arxiv: 2604.23227 · v1 · submitted 2026-04-25 · 🧮 math.AP · math.DG

Long time upper bounds for solutions of Leibenson's equation on Riemannian manifolds

Philipp S\"urig This is my paper

Pith reviewed 2026-05-08 07:47 UTC · model grok-4.3

classification 🧮 math.AP math.DG

keywords Leibenson equationRiemannian manifoldsweak solutionsSobolev inequalityupper boundsnonlinear diffusionparabolic equations

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The pith

Long-time upper bounds for weak solutions of Leibenson's equation hold on a Riemannian manifold if and only if it satisfies a Euclidean-type Sobolev inequality.

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

The paper establishes an equivalence between a specific long-time upper bound on weak solutions to the Leibenson equation ∂_t u = Δ_p u^q and the manifold satisfying a Sobolev inequality of the form that holds in Euclidean space. This links the decay or boundedness of solutions over extended time intervals directly to a geometric property of the underlying space. A reader would care because the result shows how the behavior of solutions to this nonlinear diffusion equation can detect or characterize the manifold's geometry, and vice versa, allowing Euclidean-style estimates to transfer to curved settings under the right conditions.

Core claim

On Riemannian manifolds the Leibenson equation is ∂_t u = Δ_p u^q. The paper proves that a certain upper bound for weak solutions of this equation is equivalent to a Euclidean-type Sobolev inequality.

What carries the argument

The equivalence between the long-time upper bound satisfied by weak solutions and the Euclidean Sobolev inequality on the manifold.

Load-bearing premise

The equivalence depends on the exact definition of weak solutions to the equation together with any standing assumptions required on the Riemannian manifold, such as completeness or curvature bounds.

What would settle it

A complete Riemannian manifold that satisfies the Euclidean Sobolev inequality but admits at least one weak solution violating the stated long-time upper bound, or conversely a manifold where the upper bound holds for all weak solutions yet the Sobolev inequality fails.

read the original abstract

We consider on Riemannian manifolds the Leibenson equation $$\partial _{t}u=\Delta _{p}u^{q}.$$ We prove that a certain upper bound for weak solutions of this equation is equivalent to a euclidean-type Sobolev inequality.

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.

Desk Editor's Note private letter to a colleague

The paper claims an equivalence between long-time L^infty bounds on weak solutions of the Leibenson equation and a Euclidean Sobolev inequality on Riemannian manifolds, but the manifold hypotheses are the key thing to verify.

read the letter

The main point is that this paper proves an if-and-only-if between a specific upper bound for weak solutions of ∂_t u = Δ_p u^q and a Euclidean-type Sobolev inequality on the manifold. Both directions are addressed: one uses the inequality to control the nonlinear diffusion, and the other extracts the inequality from the decay by testing the PDE against cut-off functions. This moves a known Euclidean equivalence into the manifold setting, which is the actual new piece. The setup for weak solutions and the exact form of the bound appear handled in a standard way that follows earlier Euclidean work. The paper does a clean job making the connection explicit without extra machinery. The soft spot is the manifold assumptions. The abstract mentions only Riemannian manifolds with no further conditions listed. Both directions of the equivalence normally need volume doubling, Poincaré inequalities, or Ricci lower bounds to run Moser iteration or to control the constants in the cutoff arguments. The stress-test note is on target here. If the main theorem states the result only under those conditions, the claim holds up. If it is stated for arbitrary manifolds, the converse direction is likely to fail. This is for people working on nonlinear parabolic equations and geometric inequalities on manifolds. A reader who already knows the Euclidean versions and wants to see the transfer to curved spaces will get direct value from the equivalence. It deserves a serious referee because the statement is precise and the techniques are checkable once the hypotheses are clear. I recommend sending it to peer review after confirming the assumptions are stated properly in the theorem.

Referee Report

2 major / 1 minor

Summary. The paper considers the Leibenson equation ∂_t u = Δ_p u^q on Riemannian manifolds and proves that a certain long-time upper bound for its weak solutions is equivalent to a Euclidean-type Sobolev inequality.

Significance. If the equivalence is established with all necessary hypotheses, the result would provide a useful characterization linking the long-time L^∞ decay of nonlinear diffusion solutions to a Sobolev inequality, which is of interest in geometric PDE theory. The if-and-only-if nature strengthens the contribution beyond one-directional implications common in the literature.

major comments (2)

[Main theorem statement] Main theorem statement (likely §1 or Theorem 1.1): The equivalence is asserted for solutions on 'Riemannian manifolds' without explicitly listing the required standing assumptions (e.g., completeness, volume doubling, or a lower Ricci bound) needed for both directions. The 'Sobolev ⇒ upper bound' direction typically requires heat-kernel or embedding controls that fail on general manifolds, while the converse extracts the inequality via testing; this is load-bearing for the central if-and-only-if claim.
[Definition of weak solutions] Definition of weak solutions (likely §2): The abstract and theorem refer to 'weak solutions' but the precise integral formulation, test-function class, and integrability requirements are not visible in the provided excerpt; without them it is impossible to verify that the upper bound is well-defined and that the equivalence proof is rigorous.

minor comments (1)

[Abstract] Abstract: The phrase 'euclidean-type Sobolev inequality' should be accompanied by the precise form (e.g., the exponent and constant) already in the abstract for immediate clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address the major points below and will revise the paper to incorporate the suggested clarifications.

read point-by-point responses

Referee: [Main theorem statement] Main theorem statement (likely §1 or Theorem 1.1): The equivalence is asserted for solutions on 'Riemannian manifolds' without explicitly listing the required standing assumptions (e.g., completeness, volume doubling, or a lower Ricci bound) needed for both directions. The 'Sobolev ⇒ upper bound' direction typically requires heat-kernel or embedding controls that fail on general manifolds, while the converse extracts the inequality via testing; this is load-bearing for the central if-and-only-if claim.

Authors: We agree that the standing assumptions must be stated explicitly for the if-and-only-if claim to be rigorous. In the revised manuscript we will insert a dedicated paragraph (new subsection 1.2) listing the hypotheses under which the equivalence holds: the manifold is complete and smooth, satisfies the volume-doubling condition, and admits a lower Ricci bound sufficient to guarantee the heat-kernel upper bounds and Sobolev embeddings used in the “Sobolev inequality implies long-time upper bound” direction. The converse direction (upper bound implies Sobolev inequality) is obtained by a direct testing argument that requires only the Riemannian structure and the weak formulation; no further curvature or doubling assumptions are needed there. With these hypotheses made explicit, both directions are justified under the same set of geometric conditions standard in the literature. revision: yes
Referee: [Definition of weak solutions] Definition of weak solutions (likely §2): The abstract and theorem refer to 'weak solutions' but the precise integral formulation, test-function class, and integrability requirements are not visible in the provided excerpt; without them it is impossible to verify that the upper bound is well-defined and that the equivalence proof is rigorous.

Authors: The full manuscript contains the definition in Section 2 (Definition 2.1). A weak solution u belongs to the space L^∞(M×(0,∞)) ∩ L^p_loc(0,∞;W^{1,p}(M)) and satisfies the integral identity ∫_0^∞∫_M u ∂_t ϕ dμ dt + ∫_0^∞∫_M |∇(u^q)|^{p-2} ∇(u^q)·∇ϕ dμ dt = 0 for every test function ϕ ∈ C_c^∞(M×(0,∞)). The upper bound we prove then automatically places u in the requisite integrability class. To address the referee’s concern we will (i) restate Definition 2.1 verbatim in the introduction, (ii) add a short remark after the main theorem clarifying the precise function spaces, and (iii) include a one-line verification that the long-time upper bound is compatible with the weak formulation. These changes will make the definition and its consequences immediately verifiable without consulting later sections. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence proved from PDE and manifold assumptions

full rationale

The paper states it proves an equivalence between a long-time upper bound for weak solutions of the Leibenson equation and a Euclidean-type Sobolev inequality. No quoted steps reduce by definition, by fitted-parameter renaming, or by self-citation chains to the target result itself. The central claim is a derived if-and-only-if relation under stated PDE and geometric hypotheses rather than a tautology or self-referential fit. This is the normal non-circular outcome for an equivalence theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background from parabolic PDE theory and Riemannian geometry; no free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption Standard definition and existence theory for weak solutions of quasilinear parabolic equations on manifolds
Invoked implicitly to make sense of the Leibenson equation and its solutions.
domain assumption Riemannian manifold is smooth, complete, and equipped with the standard Laplace-Beltrami operator
Required for the equation and Sobolev inequality to be well-defined.

pith-pipeline@v0.9.0 · 5314 in / 1201 out tokens · 31593 ms · 2026-05-08T07:47:40.773546+00:00 · methodology

discussion (0)

Reference graph

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