A deterministic particle method for the porous media equation

Amina Amassad; Datong Zhou

REVIEW 4 major objections 4 minor 1 cited by

A deterministic particle method for the porous media equation

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper proves a quantitative convergence rate for a deterministic particle method solving the porous media equation.

desk verdict A genuine first quantitative W2 rate for the diffusion-velocity particle method, built on a plausible commutator estimate; two regularity gaps in the written proof should be fixed before the theorem is fully rigorous. read the letter →

arxiv 2501.18745 v1 pith:DAXM5SJ3 submitted 2025-01-30 math.AP cs.NAmath-phmath.MPmath.NA

Amina Amassad , Datong Zhou This is my paper · ORCID

classification math.APcs.NAmath-phmath.MPmath.NA MSC 35K6565M7549Q22

keywords porousmediaequationdiffusion-velocityparticlemethodWassersteindistancecommutatorestimateaggregationgradientflowconvergenceratedeterministic

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 establishes the first quantitative convergence rate for a diffusion-velocity particle method applied to the porous media equation. It proves that the weak solution of the smoothed transport equation, obtained by convolving the density with a kernel $R_\varepsilon$, stays within Wasserstein-2 distance $O(\varepsilon^r)$ of the true weak solution, where $r<\frac{1}{d(4k+2)}$ and $k$ is a decay exponent of the kernel's Fourier transform. The argument compares both solutions to an intermediate scale $R_\eta \star u_\varepsilon$ and controls the difference through a new commutator estimate for the optimal-transport velocity field. A separate one-dimensional result gives the better rate $\varepsilon^{1/2}$ under a convexity condition on the kernel.

What carries the argument

The engine is a commutator estimate for the Wasserstein transport map. For two densities, Lemma 2.5 produces action-minimizing velocity fields $v_0,v_1$ with uniform $L^\infty \cap BV$ bounds, and rearrangement of the $W_2$ derivative yields one term governed by displacement convexity of the functional $E(\mu)=\int \mu^2/2\,dx$ plus three commutators $C^{(1)},C^{(2)},C^{(3)}$. These are bounded using the kernel decomposition $R_\varepsilon=R_\varepsilon^{1/2}\star R_\varepsilon^{1/2}$, the intermediate-scale kernel $L_{\varepsilon,\eta}$ defined by $\widehat{L}_{\varepsilon,\eta}=\widehat{R}_\varepsilon/\widehat{R}_\eta$, the first-moment bound $\int|x||L_{\varepsilon,\eta}(x)|dx\le C\varepsilon$, and the energy estimates of Lemma 3.1.

What would settle it

Compute $L_{\varepsilon,\eta}$ for an admissible kernel with polynomial Fourier decay, such as $\widehat{R}(\xi)\sim|\xi|^{-k}$ at infinity, and measure $\|L_{\varepsilon,\eta}\|_{L^1}$ and $\int|y|^{1/p}|L_{\varepsilon,\eta}(y)|dy$ as $\eta/\varepsilon\to0$; if the $L^1$ norm grows without bound or the fractional moment grows faster than $\varepsilon^{1/p}$, the $C^{(3)}$ estimate in Proposition 3.4 collapses. A direct numerical check of the claimed rate in dimensions $d=2$ or $3$ for several admissible kernels would also settle the issue.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: for admissible kernels $R$ satisfying Definition 1.1, the estimate $\sup_{t\in[0,T]} W_2(u,u_\varepsilon) \le C\varepsilon^r$ holds for every $r<\frac{1}{d(4k+2)}$, where $u$ is the weak solution of the porous media equation, $u_\varepsilon$ is the weak solution of the smoothed transport equation, and $C$ is independent of $\varepsilon$. If true, this gives the first quantitative convergence rate for diffusion-velocity particle methods solving diffusive equations. The proof uses the Wasserstein gradient-flow structure: the true solution and a suitably mollified version of the approximate solution are both convected by velocity fields, and the evolution inequality for the $W_2$ distance reduces the problem to three commutator terms controlled by kernel moments and energy estimates.

Load-bearing premise

The proof of the $C^{(3)}$ bound requires the fractional moment estimate $\int|y|^{1/p}|L_{\varepsilon,\eta}(y)|dy\le C\varepsilon^{1/p}$, but Definition 1.1 only states $L_{\varepsilon,\eta}\in L^1$ and the first-moment bound $\int|x||L_{\varepsilon,\eta}(x)|dx\le C\varepsilon$, so the fractional bound is guaranteed only if $\|L_{\varepsilon,\eta}\|_{L^1}$ is uniformly bounded in $\varepsilon,\eta$, a condition neither stated nor proved.

Editorial extensions

If this is right

For any admissible kernel, the diffusion-velocity particle method converges to the porous media solution with an explicit algebraic rate in the kernel width $\varepsilon$.
The Wasserstein-1 error is also controlled, and Corollary 1.3 transfers the rate to an $L^2([0,T]\times\mathbb{T}^d)$ estimate for $u-R_\varepsilon^{1/2}\star u_\varepsilon$.
In one spatial dimension, under the kernel convexity condition, the rate improves to $\varepsilon^{1/2}$.
Because the rate becomes worse as the kernel gets smoother (larger $k$), the result suggests that less regular kernels are preferable for practical particle approximations.
Combined with the known many-particle limit for fixed $\varepsilon$, the result indicates how to choose $\varepsilon\sim N^{-\alpha}$ to balance particle number and smoothing error.

Reading between the lines

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

The missing uniform $L^1$ bound on $L_{\varepsilon,\eta}$ may be provable for natural kernels with algebraically decaying Fourier transforms; if so, the $C^{(3)}$ commutator estimate closes unconditionally.
The same Wasserstein commutator strategy likely transfers to other diffusion equations with displacement-convex energies, such as the heat equation or porous media equations with general exponents $m\neq 2$, possibly with modified rates.
The predicted tradeoff between kernel smoothness and convergence speed could be tested numerically by comparing kernels with different Fourier decay exponents and measuring the empirical $W_2$ error decay.
A direct numerical check of the rate for $d=2,3$ with a few admissible kernels would confirm whether the constant $C$ is genuinely independent of $\varepsilon$ in practice.

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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

Worth knowing: this is the first quantitative Wasserstein-2 rate for the diffusion-velocity particle method applied to the porous medium equation, and the main mechanism—the commutator estimate on the Wasserstein transport map—is a real idea. The paper deserves a serious referee, but the written proof has two gaps that should be closed before I'd trust the theorem exactly as stated.

The setup is clean. They view (1.1) as the gradient flow of ∫u²/2, replace ∇u by ∇Rε*u, and compare u and uε through the W2 metric. The commutator estimate splits the error into three pieces and uses an intermediate scale Rη to convert the insufficient regularity of uε into L2 bounds on Rε^{1/2}*uε. The resulting rate r < 1/[d(4k+2)] is explicit, and the fact that it degrades as the kernel becomes smoother matches intuition. The one-dimensional theorem with rate ε^{1/2} under a convexity condition is a nice bonus. The energy estimates in Lemma 3.1 are standard but correctly handle the convolution structure. The citations to [8,9,10,18] are appropriate, and the paper does not assume the target result.

Now the soft spots, in order of seriousness. The stress-test note is right: the C^(3) bound in Section 3.3 needs ∫|y|^{1/p}|L_{ε,η}(y)|dy ≤ Cε^{1/p}, but Definition 1.1(v) only gives the first absolute moment and leaves the L1 norm of L_{ε,η} uncontrolled. Hölder gives the fractional moment only if ||L_{ε,η}||_1 is uniformly bounded in ε and η, which is neither stated nor proved. The display also drops the absolute value around L in the integrand; since L can be signed, that matters. This is a genuine gap in the proof of Proposition 3.4, but it looks fixable: add a uniform L1 condition to the admissibility definition and verify it for concrete kernels satisfying the Fourier bounds in iii.

Second, Lemma 2.4 asserts a universal L∞∩BV bound on optimal transport velocities with no proof and no stated dependence on the measures. On the torus this may be true, but as written it is doing a lot of work. They should either prove it or cite a precise statement.

Third, Proposition 2.3 is misstated—it should be a W^{1,∞} bound, not L∞—and the proof of Theorem 1.4 relies on that missing regularity. Minor, but worth fixing.

Also keep in mind what the theorem does and does not cover: the rate is for the smoothed PDE (1.2) to (1.1), not for the finite-particle system with N particles. The authors say this explicitly, so I don't count it as a flaw, just a boundary condition.

Overall: the central claim is likely correct, and the gaps are patchable rather than fatal. I'd send it to peer review, and I'd cite it once the final version appears. For a reading group, yes—this is exactly the kind of paper worth pulling apart.

Referee Report

4 major / 4 minor

Summary. The paper studies a deterministic particle method for the porous media equation (1.1) on the torus T^d, d=1,2,3. It establishes a convergence rate in the Wasserstein-2 metric between the solution u of (1.1) and the solution ũ = u_ε of the smoothed transport equation (1.2) with kernel R_ε, proving W2(u,ũ) ≲ ε^r for any r < 1/[d(4k+2)], where k is a decay exponent of the kernel. A separate one-dimensional result gives the rate ε^{1/2} under a convexity condition on the kernel. The proof uses the gradient-flow structure of the porous media equation in Wasserstein space and introduces a three-term commutator estimate involving an intermediate scale η = ε^γ. The introduction is transparent that only the continuum smoothed equation is analyzed, not the N-particle discretization.

Significance. If the proof were complete, this would be a valuable contribution: it would give the first quantitative rate for diffusion-velocity/regularized-transport approximations of the porous media equation, and the commutator structure is potentially reusable. The paper does not tune parameters against the target rate; the rate emerges from explicit estimates. However, the proof as written rests on several unproved or under-specified regularity statements. In particular, the uniform BV/L∞ bound for optimal-transport velocities (Lemma 2.4) and the integrability of the deconvolution kernel L_{ε,η} (Definition 1.1(v)) are load-bearing and need to be fixed before the claim is established.

major comments (4)

[2.2 (Lemma 2.4)] Lemma 2.4 claims a uniform bound sup_s (‖v_s‖_{L∞(T^d)} + ‖v_s‖_{BV(T^d)}) ≤ C for the Benamou-Brenier velocity between arbitrary absolutely continuous probability measures, with C independent of the measures. No proof or reference is supplied. This bound is used in every commutator bound in Section 3.3 (through v_1) and in the proof of Theorem 1.4. For measures with densities that are only L∞ and not bounded below, the optimal transport map need not be Lipschitz, and the velocity can have arbitrarily large BV norm; hence the claimed uniformity is not a standard consequence of being on a compact manifold. The authors need to either prove the lemma under the specific L∞-density assumptions or add a correct hypothesis and track the resulting dependence.
[3.3 and Definition 1.1(v)] The bound of C^(3) in Section 3.3 requires the fractional-moment estimate ∫ |y|^{1/p}|L_{ε,η}(y)| dy ≤ C ε^{1/p}. By Hölder's inequality, this estimate also requires a uniform bound on ‖L_{ε,η}‖_{L1} in ε,η, which is not part of Definition 1.1(v); the integral of L_{ε,η} equals 1, but L_{ε,η} is generally signed, so this does not control the L1 norm. Moreover, for kernels satisfying the Fourier bounds in Definition 1.1(iii), the ratio R̂(εξ)/R̂(ηξ) typically does not go to 0 as |ξ|→∞ (for a power-law tail it tends to a positive constant), so by the Riemann-Lebesgue lemma L_{ε,η} cannot be an L1 function at all; the deconvolution kernel should be treated as a finite signed measure and a uniform total-variation bound added to the definition. Until this is fixed, Proposition 3.4 and Theorem 1.2 are not proved.
[3.3 (Lemma 3.2)] The proof of Lemma 3.2 for f ∈ H^{-1} is incomplete. The sentence 'WLOG assume f = ∂_{x1} φ for some φ ∈ L2' does not cover a general H^{-1} element, and the subsequent approximation argument is not written out; the limit passage in the displayed inequalities is not justified. Since Lemma 3.2 is applied to f = ∇ũ ∈ H^{-1} in the bounds of C^(2) and C^(3), a complete proof is needed.
[2.1 (Proposition 2.3)] Proposition 2.3 states only that u ∈ L∞([0,T], L∞(T^d)), which is already contained in Proposition 2.1 and is not the regularity used in the proof of Theorem 1.4. The proof of Theorem 1.4 needs u ∈ L∞([0,T], W^{1,∞}(T)) to justify the estimate |C_{u,ũ}| ≤ C ε. The statement should be corrected or the proof amended.

minor comments (4)

[3.3 (C^(2), C^(3))] In the displayed estimates for C^(2) and C^(3), the integrals involving L_{ε,η}(y) should have absolute values around L_{ε,η}; the text writes e.g. '∫ |y| L_{ε,η}(y) dy' without the absolute value on L.
[3.3 (Proposition 3.4)] In the proof of Proposition 3.4, the summary term omits the C^(2) contribution ε(ε/η)^{2k}; it is dominated by ε^{1/p}(ε/η)^{2k} for the chosen γ, but the omission should be explained.
[Abstract] The abstract's phrase 'first quantitative rate for diffusion-velocity particle methods' is stronger than the statements in the paper, since the theorems concern the smoothed transport equation (1.2) and not the N-particle ODE system (1.3). The introduction is transparent about this, but the abstract should be qualified.
[Throughout] There are several LaTeX/OCR artifacts in the displayed text, e.g., '/BD' in the Fourier integrals in Lemma 3.2 and 'P ARTICLE' in the title; these should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the convergence rate is derived from explicit kernel and energy assumptions via commutator estimates, with no fitted parameters and no load-bearing self-citation.

full rationale

Theorem 1.2 is not obtained by assuming the target rate. The admissible-kernel Definition 1.1 states a fixed set of hypotheses (nonnegativity, normalization, Fourier decay, gradient domination, and a first-moment bound on the deconvolution kernel L_{ε,η}) before the main result, and Proposition 3.4 derives the W2^2 bound from those hypotheses together with Lemmas 2.4–2.6, 3.1, 3.2, and 3.3. The rate appears only after optimizing the intermediate scale η = ε^γ; no constant is tuned to match the displayed power, and no data are fitted. The citations to prior work [8,9,10,18] supply background, existence theory, and the known N→∞ limit for fixed ε; they are not used to assert the ε-rate and are not self-citations of Amassad and Zhou. The skeptical concern is a genuine proof gap but not a circular one: in the bound of C^(3) in Section 3.3 the paper passes from the first-moment condition (1.9) to the fractional-moment estimate ∫ |y|^{1/p}|L_{ε,η}(y)| dy ≤ C ε^{1/p}, which requires a uniform L1 bound on L_{ε,η} that Definition 1.1(v) does not state, and it writes L_{ε,η} without absolute values despite the kernel being possibly signed. This is an omitted or insufficiently supported estimate inside the proof, not an input defined in terms of the output; the claimed rate is not equivalent by construction to any fitted parameter or self-citation. Therefore the circularity score is 0, with correctness risk assessed separately.

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

The proof rests on standard optimal transport and gradient flow theory, plus a bespoke class of admissible kernels. The main nonstandard input is the uniform regularity of optimal transport velocities and the kernel deconvolution estimates, including an unstated uniform L1 bound on L_{ε,η}.

assumptions (5)

domain assumption The Benamou-Brenier optimal transport velocity v_s between any two probability measures on T^d, with at least one absolutely continuous, satisfies a uniform L∞∩BV bound (Lemma 2.4).
Stated in Lemma 2.4 and used throughout Sections 2.3 and 3.3; no proof or precise reference is given, and it is a strong global regularity assertion for optimal transport geodesics.
standard math The functional E(µ)=∫µ²/2 is displacement convex in W2, so the W2 distance between two solutions of the porous media equation is non-increasing (Lemmas 2.5 and 2.6).
Standard Otto calculus result, cited to [4,19].
domain assumption Admissible kernels in Definition 1.1 exist and satisfy the Fourier bounds (iii), the gradient domination (iv), and the first-moment bound (v).
The theorem is conditional on this kernel class; no explicit example is constructed, and existence for arbitrary k is only sketched via a necessary condition.
ad hoc to paper The deconvolution kernel L_{ε,η} has L1 norm uniformly bounded independent of ε and η.
Not stated in the paper; required to pass from ∫|x||L_{ε,η}|dx≤Cε to ∫|y|^{1/p}|L_{ε,η}|dy≤Cε^{1/p} by Hölder interpolation in the C^(3) estimate.
standard math Well-posedness of (1.1) and (1.2) (Propositions 2.1 and 2.2) and one-dimensional W^{1,∞} regularity of solutions to (1.1) (Proposition 2.3).
Cited to [5,20]; the statement of Proposition 2.3 appears to have a typo, saying L∞ instead of W^{1,∞}.

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Pith. "Pith review of A deterministic particle method for the porous media equation." pith.science (2026). https://pith.science/paper/DAXM5SJ3

@misc{pith2026250118745,
  author       = {Pith},
  title        = {Pith review of: A deterministic particle method for the porous media equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DAXM5SJ3}},
  note         = {Machine review of arXiv:2501.18745}
}

read the original abstract

This paper deals with the deterministic particle method for the equation of porous media (with p = 2). We establish a convergence rate in the Wasserstein-2 distance between the approximate solution of the associated nonlinear transport equation and the solution of the original one. This seems to be the first quantitative rate for diffusion-velocity particle methods solving diffusive equations and is achieved using a novel commutator estimate for the Wasserstein transport map.

Discussion (0). Continue with ORCID to comment.

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Cited by 1 Pith paper

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

Rate of Convergence for a Nonlocal-to-local Limit in One Dimension
math.AP 2025-05 conditional novelty 5.0 of 10

In one dimension, the 2-Wasserstein distance between the nonlocal and local porous medium solutions is bounded by a constant times the square root of the smoothing parameter.

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Works this paper leans on

21 extracted references · 20 canonical work pages · cited by 1 Pith paper

[1]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar ´e, Gradient ﬂows: in metric spaces and in the space of probability measures , Springer Science & Business Media, 2005

work page 2005
[2]

Aronson and J

D. Aronson and J. V azquez , The porous medium equation as a ﬁnite-speed approximation to a hamilton-jacobi equation , Annales de l’Institut Henri Poincar´ e C, 4 (1987), pp. 203– 230

work page 1987
[3]

D. G. Aronson , Regularity propeties of ﬂows through porous media , SIAM Journal on Ap- plied Mathematics, 17 (1969), pp. 461–467

work page 1969
[4]

Benamou and Y

J.-D. Benamou and Y. Brenier , A computational ﬂuid mechanics solution to the monge- kantorovich mass transfer problem , Numerische Mathematik, 84 (2000), pp. 375–393

work page 2000
[5]

B ´enilan, M

P. B ´enilan, M. G. Crandall, and M. Pierre , Solutions of the porous medium equation in rN under optimal conditions on initial values , Indiana University mathematics journal, 33 (1984), pp. 51–87

work page 1984
[6]

Burger and A

M. Burger and A. Esposito , Porous medium equation and cross-diﬀusion systems as limit of nonlocal interaction , Nonlinear Analysis, 235 (2023), p. 113347

work page 2023
[7]

L. A. Caffarelli, J. L. V ´azquez, and N. I. Wolanski , Lipschitz continuity of solutions and interfaces of the n–dimensional porous medium equation , Indiana University mathematics journal, 36 (1987), pp. 373–401

work page 1987
[8]

J. A. Carrillo, K. Craig, and F. S. Patacchini , A blob method for diﬀusion , Calculus of Variations and Partial Diﬀerential Equations, 58 (2019), p p. 1–53

work page 2019

Show all 21 references

[9]

J. A. Carrillo, A. Esposito, J. Skrzeczkowski, and J. S.-H. Wu , Nonlocal particle ap- proximation for linear and fast diﬀusion equations , arXiv preprint arXiv:2408.02345, (2024)

2024 arXiv
[10]

J. A. Carrillo, A. Esposito, and J. S.-H. Wu , Nonlocal approximation of nonlinear diﬀu- sion equations , Calculus of Variations and Partial Diﬀerential Equations , 63 (2024), p. 100

2024
[11]

J. A. Carrillo, S. Jin, and Y. Tang , Random batch particle methods for the homogeneous landau equation , Communications in Computational Physics, 31 (2021)

2021
[12]

J. A. Carrillo, R. J. McCann, and C. Villani , Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass t ransportation estimates, Revista Matematica Iberoamericana, 19 (2003), pp. 971–1018

2003
[13]

Chertock, A practical guide to deterministic particle methods , in Handbook of numerical analysis, vol

A. Chertock, A practical guide to deterministic particle methods , in Handbook of numerical analysis, vol. 18, Elsevier, 2017, pp. 177–202

2017
[14]

Degond and F.-J

P. Degond and F.-J. Mustieles , A deterministic approximation of diﬀusion equations using particles, SIAM Journal on Scientiﬁc and Statistical Computing, 11 (1 990), pp. 293–310
[15]

Di Francesco, A

M. Di Francesco, A. Esposito, and M. Schmidtchen , Many-particle limit for a system of interaction equations driven by newtonian potentials , Calculus of Variations and Partial Diﬀerential Equations, 60 (2021), p. 68

2021
[16]

Lacombe, Analyse d’une ´ equation de vitesse de diﬀusion, Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics, 329 (1999), pp

G. Lacombe, Analyse d’une ´ equation de vitesse de diﬀusion, Comptes Rendus de l’Acad´ emie des Sciences-Series I-Mathematics, 329 (1999), pp. 383–38 6. A DETERMINISTIC PARTICLE METHOD FOR THE POROUS MEDIA EQUATI ON 17

1999
[17]

Lacombe and S

G. Lacombe and S. Mas-Gallic , Presentation and analysis of a diﬀusion-velocity method , in ESAIM: Proceedings, vol. 7, EDP Sciences, 1999, pp. 225–2 33

1999
[18]

Lions and S

P.-L. Lions and S. Mas-Gallic , Une m´ ethode particulaire d´ eterministe pour des ´ equations diﬀusives non lin´ eaires, Comptes Rendus de l’Acad´ emie des Sciences-Series I-Math ematics, 332 (2001), pp. 369–376

2001
[19]

Otto , The geometry of dissipative evolution equations: the porou s medium equation , Comm

F. Otto , The geometry of dissipative evolution equations: the porou s medium equation , Comm. Partial Diﬀerential Equations, 26 (2001), pp. 101–17 4

2001
[20]

J. L. V ´azquez, The porous medium equation: mathematical theory , Oxford University Press, 2007

2007
[21]

Villani , Optimal Transport: Old and New , vol

C. Villani , Optimal Transport: Old and New , vol. 338, Springer, 2009

2009

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar ´e, Gradient ﬂows: in metric spaces and in the space of probability measures , Springer Science & Business Media, 2005

work page 2005

[2] [2]

Aronson and J

D. Aronson and J. V azquez , The porous medium equation as a ﬁnite-speed approximation to a hamilton-jacobi equation , Annales de l’Institut Henri Poincar´ e C, 4 (1987), pp. 203– 230

work page 1987

[3] [3]

D. G. Aronson , Regularity propeties of ﬂows through porous media , SIAM Journal on Ap- plied Mathematics, 17 (1969), pp. 461–467

work page 1969

[4] [4]

Benamou and Y

J.-D. Benamou and Y. Brenier , A computational ﬂuid mechanics solution to the monge- kantorovich mass transfer problem , Numerische Mathematik, 84 (2000), pp. 375–393

work page 2000

[5] [5]

B ´enilan, M

P. B ´enilan, M. G. Crandall, and M. Pierre , Solutions of the porous medium equation in rN under optimal conditions on initial values , Indiana University mathematics journal, 33 (1984), pp. 51–87

work page 1984

[6] [6]

Burger and A

M. Burger and A. Esposito , Porous medium equation and cross-diﬀusion systems as limit of nonlocal interaction , Nonlinear Analysis, 235 (2023), p. 113347

work page 2023

[7] [7]

L. A. Caffarelli, J. L. V ´azquez, and N. I. Wolanski , Lipschitz continuity of solutions and interfaces of the n–dimensional porous medium equation , Indiana University mathematics journal, 36 (1987), pp. 373–401

work page 1987

[8] [8]

J. A. Carrillo, K. Craig, and F. S. Patacchini , A blob method for diﬀusion , Calculus of Variations and Partial Diﬀerential Equations, 58 (2019), p p. 1–53

work page 2019