffusion regime q(p-1)<1, the author introduces a pressure variable v, assumes the two-sided bound Λmin ≤ |∇v|^{p-2}v ≤ Λmax, and uses a Bochner formula, an auxiliary function φ, a Caccioppoli inequality, and Moser iteration to bound the quantity Fα = |∇v|^p/v ∓ α(∂_t v)/v in geodesic cylinders and then globally as R→∞. The main results are Theorem 1.1 (slow diffusion), Theorem 1.3 (fast diffusion), and their global corollaries.

Significance. If the proof were complete, the paper would provide the first local and global gradient estimates of Li-Yau type for the general Leibenson equation on noncompact manifolds with negative Ricci lower bound, extending earlier porous-medium and p-Laplacian results. The paper is clearly written, compares its constants with known estimates, and treats the fast-diffusion case under a dimension condition p-nD>0. No fitted parameters appear, and the final estimates are explicit. However, the central Moser-iteration argument currently contains a sign/positivity gap, and several displayed constants are algebraically inconsistent, so the main result is not yet established as written.

major comments (3)
  1. [§2.1, Lemmas 2.2 and 2.4, and Theorem 2.9] The proof of (2.23) uses the inequality 2c1fφ ≥ 0, which is valid only when f≥0, and the subsequent iteration in Lemmas 2.4-2.8 forms powers f^{λ-1}, f^{λ/2}, f^λ, and f^{λ+1} for real λ, so the whole Moser iteration requires f to be nonnegative. No such lower bound is proved from the assumptions. The theorem's hypotheses do not supply one: for p=2, q>1, K=0, the pressure variable satisfies v_t = (q-1)v Δv + |∇v|², and the local solution with v(0,x)=1+|x|² has v=1, ∇v=0, and v_t=2n(q-1)>0 at x=0. Since φ≡0 for K=0, f=Fα=-α v_t/v<0 there, while (1.6) holds with suitable Λmin<1<Λmax. Thus the Caccioppoli inequality (2.26), the mean-value inequality (2.35), and Lemmas 2.8-2.9 do not apply to the actual f of the theorem. A standard repair would be to run the iteration on the positive part f_+ and prove the required differential inequality for f_+; as written this is a load-bearing gap.
  2. [§2.1, Eq. (2.25), and Theorem 1.1 / Corollary 1.2] The KΛmax coefficient in the final estimates is computed incorrectly. From cδ=pδ/(p-1), c2=2(p-1)p(α-1)/(nδα²), and φ≤2b/a, the bound (2.25) should read φ(t) ≤ cδKΛmax/c2 = nδ²α²KΛmax/[2(p-1)²(α-1)], not α²nKδ 2Λmax/((p-1)(α-1)). The incorrect value is then propagated into (2.38), (2.39), (1.7), and (1.8), so the displayed explicit constants in the main theorems are not correct as printed. Additionally, the prefactor and exponent in (1.7) are inconsistent with those in (2.38): (1.7) places C0 in the exponent and omits the C1/c1 prefactor, while (2.38) has no C0 in the exponent and includes C1/c1 multiplicatively. The two statements are not equivalent as written.
  3. [§3, Lemma 3.3 and Theorem 3.5] The same positivity gap appears in the fast-diffusion part. In the proof of Lemma 3.3, after choosing the coefficients positive, the inequality is reduced to (3.47) by discarding the cross term involving fφ; this reduction is valid only if fφ≥0 (hence, since φ≥0, if f≥0). The subsequent estimates also use f^{λ}, f^{λ+1}, and f^{λ/2} for real λ. No pointwise lower bound for f is established from the assumptions (1.14), so the proof of Theorem 3.5 has the same unfilled hypothesis as the slow-diffusion argument.
minor comments (4)
  1. [§1, page 3] The word 'summond' should be 'summand'.
  2. [Table of contents] The entry 'F ast diffusion case' contains a spacing typo and should read 'Fast diffusion case'.
  3. [§1, page 2 and Corollary 2.10] The constant C0 in (1.8) is said to be as in Theorem 1.1, while Corollary 2.10 writes C1/c1; the two constants are equal in magnitude but the reader has to compare (2.39) and (1.8) to see this. It would be clearer to use one notation consistently.
  4. [§2.3, Lemma 2.6] The notation B=B(x,r1) and B'=B(x,r2) for the two balls is slightly confusing because B was already used in the introduction for a geodesic ball of radius R; renaming the two balls B1 and B2 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the gradient estimate is derived from a Bochner formula and Moser iteration, not from its own conclusion.

full rationale

The paper's central estimate is derived by direct Moser iteration: define f = F_alpha - phi, derive the differential inequality (2.23) from the cited Bochner formula, integrate against f^{lambda-1} eta^2 to get a Caccioppoli inequality (2.26), then combine a mean-value inequality with an L^lambda bound to obtain Theorem 2.9. None of these steps fits a parameter, renames an input as a conclusion, or invokes a uniqueness theorem from the author's own prior work. The self-citations to [5] are for a parabolic Sobolev/Moser inequality and an elementary iteration lemma, both auxiliary and standard, and the Bochner inequality is cited from the external published work of Wang-Chen [17]. The proof's use of the unproved sign condition f >= 0, e.g. '2 c1 f phi >= 0' in Lemma 2.2 and the real powers f^lambda in the Moser iteration, is a genuine correctness gap but is not a circular reduction: the theorem is not defined in terms of f >= 0, and the gap does not make the conclusion equal to an input. Therefore the paper shows no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new physical or geometric entities. The constants C, C', C0 and C1 are structural dependencies, not empirical fits. The main axioms are the Ricci lower bound, smooth positive solutions, a two-sided nondegeneracy bound, the imported Bochner inequality, and standard Sobolev and Moser inequalities.

assumptions (6)
  • domain assumption Ric_B ≥ -K for some K ≥ 0 on a geodesic ball B
    Used throughout to obtain the nonlinear Bochner inequality (Lemmas 2.1 and 3.1) and the Sobolev constant bound (2.30).
  • domain assumption u is a positive smooth solution of ∂_t u = Δ_p u^q
    The proof works only for smooth positive solutions; regularity of weak solutions is not addressed.
  • domain assumption Uniform ellipticity bound Λmin ≤ |∇v|^{p-2}v ≤ Λmax holds pointwise
    The lower bound controls denominators in Moser iteration and the upper bound is used in Lemmas 2.2 and 3.3; the paper states as an open question whether the lower bound can be removed.
  • standard math Nonlinear Bochner inequality in Lemma 2.1 and Lemma 3.1 from Wang-Chen [17]
    Imported without proof; this inequality is the key differential estimate driving all subsequent bounds.
  • standard math Moser inequality (2.31) and iteration lemma (4.5) from Grigor'yan-Sürig [5]
    Used to derive the mean value inequality and the iteration bound; self-cited but auxiliary and standard in nature.
  • standard math Sobolev inequality on geodesic balls with SB ≤ C e^{Cn√KR} R² / μ(B)^ν
    Standard result of Buser, Grigor'yan and Saloff-Coste, used in Theorems 2.9 and 3.5 to pass from Sobolev constants to explicit R and K dependence.

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Cite this review

Pith. "Pith review of Gradient estimates for Leibenson's equation on Riemannian manifolds." pith.science (2026). https://pith.science/paper/VSL4SSEV

@misc{pith2026250607221,
  author       = {Pith},
  title        = {Pith review of: Gradient estimates for Leibenson's equation on Riemannian manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VSL4SSEV}},
  note         = {Machine review of arXiv:2506.07221}
}
abstract

We consider on Riemannian manifolds solutions of the Leibenson equation \begin{equation*} \partial _{t}u=\Delta _{p}u^{q}. \end{equation*} This equation is also known as doubly nonlinear evolution equation. We prove gradient estimates for positive solutions $u$ under the condition that the Ricci curvature on $M$ is bounded from below by a non-positive constant. We distinguish between the case $q(p-1)>1$ (slow diffusion case) and the case $q(p-1)<1$ (fast diffusion case).

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Existence results for Leibenson's equation on Riemannian manifolds

    math.AP 2026-01 unverdicted novelty 6.0 of 10

    The Cauchy problem for ∂t u = Δp u^q on Riemannian manifolds admits a unique weak solution when p>1, q>0, pq≥1 for any initial data in L1(M) ∩ L∞(M).

Reference graph

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