Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below

Carlo Orrieri; Matteo Muratori; Nicol\`o De Ponti

REVIEW 1 major objections 6 minor 41 references

Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that on manifolds with Ricci curvature bounded below by $-K$, the Wasserstein distance between two porous medium-type solutions stays controlled by an explicit exponential factor, and that this estimate is sharp in…

desk verdict A genuinely new Wasserstein stability estimate under negative Ricci bounds, but a sign typo in Lemma 5.5 currently breaks the main proof—fixable, and the paper deserves peer review. read the letter →

arxiv 1908.03147 v2 pith:IBORGARO submitted 2019-08-08 math.AP math.FAmath.MG

Nicolò De Ponti , Matteo Muratori , Carlo Orrieri This is my paper · ORCID

classification math.APmath.FAmath.MG MSC 35K6549Q2253C2158J3535B35

keywords porousmediumequationWassersteindistanceRiccicurvatureboundsstabilityestimateshyperbolicspacesmoothingeffectnonlineardiffusionoptimaltransport

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

On a complete Riemannian manifold whose Ricci curvature is bounded below by $-K$, the paper proves a quantitative 2-Wasserstein stability estimate for porous medium-type equations: the distance between two solutions at time $t$ is controlled by the distance between their initial data times an explicit exponential factor. The exponent is $K c_1 C_m[(tM^{m-1})^{2/(2+n(m-1))} \lor tM^{m-1}]$, so short times are governed by a nonlinear power and long times by a linear term. This extends the known contraction property for $K=0$ to arbitrary negative curvature bounds, for $m>1$ and general nonlinearities satisfying the McCann-type condition. A matching lower bound in hyperbolic space shows the short-time power is sharp and rules out a Wasserstein gradient-flow description of the porous medium equation on negatively curved manifolds.

What carries the argument

The load-bearing mechanism is the Hamiltonian functional $E_{\rho(t)}[\varphi(t)] = \int_{M^n} |\nabla \varphi(t)|^2 \rho(t)\, dV$, evaluated on a solution $\rho$ and on a solution $\varphi$ of the backward adjoint equation $\partial_t \varphi = -P'_\varepsilon(\rho)\Delta\varphi$. Along the flow it obeys the differential inequality $\frac12 \frac{d}{dt} E_{\rho(t)}[\varphi(t)] \ge -K \int \Gamma(\varphi(t)) P_\varepsilon(\rho(t))\, dV$; the paper closes this inequality using the quantitative $L^1$–$L^\infty$ smoothing estimate from Proposition 4.3, which bounds the sup norm by a power of $t^{-1}$ times the initial mass. The lower-bound proof uses a different geometric ingredient: the small-time comparison of the hyperbolic-space Barenblatt solution with the Euclidean Barenblatt profile, combined with a distance expansion for nearby spheres.

What would settle it

Compute, for a concrete case such as $n=3$, $m=2$, $M=1$ in $H^3_K$, the $W_2$ distance between solutions starting from two Dirac masses separated by a small $\delta$, at times spanning several scales. The paper's lower bound predicts logarithmic growth with exponent $2/(2+n(m-1))=2/5$ in $t$; observing a different exponent, or a prefactor not proportional to $K$ as $t\downarrow 0$, would refute the sharpness claim. For the upper bound, a direct check is to verify that the ratio $W_2(\rho(t),\hat\rho(t))/W_2(\mu_0,\hat\mu_0)$ never exceeds the stated exponential.

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Extended reading notes

Core claim

The paper's central claim is that solutions of $\partial_t \rho = \Delta P(\rho)$ started from measures of mass $M$ on a complete manifold with $\mathrm{Ric} \ge -K$ and a Sobolev inequality satisfy the stability estimate (2.6), with the explicit exponential factor above. The same theorem gives the quantitative smoothing bound (2.5), $\|\rho(t)\|_{L^\infty} \le C(t^{-n/(2+n(m-1))} M^{2/(2+n(m-1))} + M)$. The optimality theorem states that in hyperbolic space $H^n_K$, for two close Dirac initial data, the distance ratio is at least $1 + K\kappa(tM^{m-1})^{2/(2+n(m-1))}$ for small times; hence the exponent cannot be improved, and the flow is not a gradient flow of a $\lambda$-convex functional in the Wasserstein metric when $K>0$.

Load-bearing premise

The sharpness result depends on the external fact that, in hyperbolic space, the porous-medium solution from a point mass stays below the Euclidean Barenblatt profile with the Euclidean time scale; the main stability upper bound does not rely on this comparison.

Editorial extensions

If this is right

For the model case $P(\rho)=\rho^m$, the bound gives an explicit quantitative replacement for the contraction that was previously available only when $\mathrm{Ric}\ge 0$.
Letting $m\downarrow 1$ recovers the linear heat-flow estimate $W_2(\rho(t),\hat\rho(t)) \le e^{Kt} W_2(\mu_0,\hat\mu_0)$.
The hyperbolic-space lower bound implies the porous medium equation is not, for $K>0$, a gradient flow of a $\lambda$-convex energy with respect to $W_2$ in the evolutionary variational inequality sense.
Under a Euclidean Sobolev inequality in place of the lower-dimensional one, the long-time linear term disappears and the exponent reduces to just the power $(tM^{m-1})^{2/(2+n(m-1))}$.
The estimates cover a whole class of nonlinearities $P$ satisfying the growth conditions (H4)–(H5), not only the pure power law.

Reading between the lines

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

One extension left implicit: the smoothing-plus-Hamiltonian mechanism suggests the same kind of bound should hold for the fast-diffusion range $m\in(0,1)$ up to the extinction time, but the compact-support and smoothing estimates would need a different treatment.
The optimality theorem is proved only in constant-curvature hyperbolic space; a natural test is whether the same small-time exponent is sharp on every Cartan-Hadamard manifold, where the Barenblatt comparison is also available.
The proof's reliance on PDE smoothing suggests that adapting the estimate to metric-measure spaces would require a quantitative regularization estimate in that setting; the paper singles this out as an obstruction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper proves a quantitative W2 stability bound for porous medium-type equations on complete manifolds with Ric ≥ −K, with an explicit time-dependent exponent, and shows sharpness on hyperbolic space. That is a real advance over the known Ric ≥ 0 contraction results of Sturm and Otto–Westdickenberg, and the combination of L1–L∞ smoothing with the Hamiltonian approach is well suited to the problem.

The paper does a lot well. The smoothing estimate is proved self-containedly via Moser iteration, with careful control of constants as m ↓ 1. The use of the Ambrosio–Mondino–Savaré machinery is appropriate, and the optimality construction in hyperbolic space is clever, turning an Ollivier-type geodesic expansion plus Barenblatt comparison into a matching lower bound. The main theorem is new and the sharpness example is convincing.

The soft spot is exactly where the reader put it: Lemma 5.5. The displayed definition of ψ_{s,t} has the wrong sign in the exponential. With ψ = e^{+A}φ, the equality after (5.15) does not hold; you need ψ = e^{-A}φ for the E_ ho factor to cancel. I checked the algebra and the stress-test note is right. This is almost certainly a sign typo, but it sits in the load-bearing step that converts the smoothed Hamiltonian lower bound into the Wasserstein estimate, so as printed the proof of Theorem 2.4 has a genuine gap. It should be an easy fix.

Two smaller things are worth mentioning. First, the optimality proof relies on the comparison (5.47) between the hyperbolic-space PME solution and the Euclidean Barenblatt profile from [21] and [38]. That is an external result, but a credible one; the upper bound (2.6) does not depend on it. Second, Remark 2.6 states the non-gradient-flow conclusion for “general negatively-curved manifolds” when the proof only covers hyperbolic space. The logic is acceptable—if a universal gradient-flow structure existed it would hold on hyperbolic space—but the phrasing could be tightened.

Overall: this is a solid paper with one correctable typo. I would send it to a serious referee. If the sign in Lemma 5.5 is fixed and the remark is reworded, I would expect it to be accepted.

Referee Report

1 major / 6 minor

Summary. The paper studies the Cauchy problem for porous medium-type equations ∂_t ρ = ΔP(ρ) on complete Riemannian manifolds with Ricci curvature bounded below (Ric ≥ -K) and a Sobolev inequality. The main result (Theorem 2.4) establishes a quantitative 2-Wasserstein stability estimate: for two solutions with initial data of mass M, W_2(ρ(t), ρ̂(t)) ≤ exp{K c_1 C_m[(tM^{m-1})^{2/(2+n(m-1))} ∨ (tM^{m-1})]} W_2(µ_0, µ̂_0). The proof combines the Hamiltonian/Eulerian approach of Ambrosio–Mondino–Savaré with a new quantitative L^1–L^∞ smoothing estimate obtained by Moser iteration, and a compact-support property. Theorem 2.5 shows that, in hyperbolic space, the time exponent in the estimate is sharp for small times, and consequently the PME cannot be a gradient flow of a λ-convex functional in the EVI sense. The paper also provides existence, uniqueness, regularity, moment propagation, and continuity properties of the solutions.

Significance. If the results are correct, they constitute a substantial advance: they provide the first quantitative Wasserstein contraction-type estimate for nonlinear diffusion on manifolds beyond the Ric ≥ 0 regime, with explicit constants and a sharp small-time exponent. The proof is largely self-contained, including the smoothing estimate and the compact-support argument, and the treatment of both noncompact and compact manifolds is careful. The optimality result is a valuable contribution, showing that the PME does not admit a gradient-flow/EVI structure on negatively curved manifolds. The paper also gives credit to previous work and clearly delineates the external inputs (e.g., the hyperbolic-space comparison (5.47)).

major comments (1)

[§5.2, Lemma 5.5, display following (5.15)] The definition of ψ_{s,t} has the wrong sign of the exponential, and this makes the displayed equality in (5.15) algebraically false. Setting A = 2K c_1 C_m[(tM^{m-1})^{2/(2+n(m-1))} ∨ (tM^{m-1}) + ε t/(c_1 C_m)], the bracket on the preceding line is -1/2 e^{-A} E_{ρ^s_0}[φ_s(0)] + ⟨d/ds ρ^s_0, φ_s(0)⟩. To factor out e^A and obtain an expression of the form -1/2 E[ψ] + ⟨·, ψ⟩, one must set ψ_{s,t} = e^{-A} φ_s(0), because E is quadratic in its argument. With the printed definition ψ_{s,t} = e^{A} φ_s(0), the resulting integrand becomes e^A[-1/2 e^{2A}E[φ_s(0)] + e^{A}⟨·,φ_s(0)⟩] = -1/2 e^{3A}E[φ_s(0)] + e^{2A}⟨·,φ_s(0)⟩, which does not equal the original bracket and is not bounded by the claimed Fenchel-dual term. Since this step is essential for passing from the smoothed Hamiltonian lower bound to the Wasserstein estimate (5.14)–(5.20), the proof of (2.6) as written contains a genuine algebraic gap. The error appears to be a sign typo, but it must be corrected and the subsequent estimate rechecked before the proof is sound.

minor comments (6)

[Abstract and title] There are several typos, e.g., 'por ous' and 'W ASSERSTEIN ST ABILITY' in the title; the manuscript would benefit from a careful proofreading pass.
[Theorem 2.4 and Lemma 5.3] The notation 'Cm := Cm−1 2m−2 [2 + n(m − 1)]' is ambiguous; it should be typeset as C^{m-1} 2^{m-2} [2 + n(m-1)] to avoid confusion.
[§5.4, proof of Theorem 2.5] The sharpness result relies crucially on the comparison (5.47) with the Euclidean Barenblatt profile, imported from [21, Remark 2.12] and [38]. Since the lower bound (5.50) depends on this comparison, it would be helpful to state explicitly the conditions under which (5.47) is known to hold and whether the constants D and k depend only on n and m.
[§5.3, compact case] The discussion of the spaces V'_E and D'_E is rather brief; adding precise references to the relevant statements in [4] would make the compact-case argument easier to verify.
[§5.4, Lemma 5.6] The notation 'E := exp_x v^⊥' is a slight abuse; it would be clearer to write 'E := exp_x(v^⊥)' or 'E := {exp_x w : w ∈ v^⊥}'.
[§5.4, proof of Lemma 5.7] In inequality (5.46), the constant '3κ' appears without explanation; it would be clearer to introduce a new constant (e.g., κ') at that point.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Wasserstein stability estimate is derived from an in-paper smoothing estimate and independent Hamiltonian calculus, with no fitted input called a prediction.

full rationale

The main deliverable (2.6) is obtained by a genuine derivation chain: Lemma 5.2 converts the Ricci lower bound into the Hamiltonian differential inequality; Lemma 5.3 integrates this using the L1–L∞ smoothing effect, which is proved in Proposition 4.3 by Moser iteration; Lemma 5.5 then passes from the Hamiltonian lower bound to the Wasserstein estimate through the Fenchel dual and the identity (3.21). None of these steps fits a parameter to the target W2 bound: the exponential prefactor is an explicit function of K, c1, Cm, t and M, obtained from the smoothing constants, and the target Wasserstein quantity appears only at the end as the output, not as an input. The optimality proof (2.7) uses the external comparison (5.47) between the hyperbolic-space PME profile and the Euclidean Barenblatt profile, cited to [21, Remark 2.12] and [38]; although one of the authors is among the authors of [21], that cited bound is an independent published result about the solution profile and does not assume the stability estimate being proved. The paper also proves its own smoothing estimate rather than citing it, and the curvature-to-Hamiltonian step is standard Bakry–Émery calculus. Any algebraic sign issue in the displayed derivation of Lemma 5.5 would be a correctness defect, not a circularity, since the printed claim does not reduce by construction to its input. Overall, no load-bearing step is equivalent to its own inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; all constants are explicit or depend only on n, m, C_S, c0, c1. No new mathematical or physical entities are introduced. The derivation relies on standard or explicitly cited background results listed above; the smoothing estimate is derived in the paper.

assumptions (5)

domain assumption The manifold supports the Sobolev inequality (H2), i.e., ||f||_{L^{2*}} ≤ C_S(||∇f||_2 + ||f||_2), and hence its Gagliardo-Nirenberg form (4.32).
Assumption (H2) in Hypotheses 2.1 is used in Proposition 4.3 to run the Moser iteration for the smoothing estimate. It is satisfied under Ric≥-K plus noncollapse, but is not a consequence of curvature alone.
domain assumption The Euclidean Barenblatt comparison (5.47): the hyperbolic-space PME solution from a Dirac mass is bounded above by the Euclidean Barenblatt profile.
Used in the proof of Theorem 2.5 to derive the lower bound on the density (5.50) and then the Wasserstein lower bound. Cited to [21, Remark 2.12] and [38], not proved in this paper.
standard math The Hamiltonian/Eulerian calculus results of Ambrosio-Mondino-Savaré [4]: identity (3.21) for regular curves, well-posedness of the forward linearized and backward adjoint equations (Theorems 4.9-4.10), and the differentiation formula for the Hamiltonian (Lemma 5.1).
These are quoted from [4] and underlie Lemmas 5.1, 5.4 and 5.5. They are published results, but are the technical engine of the proof.
standard math Ollivier's distance expansion (5.28) for hyperbolic space, together with the double exponential map estimates from [18].
Used in Lemma 5.6 to compute the distance to the totally geodesic hyperplane E, the key geometric input for the sharpness construction.
standard math The equivalence of the Bakry-Émery condition BE(λ,n) with Ric≥λ on smooth Riemannian manifolds (Section 3.1).
Used to translate the curvature bound (H1) into the iterated carré du champ inequality (3.6), which is the starting point of Lemma 5.2.

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Pith. "Pith review of Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below." pith.science (2026). https://pith.science/paper/IBORGARO

@misc{pith2026190803147,
  author       = {Pith},
  title        = {Pith review of: Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IBORGARO}},
  note         = {Machine review of arXiv:1908.03147}
}

abstract

Given a complete, connected Riemannian manifold $ \mathbb{M}^n $ with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance. We produce (sharp) stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm and Otto-Westdickenberg. The strategy of the proof mainly relies on a quantitative $L^1-L^\infty$ smoothing property of the equation considered, combined with the Hamiltonian approach developed by Ambrosio, Mondino and Savar\'e in a metric-measure setting.

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [21]

Grillo, M

G. Grillo, M. Muratori, F. Punzo, The porous medium equation with measure data on negatively c urved Rie- mannian manifolds , J. Eur. Math. Soc. (JEMS) 20 (2018), 2769–2812

work page 2018

[2] [38]

Vázquez, Fundamental solution and long time behavior of the porous me dium equation in hyperbolic space , J

J.L. Vázquez, Fundamental solution and long time behavior of the porous me dium equation in hyperbolic space , J. Math. Pures Appl. 104 (2015), 454–484. 40 NICOLÒ DE PONTI, MATTEO MURATORI, CARLO ORRIERI

work page 2015

[3] [1]

Gradient Flows in Metr ic Spaces and in the Space of Probability Measures

L. Ambrosio, N. Gigli, G. Savaré, “Gradient Flows in Metr ic Spaces and in the Space of Probability Measures”. Second Edition. Lectures in Mathematics ETH Zürich. Birkhä user Verlag, Basel, 2008

work page 2008

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Ambrosio, N

L. Ambrosio, N. Gigli, G. Savaré, Calculus and heat ﬂow in metric measure spaces and applicati ons to spaces with Ricci bounds from below , Invent. Math. 195 (2014), 289–391

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Ambrosio, N

L. Ambrosio, N. Gigli, G. Savaré, Bakry-Émery curvature-dimension condition and Riemannia n Ricci curvature bounds, Ann. Probab. 43 (2015), 339–404

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