Homogenization of Time-Discrete Gradient Flows

Li Wang; Yuan Gao; Yunhong Deng

REVIEW 3 major objections 4 minor 41 references

Homogenization of Time-Discrete Gradient Flows

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For Fokker–Planck equations with rapidly oscillating diffusion, the choice of metric in a time-discrete gradient flow determines whether homogenization and the time-step limit commute: the weighted L2 metric recovers the effective dynamics

desk verdict The weighted-L2 half is solid and worth knowing; the JKO half is a credible but conditional story that the abstract overstates. read the letter →

arxiv 2607.25206 v1 pith:SZNAODGC submitted 2026-07-28 math.AP cs.NAmath-phmath.MPmath.NA

Yunhong Deng , Yuan Gao , Li Wang This is my paper · ORCID

classification math.APcs.NAmath-phmath.MPmath.NA MSC 35A1535B2735Q8437M1546N4049Q22

keywords homogenizationFokker-PlanckequationgradientflowsminimizingmovementsJKOschemeweightedL2metriceffectivediffusionGamma-convergence

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

This paper asks whether natural time-discrete approximations of a Fokker–Planck equation with rapidly oscillating diffusion survive the homogenization limit ε→0. Two candidate schemes are compared: a minimizing-movement (JKO) scheme built on a transport-type metric, and a scheme built on a weighted L2 metric. The paper proves that the choice matters: homogenizing the weighted L2 scheme and then sending the time step τ to zero recovers the classical effective Fokker–Planck equation with coefficient A*, whereas the homogenized JKO scheme converges to a quasilinear equation with a different coefficient, equal in one dimension to (∫ A^{-1/2})^{-2} instead of A* = (∫ A^{-1})^{-1}. Therefore, among the two schemes studied, only the weighted L2 metric is asymptotic-preserving. If correct, this is a caution for the widespread use of JKO-type variational discretizations for multiscale gradient-flow problems, and it isolates the cause: the two limits ε→0 and τ→0 do not commute.

What carries the argument

The load-bearing objects are the two metrics and their homogenized limits. For the weighted L2 scheme, the distance Kε(μ,ν) = (∫(μ−ν)²/πε dx)^{1/2} and the Dirichlet energy Qε homogenize to their counterparts with the classical elliptic-corrector coefficient A* = ∫(A + A∇χ). For the JKO scheme, the transport cost cε(x,y) = min∫⟨A(z/ε)^{-1}ż, ż⟩dt homogenizes (as a Lagrangian) to A_hom; the limiting equation's diffusion coefficient is then (1/2∇A_hom)^{-1}, obtained by Legendre duality from the homogenized action. The mismatch between A* and (1/2∇A_hom)^{-1} is the quantitative expression of the non-commutativity.

What would settle it

Run the 1D numerical example from the paper (A(x) = (1 + ½ sin 2πx)², ε = 10⁻⁵, small τ) and measure the effective diffusion coefficient of the homogenized JKO solutions as τ→0. If the spreading follows A* = (∫ A^{-1})^{-1} ≈ 0.65 rather than (∫ A^{-1/2})^{-2} ≈ 0.75, the central claim fails; conversely, reproducing ≈ 0.75 confirms it. A proof or counterexample of the W^{1,1} compactness of the homogenized JKO trajectories would directly settle the gap left open in the paper.

Watch

Extended reading notes

Core claim

For the Fokker–Planck equation ∂ₜμ − ∇·(A(x/ε)(∇μ + μ∇V)) = 0 on the flat torus with 1-periodic A, the paper proves (for fixed τ) Γ-convergence of both time-discrete schemes: the weighted L2 scheme homogenizes to a backward-Euler scheme for the effective equation with coefficient A* (Theorem 1.4), while the ε-JKO scheme homogenizes to a JKO scheme whose transport cost is generated by the homogenized action A_hom(ξ) = liminf_T (1/T)∫⟨A(tξ+φ)^{-1}(ξ+φ′), ξ+φ′⟩dt (Theorem 1.5). Sending τ→0, the homogenized weighted-L2 scheme gives the effective Fokker–Planck equation (1.2) (Theorem 1.7); the homogenized JKO scheme gives, under a W^{1,1} compactness assumption, the quasilinear equation (1.23) wi

Load-bearing premise

The conclusion that the homogenized JKO scheme converges to the quasilinear equation (1.23) rests on the unproved W^{1,1}(T^d) compactness of its time-discrete trajectories as τ→0 (explicitly left open in Remark 1.9) together with the assumed uniform strong convexity of the homogenized action A_hom; if either fails, the claimed limiting equation — and hence the claim that JKO fails to recover the effective dynamics — is not established.

Editorial extensions

If this is right

The weighted L2 scheme is asymptotic-preserving: its homogenized limit at fixed τ is the backward-Euler discretization of the effective Fokker–Planck equation, and its τ→0 limit is exactly (1.2).
The homogenized JKO scheme is not asymptotic-preserving: sending τ→0 yields the quasilinear equation (1.23) with coefficient (1/2∇A_hom)^{-1}, which differs from A* in general.
In one dimension the mismatch is explicit: the JKO limit gives effective diffusivity (∫ A^{-1/2})^{-2} whereas the correct effective diffusivity is A* = (∫ A^{-1})^{-1}; the two limits ε→0 and τ→0 therefore do not commute for the JKO scheme.
The mechanism is identified as the blow-up, as ε→0, of the second derivative of the exponential map of the transport metric, which makes the JKO consistency constant diverge while the weighted L2 constant stays bounded.

Reading between the lines

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

A byproduct of the 1D formula is a cheap quantitative bench test: for any periodic A, a multiscale scheme that claims asymptotic preservation should reproduce A*, whereas a JKO-type transport scheme will produce (∫ A^{-1/2})^{-2}; for A(x) = (1 + ½ sin 2πx)² these two numbers already differ by roughly 15%.
The same mechanism — trajectory-level homogenization of the Lagrangian versus PDE-level homogenization — likely affects other Wasserstein gradient flows (e.g., aggregation–diffusion or porous-medium type) once the entropy or drift is modified, so the conclusion may extend well beyond the linear Fokker–Planck case studied here.
If the W^{1,1} compactness left open in Remark 1.9 fails for some initial data, the homogenized JKO scheme might have no well-defined τ→0 limit at all — an outcome that would strengthen rather than weaken the paper's warning that transport-step schemes need scrutiny.

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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

First thing to know: the paper has a solid half and a conditional half. The solid half is the weighted-L2 scheme: fixed-τ homogenization via Γ-convergence and H1 compactness, then τ→0 by backward-Euler convergence. That chain is in good shape. The conditional half is the JKO story: Theorem 1.8's limiting equation (1.23) rests on W1,1 compactness the authors explicitly say they don't prove, and on uniform strong convexity of Ahom that Assumption 1.1 doesn't guarantee. The abstract's "only the weighted L2 recovers correct effective dynamics" is stronger than what is proved.

What's genuinely new: earlier work identified the continuous Wasserstein gradient-flow homogenization and the metric limit Wε→WGH. This paper asks what happens at the time-discrete level and shows the answer depends on the metric. The weighted-L2 chain is rigorous and clean, and the comparison between the two schemes is a real contribution. The 1D computation showing the JKO coefficient differs from A* is nicely concrete.

Soft spots, in proportion: Lemma 4.5 jumps from pointwise cost convergence to uniform control of the integral; pointwise convergence on a compact set is not enough without an equicontinuity argument. Likely fixable, but as written it's a gap. The bigger issue is Theorem 1.8. A uniform W1,1 bound alone does not give compactness in W1,1, and the paper concedes in Remark 1.9 that the needed compactness is beyond its scope. Uniform strong convexity of Ahom is also an assumption, not a consequence of the stated hypotheses. So the advertised failure of JKO is established only conditionally. The qualitative message is credible and probably true—especially in 1D—but the abstract should not state it as an unconditional theorem.

The citation pattern looks honest: the targets come from independent published work, no parameters are fitted, and the paper doesn't disguise what it inherits from [20]. No code or data is shipped; the numerical figure is illustrative.

Who is this for: applied analysts and numerical analysts working on asymptotic-preserving schemes for Fokker-Planck equations and on discrete gradient flows. It deserves a serious referee. I'd send it out, but with the expectation of major revision: either close the W1,1 compactness gap or rewrite the main claims so the conditional status is explicit. The weighted-L2 half can stand; the JKO half needs its caveats in the title or abstract.

Referee Report

3 major / 4 minor

Summary. The paper studies homogenization limits of two time-discrete minimizing movement schemes for Fokker–Planck equations on the flat torus with rapidly oscillating diffusion A(x/ε). The first scheme uses a weighted L2 metric (backward Euler form); the second is an ε-JKO scheme with a Wasserstein-type distance induced by A. For fixed τ, Theorem 1.4 homogenizes the weighted L2 scheme to a scheme using the effective diffusion A*, while Theorem 1.5 homogenizes the ε-JKO scheme to a scheme using the Gromov–Hausdorff limiting cost WGH and the associated Finsler action Ahom. The paper then identifies τ→0 limits: Theorem 1.7 states the homogenized weighted L2 scheme converges to the effective Fokker–Planck equation (1.2); Theorem 1.8 states that, conditional on uniform strong convexity of Ahom and an unproved W^{1,1} compactness assumption, the homogenized JKO scheme converges to the quasilinear equation (1.23) with coefficient (1/2∇Ahom)^{-1}. The paper concludes that only the weighted L2 scheme is asymptotic-preserving.

Significance. The question addressed is timely and important: whether variational time discretizations preserve homogenization limits is directly relevant to asymptotic-preserving numerical schemes and to the theory of gradient flows in inhomogeneous media. The weighted-L2 half of the paper is clean: the Γ-convergence of the Dirichlet energy and the convergence of the weighted L2 distance are proved in detail, and the H1 compactness argument is sound. The homogenization of the ε-JKO scheme for fixed τ is also a worthwhile contribution, and the numerical experiment in 1D illustrates the claimed discrepancy. The paper makes good use of independent published results ([20], [9], [41]) rather than fitting parameters, and the conditional theorem is stated honestly. However, the central comparative claim — that only the weighted L2 scheme recovers the correct effective dynamics — rests on Theorem 1.8, whose hypotheses are not established in the paper. The abstract and Section 1.5 present this claim unconditionally, which is not supported by the proofs as written.

major comments (3)

[Theorem 1.8 / Remark 1.9 / §4.2.2] The load-bearing conclusion is conditional. Passing τ→0 inside (1/2∇Ahom)^{-1} in (4.25) requires both uniform strong convexity of Ahom and the assumed W^{1,1}(Td) convergence µτ→µ; the latter is explicitly left unproved in Remark 1.9 ('lies beyond the scope of this work'). A uniform W^{1,1} bound, which is all that §4.2.2 establishes, does not imply W^{1,1} compactness because W^{1,1} is not reflexive and no equicontinuity of gradients is supplied. Moreover, uniform strong convexity of Ahom is not a consequence of Assumption 1.1 in d>1; homogenized Finsler action functions can develop flat directions. Therefore the abstract's unconditional statement that 'only the scheme based on the weighted L2 metric recovers the correct effective dynamics' is not established. The theorem should either be proven under additional verifiable hypotheses, or the claims and abstract should be rephrased as
[Lemma 4.5 / §4.1.2] The proof of convergence of Wasserstein distances uses pointwise convergence of costs cε(x,y)→chom(x,y) and then asserts: 'Since Td×Td is compact, ε(δ) can be made independent of (x,y)'. This is not valid in general: pointwise convergence of continuous functions on a compact set does not imply uniform convergence. The lemma needs an equicontinuity/Dini-type argument for the family (cε) or an alternative stability theorem for optimal transport under Γ-convergent costs. Without this, the proof of Theorem 1.5 has a gap, although the statement is likely repairable.
[Theorem 1.7 / Proposition 3.2] The weighted-L2 half of the paper's central claim depends on the τ→0 convergence of the backward Euler scheme. This is stated in Proposition 3.2 'without proof', and the proof of Theorem 1.7 is deferred to 'the same argument'. Since this is the half that is claimed to be correct and asymptotic-preserving, a proof or a precise reference with matching hypotheses should be included. The standard numerical-analysis references may not cover the exact weak formulation used here (with Vε and the H1 setting); a short proof would remove the asymmetry in rigor between the two halves.

minor comments (4)

[Figure 1] Typo: 'weigthed L2 scheme' should be 'weighted L2 scheme'.
[Remark 1.9] Remark 1.9 lists [26, 36, 14] as stronger convergence results for JKO schemes; these references are for standard Fokker–Planck or related equations and do not cover the homogenized Finsler metric considered here. Please clarify that the cited results do not imply the required W^{1,1} compactness for the present WGH metric.
[Lemma 4.10 / (4.16)] In the displayed inequality before (4.16), the factor τ is written as 'τ × Cε(φ)/(2τ)' which simplifies to Cε/2; the final bound should make explicit that the constant depends on φ (and is O(τ) after using Lemma 4.9). The current display is dimensionally confusing.
[General] Minor typos: 'Boreal sets' should be 'Borel sets'; 'Monge-Kantrovich' should be 'Monge–Kantorovich'; 'trajecotry' in Section 4.1; 'the prove' in Section 3.2 should be 'the proof'.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the homogenization derivations are self-contained, with only a minor self-citation and an explicitly flagged compactness gap.

full rationale

The paper's central derivations do not reduce to their inputs by construction. For the weighted L2 scheme (Section 3), Qε Γ-converges to Q via classical elliptic homogenization (Lemma 3.8), Kε→K via uniform convergence πε→π*, and compactness is supplied by H1 estimates; A* is fixed by the corrector problem (1.3), not by the scheme's output. For the JKO scheme (Section 4.1), the limit WGH is derived through pointwise cost convergence cε→chom (Lemma 4.6) using Γ-convergence of Eε to E* with independent external citations [41,9,19], and the Wasserstein-distance convergence follows by stability of optimal transport. The τ→0 limit (Theorem 1.8) is then obtained from the first-order optimality condition (Theorem 4.14) and the consistency estimate (Lemma 4.16), not by postulating (1.23). No free parameter is fitted, and the comparison between (1.2) and (1.23) is not definitionally forced. The genuinely load-bearing borrowed item is the 1D identity comparing (1/2∇Ahom)^{-1} with A*, taken from [20], which is co-authored by Yuan Gao; this is a minor self-citation, but [20] is a separately published derivation and the present paper independently re-derives the homogenized WGH structure. The real caveat is Remark 1.9: Theorem 1.8 is conditional on W^{1,1} compactness of the homogenized JKO trajectories and on uniform strong convexity of Ahom, both unproved; thus the abstract's unconditional 'only the weighted L2 metric recovers the correct effective dynamics' is partly conditional. This is a completeness/correctness gap, not a circularity by reduction.

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

No constants are fitted to data; ε and τ are asymptotic parameters. The results depend on standard homogenization theorems, cited external results (especially [20] for WGH and Ahom), and two explicit ad hoc assumptions for the JKO time-continuous limit: strong convexity of Ahom and W^{1,1} compactness. No new physical or mathematical entities are introduced.

assumptions (6)

standard math Classical elliptic periodic homogenization Γ-convergence for Fε to F with effective coefficient A* (Lemma 3.8)
Invoked in the proof of Lemma 3.7; cited from [30, 32, 6, 2].
standard math Homogenization of vector-valued action functionals: Γ-lim Eε = E* with integrand Ahom
Used in Lemma 4.6 to prove cε→chom; cited from [9, 41].
domain assumption WGH is the W2-limit of Wε and the 1D coefficient formula (∫ A^{-1/2})^{-2} are as given in [20]
External result by Gao-Yip (co-author Gao); used to identify the homogenized JKO metric and the non-equality with A*.
ad hoc to paper Ahom is uniformly strongly convex
Assumed in Theorem 1.8; not derived from assumptions on A. Needed to invert ∇Ahom and obtain the quasilinear equation (1.23).
ad hoc to paper Homogenized JKO time-discrete solutions µτ converge in W^{1,1}(Td) as τ→0
Explicitly assumed in Theorem 1.8; Remark 1.9 says this is a technical issue beyond the scope of the paper.
standard math Uniform H1 estimates, weighted Poincaré inequalities, and compactness for the weighted-L2 scheme
Used to prove compactness of µε,τ in L2 and to apply the fundamental theorem of Γ-convergence.

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Cite this review

Pith. "Pith review of Homogenization of Time-Discrete Gradient Flows." pith.science (2026). https://pith.science/paper/SZNAODGC

@misc{pith2026260725206,
  author       = {Pith},
  title        = {Pith review of: Homogenization of Time-Discrete Gradient Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZNAODGC}},
  note         = {Machine review of arXiv:2607.25206}
}

read the original abstract

In this work, we study the homogenization limit for Fokker--Planck equations on the flat torus with rapidly oscillating diffusion coefficients at the time-discrete level. We discretize the equation in time using a minimizing movement scheme and compare two choices of metric: a transport-type metric and a weighted L2 metric. We prove that this choice is crucial: the two schemes lead to different homogenization limits as the scaling parameter tends to zero, and consequently to different evolution equations as the time step is sent to zero. In particular, only the scheme based on the weighted L2 metric recovers the correct effective dynamics.

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Works this paper leans on

41 extracted references · 3 linked inside Pith

[20]

Gao and N

Y. Gao and N. K. Yip , Homogenisation of Wasserstein gradient flows , European J. Appl. Math., 37 (2026), p. 586–613

2026
[9]

Braides, G

A. Braides, G. Buttazzo, and I. Fragala , Riemannian approximation of finsler metrics , Asymptot. Anal., 31 (2002), pp. 177–187

2002
[41]

Weinan, A class of homogenization problems in the calculus of variations , Commun

E. Weinan, A class of homogenization problems in the calculus of variations , Commun. Pure Appl. Math., 44 (1991), pp. 733–759

1991
[1]

Allaire and R

G. Allaire and R. Orive , Homogenization of periodic non self-adjoint problems with large drift and potential , ESAIM Control Optim. Calc. Var., 13 (2007), pp. 735–749

2007
[2]

Alouges, Introduction to periodic homogenization, Interdiscip

F. Alouges, Introduction to periodic homogenization, Interdiscip. Inf. Sci., 22 (2016), pp. 147– 186

2016
[3]

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, 2008

2008
[4]

Bedrossian and I

J. Bedrossian and I. C. Kim , Global existence and finite time blow-up for critical Patlak– Keller–Segel models with inhomogeneous diffusion , SIAM J. Math. Anal., 45 (2013), pp. 934–964

2013
[5]

Bedrossian and N

J. Bedrossian and N. Rodr´ıguez, Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in Rd , arXiv preprint arXiv:1108.5167, (2011)

arXiv 2011

Show all 41 references

[6]

Bensoussan, Asymptotic analysis for periodic structures, vol

A. Bensoussan, Asymptotic analysis for periodic structures, vol. 374 , American Mathematical Soc., (2011)

2011
[7]

Braides, Gamma-convergence for Beginners, vol

A. Braides, Gamma-convergence for Beginners, vol. 22, Clarendon Press, 2002

2002
[8]

Braides, Local minimization, variational evolution and Γ-convergence, vol

A. Braides, Local minimization, variational evolution and Γ-convergence, vol. 2094, Springer, 2014
[10]

J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent, and D. Slep ˇcev, Global-in- time weak measure solutions and finite-time aggregation for nonlocal interaction equations, (2011)

2011
[11]

J. A. Carrillo, H. Liu, and H. Yu, Positivity-preserving and energy-dissipating discontinuous galerkin methods for nonlinear nonlocal Fokker–Planck equations , Commun. Appl. Ind. Math., 16 (2025)

2025
[12]

J. A. Carrillo, R. J. McCann, and C. Villani , Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates , Rev. Mat. Iberoam., 19 (2003), pp. 971–1018. HOMOGENIZATION OF TIME-DISCRETE GRADIENT FLOWS 35

2003
[13]

J. A. Carrillo, R. J. McCann, and C. Villani , Contractions in the 2-Wasserstein length space and thermalization of granular media , Arch. Ration. Mech. Anal., 179 (2006), pp. 217–263

2006
[14]

Coudreuse, Li-Yau-Hamilton inequality on the JKO scheme for the granular-medium equa- tion, arXiv preprint arXiv:2510.09231, (2025)

F. Coudreuse, Li-Yau-Hamilton inequality on the JKO scheme for the granular-medium equa- tion, arXiv preprint arXiv:2510.09231, (2025)

2025
[15]

Dacorogna, Direct methods in the calculus of variations , Springer, 2008

B. Dacorogna, Direct methods in the calculus of variations , Springer, 2008

2008
[16]

Dal Maso , An introduction to Γ-convergence, vol

G. Dal Maso , An introduction to Γ-convergence, vol. 8, Springer Science & Business Media, 2012

2012
[17]

Erbar, The heat equation on manifolds as a gradient flow in the Wasserstein space, Annales de l’IHP Probabilit´ es et statistiques, 46 (2010), pp

M. Erbar, The heat equation on manifolds as a gradient flow in the Wasserstein space, Annales de l’IHP Probabilit´ es et statistiques, 46 (2010), pp. 1–23

2010
[18]

Forkert, J

D. Forkert, J. Maas, and L. Portinale , Evolutionary Γ-convergence of entropic gradient flow structures for Fokker–Planck equations in multiple dimensions , SIAM J. Math. Anal., 54 (2022), pp. 4297–4333

2022
[19]

Gangbo and A

W. Gangbo and A. Tudorascu , Homogenization for a class of integral functionals in spaces of probability measures, Adv. Math., 230 (2012), pp. 1124–1173

2012
[21]

Gladbach, E

P. Gladbach, E. Kopfer, and J. Maas , Scaling limits of discrete optimal transport , SIAM J. Math. Anal., 52 (2020), pp. 2759–2802

2020
[22]

Jacobs, W

M. Jacobs, W. Lee, and F. L ´eger, The back-and-forth method for Wasserstein gradient flows, ESAIM Control Optim. Calc. Var., 27 (2021), p. 28

2021
[23]

Jacobs and F

M. Jacobs and F. L ´eger, A fast approach to optimal transport: The back-and-forth method , Numer. Math., 146 (2020), pp. 513–544

2020
[24]

Jordan, D

R. Jordan, D. Kinderlehrer, and F. Otto , The variational formulation of the Fokker– Planck equation, SIAM J. Math. Anal., 29 (1998), pp. 1–17

1998
[25]

Jost, Riemannian geometry and geometric analysis , Springer, 2005

J. Jost, Riemannian geometry and geometric analysis , Springer, 2005

2005
[26]

P. W. Lee , On the Jordan–Kinderlehrer–Otto scheme , J. Math. Anal. Appl., 429 (2015), pp. 131–142

2015
[27]

Lisini, Nonlinear diffusion equations with variable coefficients as gradient flows in Wasser- stein spaces, ESAIM Control Optim

S. Lisini, Nonlinear diffusion equations with variable coefficients as gradient flows in Wasser- stein spaces, ESAIM Control Optim. Calc. Var., 15 (2009), pp. 712–740

2009
[28]

Liu and A

H. Liu and A. E. Tzavaras , Variational structure of Fokker-Planck equations with variable mobility, arXiv preprint arXiv:2505.10676, (2025)

2025 arXiv
[29]

J.-G. Liu, L. W ang, and Z. Zhou , Positivity-preserving and asymptotic preserving method for 2d Keller-Segal equations , Math. Comput., 87 (2018), pp. 1165–1189

2018
[30]

Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann

P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., 117 (1978), pp. 139–152

1978
[31]

Otto, The geometry of dissipative evolution equations: the porous medium equation, (2001)

F. Otto, The geometry of dissipative evolution equations: the porous medium equation, (2001)

2001
[32]

G. A. Pavliotis and A. Stuart, Multiscale methods: averaging and homogenization , vol. 53, Springer Science & Business Media, 2008

2008
[33]

Pechstein and R

C. Pechstein and R. Scheichl , Weighted Poincar´ e inequalities, IMA J. Numer. Anal., 33 (2013), pp. 652–686

2013
[34]

Rankin and T.-K

C. Rankin and T.-K. L. Wong , JKO schemes with general transport costs , arXiv preprint arXiv:2402.17681, (2024)

2024 arXiv
[35]

Sandier and S

E. Sandier and S. Serfaty , Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Commun. Pure Appl. Math., 57 (2004), pp. 1627–1672

2004
[36]

Santambrogio and G

F. Santambrogio and G. Toshpulatov , Strong L2H 2 convergence of the JKO scheme for the Fokker–Planck equation , Arch. Ration. Mech. Anal., 248 (2024), p. 99

2024
[37]

Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applica- tions, Discrete Contin

S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applica- tions, Discrete Contin. Dyn. Syst, 31 (2011), pp. 1427–1451

2011
[38]

J. C. Strikwerda , Finite difference schemes and partial differential equations , SIAM, 2004

2004
[39]

Struwe, Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems , Springer, 2008

M. Struwe, Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems , Springer, 2008

2008
[40]

Villani, Optimal transport: old and new , vol

C. Villani, Optimal transport: old and new , vol. 338, Springer, 2008

2008

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [20]

Gao and N

Y. Gao and N. K. Yip , Homogenisation of Wasserstein gradient flows , European J. Appl. Math., 37 (2026), p. 586–613

2026

[2] [9]

Braides, G

A. Braides, G. Buttazzo, and I. Fragala , Riemannian approximation of finsler metrics , Asymptot. Anal., 31 (2002), pp. 177–187

2002

[3] [41]

Weinan, A class of homogenization problems in the calculus of variations , Commun

E. Weinan, A class of homogenization problems in the calculus of variations , Commun. Pure Appl. Math., 44 (1991), pp. 733–759

1991

[4] [1]

Allaire and R

G. Allaire and R. Orive , Homogenization of periodic non self-adjoint problems with large drift and potential , ESAIM Control Optim. Calc. Var., 13 (2007), pp. 735–749

2007

[5] [2]

Alouges, Introduction to periodic homogenization, Interdiscip

F. Alouges, Introduction to periodic homogenization, Interdiscip. Inf. Sci., 22 (2016), pp. 147– 186

2016

[6] [3]

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, 2008

2008

[7] [4]

Bedrossian and I

J. Bedrossian and I. C. Kim , Global existence and finite time blow-up for critical Patlak– Keller–Segel models with inhomogeneous diffusion , SIAM J. Math. Anal., 45 (2013), pp. 934–964

2013

[8] [5]

Bedrossian and N

J. Bedrossian and N. Rodr´ıguez, Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in Rd , arXiv preprint arXiv:1108.5167, (2011)

arXiv 2011