Pith. sign in

REVIEW 3 minor 36 references

Coulomb MMD flow relaxes to target on torus and in plane, with exponential rates under coercivity

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 06:25 UTC pith:MU3LE7B2

load-bearing objection A serious paper that largely delivers on its advertised results; the only significant caveat—non-uniqueness for measure initial data—is openly acknowledged rather than hidden.

arxiv 2607.12579 v2 pith:MU3LE7B2 submitted 2026-07-14 math.AP stat.ML

Wasserstein gradient flows for Coulomb discrepancies

Antonin Chodron de Courcel , Matthew Rosenzweig This is my paper
classification math.AP stat.ML MSC 35A0135B4049Q2235D30
keywords Wasserstein gradient flowMaximum Mean DiscrepancyCoulomb potentialPolyak–Łojasiewicz inequalitylong-time behaviornonlocal transport equationLagrangian critical pointplanar tightness
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) with a Coulomb kernel, a nonlocal transport equation that pushes a probability density toward a target density. The main message: from arbitrary Borel initial measures the flow exists globally, instantly becomes bounded when the target is bounded, and, under coercivity conditions, relaxes to the target at an exponential rate. On the torus, a defective Polyak–Łojasiewicz inequality yields exponential decay for uniformly positive targets even when the evolving density has vacuum regions. On the whole space the situation is more delicate: radial sources supported inside the target converge exponentially, but no uniform decay rate or global PL inequality holds when the source may start far away. In dimension two, finite Coulomb energy at a positive time forces tightness, and every constructed solution converges to the target in several weak topologies.

Core claim

The central discovery is that the squared-MMD Coulomb discrepancy functional, though nonconvex, has a Polyak–Łojasiewicz dissipation-to-energy structure in exactly the regimes where transport is not obstructed. The paper proves a global metric PL inequality on the torus for all finite-energy sources when the target is nearly uniform, and a defective PL inequality for uniformly positive targets whose vacuum defect decays exponentially; these yield exponential decay of the squared MMD for every constructed solution after any waiting time. It also proves rigidity of Lagrangian critical points — if (ρ−μ)+ is absolutely continuous and the Coulomb field vanishes on it, then ρ=μ — and, in the plane

What carries the argument

The Polyak–Łojasiewicz (PL) inequality D(ρ|μ) ≥ c MMD²(ρ,μ), comparing the dissipation D(ρ|μ)=∫|∇g*(ρ−μ)|² dρ to the squared MMD energy, is the main coercivity mechanism; where the classical inequality fails (vacuum regions, distant sources), the paper uses a defective PL inequality with an explicit exponentially decaying defect, and a metric-slope formulation to handle singular measures. The rigidity of Lagrangian critical points — ∇h=0 on (ρ−μ)+ forces ρ=μ when (ρ−μ)+ is absolutely continuous — supplies the final identification in the planar LaSalle argument, together with a logarithmic-capacity estimate that turns finite Coulomb energy into uniform tightness in two dimensions.

Load-bearing premise

The long-time convergence results for arbitrary Borel initial measures apply to a solution obtained by a specific smoothing procedure, and the paper does not prove that this limiting solution is unique; if a different regularization produced a different trajectory, the convergence statements would not characterize the equation itself.

What would settle it

Take a singular initial measure and two sequences of smooth mollifications that converge to it; if the corresponding global weak solutions differ at positive times or have different long-time limits, the paper's convergence theorems for arbitrary initial data would be regularization-dependent rather than intrinsic. Alternatively, construct a Lagrangian critical point with (ρ−μ)+ absolutely continuous whose Coulomb field vanishes on the positive part but ρ≠μ; this would directly contradict the rigidity theorem.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • On the torus, every solution constructed from an arbitrary probability measure and targeting a bounded, uniformly positive density attains exponential decay of the squared MMD after any positive time, with any rate below 2λ/3 where λ is the target's essential infimum.
  • Global PL coercivity holds on the torus for all finite-energy sources when the target is close to uniform, and in dimension one for every bounded uniformly positive target; this gives quantitative transport-information bounds independent of the source's lower bound.
  • In the plane, finite Coulomb energy at one positive time is enough to guarantee convergence of the constructed solution to the target in narrow, L∞ weak-*, and H^{-α} topologies; no moment or support assumptions are needed.
  • On the whole space, any uniform exponential decay modulus or global PL inequality fails: a localized source separated from a compactly supported target by distance D retains a positive fraction of its initial squared MMD for times proportional to D.
  • Radial sources with connected target support and source-support inclusion converge exponentially on R^d, d≥2, with explicit constant max{1/λ, d²A/(2λ²)}.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The defective PL mechanism suggests that for degenerate targets (vanishing on a set), the relaxation rate may be governed by local Łojasiewicz exponents tied to the order of vanishing; the paper leaves this as an open problem, and a natural next step is to compute such rates for targets vanishing like |x|^k.
  • Because uniqueness for singular initial data is unresolved, the planar convergence theorem is a statement about the particular approximation-selected solution; if selection is later shown to matter, the theorem would need reinterpretation as a property of a canonical semigroup rather than the equation itself.
  • The travel-time obstruction at spatial infinity implies that any data-independent convergence rate for the whole-space problem must encode location or tail information; in sampling and generative-modeling applications, this quantifies the cost of initializing far from the target.
  • The rigidity theorem classifies Lagrangian critical points with absolutely continuous positive part, but singular positive parts (e.g., atoms) can yield non-target critical points for suitable field representatives, so the critical-point landscape outside the absolutely continuous class remains open.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper studies the long-time behavior of the Wasserstein gradient flow of the squared Coulomb Maximum Mean Discrepancy, i.e. the nonlocal transport equation ∂tρ − div(ρ∇g∗(ρ−µ)) = 0 on the flat torus and on R^d. It constructs global weak solutions from arbitrary Borel initial measures when the target µ is bounded, with instantaneous L∞ regularization and explicit Lp estimates, proves uniqueness in the bounded initial-data class by an Osgood/modulated-energy argument, and shows that Hölder seminorms can grow exponentially. On T^d it proves a global metric PL inequality under a target density-ratio condition, a defective PL inequality yielding exponential decay of the squared MMD for uniformly positive targets, and two counterexamples showing limits of PL coercivity. On R^d it proves a radial PL inequality under source-support inclusion, and whole-space travel-time obstructions showing that no uniform convergence rate or global PL inequality can hold on the unrestricted class. Finally, it proves that every Lagrangian critical point with (ρ−µ)+ absolutely continuous equals the target, and combines this with a logarithmic-capacity tightness estimate to obtain narrow, weak-* L∞, and H^{-α} convergence in dimension two whenever the Coulomb energy is finite at some positive time.

Significance. If the results hold, this is a substantial contribution to the analysis of MMD gradient flows and Wasserstein gradient flows of singular interaction energies. The paper's strengths are notable: the Cauchy theory handles arbitrary Borel initial measures; the torus PL inequality is proved by an intricate variational barrier argument rather than by fitting constants; the defective PL proof gives explicit exponential rates without a lower bound on the evolving density; the radial Euclidean PL result is proved by a clean quantile-shell decomposition; and the planar convergence theorem uses a genuine logarithmic-capacity tightness mechanism. The negative results—PL failure for a target vanishing at one point, non-uniformity over targets with a fixed lower bound, and the travel-time obstruction on R^d—are constructive and clarify the sharpness of the hypotheses. The acknowledged non-uniqueness for measure initial data limits the interpretation of Theorems 1.7 and 1.22 as statements about the particular constructed solutions rather than an intrinsic semigroup, but the paper states this caveat explicitly and the theorems are correctly quantified over constructed solutions. I found no intern

minor comments (3)
  1. [Eq. (3.34)] The display reads “2w − εΦ′(ρ̄) = κρ̄ dx-a.e.”, but the derivation from (3.33) gives the density (2w − εΦ′(ρ̄) − κ)ρ̄ = 0, and since ρ̄ > 0 a.e. the conclusion should be “2w − εΦ′(ρ̄) = κ dx-a.e.” The subsequent Euler-equation argument uses the correct form, so this is a harmless typo, but it should be corrected.
  2. [Definition 1.1 and Theorem 1.2] Definition 1.1 requires ρ_t ∈ L∞ for a.e. t > 0, while Theorem 1.2 states the bound for every t > 0. This is a minor mismatch between the weak-solution definition and the regularity actually proved; the authors may wish to harmonize the wording or explicitly state that the constructed solutions have the stronger pointwise-in-time regularity.
  3. [Section 2, before §2.1] The non-uniqueness caveat for arbitrary Borel initial data is stated in Section 2, but it is also relevant to the long-time theorems in the introduction. Adding a sentence near Theorems 1.7 and 1.22 reminding the reader that these results concern solutions constructed by smooth approximation would improve the exposition, even though the theorems are already quantified correctly.

Circularity Check

0 steps flagged

No significant circularity: central derivations are self-contained; only minor non-load-bearing self-citations appear.

full rationale

The paper's central claims—the Cauchy theory, ultracontractivity, torus PL inequalities, defective-PL exponential decay, radial whole-space PL, whole-space obstructions, and planar convergence—are derived from the equation's structure and proven estimates, not from fitted parameters or target-derived constants. The PL constants arise from explicit inequalities (e.g., Theorem 1.3's m - (d-1)/d M; Theorem 1.7's gamma < 2*lambda/3), and the defective PL inequality in Lemma 3.3 is proven by splitting the dissipation and bounding the defect, then showing the defect decays in Lemma 3.4. The rigidity Theorem 1.21 uses the external level-set theorem [APR20, Thm 1.1], not the authors' own results. The only self-citations, [RS26] and [CdCRS25], provide commutator estimates in the stability/uniqueness proof (Proposition 2.4); these are published, parameter-free external tools and are not the source of the convergence conclusions. The paper explicitly acknowledges non-uniqueness for arbitrary Borel initial data ('We do not know whether uniqueness holds for such global weak solutions,' Section 2), and consequently Theorems 1.7 and 1.22 are formulated for any solution constructed in Theorem 1.2. This is a scope limitation, not circularity: the proofs use only properties shared by every constructed solution. No equation is shown to be equivalent to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters are fitted; the paper is purely theorem-proof. The main background inputs are standard elliptic/PDE tools plus two deep external theorems ([APR20], [RS26]) and one acknowledged unsolved selection problem.

axioms (6)
  • standard math Calderón–Zygmund and elliptic regularity estimates for the Coulomb Poisson equation on R^d and T^d.
    Used throughout the compactness and field-identification arguments (Section 2), and in the proof of Theorem 1.3.
  • standard math Level-set rigidity theorem [APR20, Theorem 1.1]: the absolutely continuous part of the distributional Laplacian vanishes a.e. on vector level sets of the gradient.
    Core input in proof of Theorem 1.21 (Section 6.1).
  • domain assumption Osgood uniqueness/flow representation for continuity equations with log-Lipschitz vector fields [AB08].
    Used to obtain uniqueness in L∞ class and support confinement/persistence in Section 5.
  • standard math Coulomb commutator estimate [RS26, Prop. 3.2] and periodic analogue [CdCRS25, §6.2].
    Used in uniqueness/stability proof (Section 2.2); self-cited but published external tools.
  • standard math Hilbert-space chain rule in H^{-1} for absolutely continuous curves of measures with bounded L² fluxes.
    Justifies energy dissipation identities in Sections 2-3 and 6.
  • domain assumption Smooth well-posedness for the approximated PDE (2.1) follows by adaptation of [LZ00].
    Starting point for the compactness construction; relies on prior theory.

pith-pipeline@v1.3.0-alltime-deepseek · 43641 in / 13070 out tokens · 127278 ms · 2026-08-02T06:25:06.834944+00:00 · methodology

0 comments
read the original abstract

We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $\rho$ and a target measure $\mu$, where the underlying kernel is given by a Coulomb potential. For $L^\infty$ target densities $\mu$, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove that the density $\rho_t$ belongs to $L^\infty$ for any $t>0$. We also show that the H\"older norm can grow exponentially in time. On the flat torus ${\mathbb{T}}^\mathsf{d}$, we prove a global metric PL inequality for every finite-Coulomb-energy source and nearly uniform target. For general bounded, uniformly positive targets, we prove exponential decay of the squared MMD without requiring a lower bound on the initial data, using a defective PL inequality. We also prove that the usual PL inequality may fail when the target vanishes only at one point and that, when $\mathsf{d}\ge2$, no PL constant can hold uniformly over all targets satisfying a prescribed lower bound. On ${\mathbb{R}}^\mathsf{d}$, for $\mathsf{d}\ge2$, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence. On the unrestricted whole-space class, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold. Finally, in every dimension and in both spatial settings, we prove that every Lagrangian critical point coincides with the target when $(\rho-\mu)^+$ is absolutely continuous. In dimension two, the energy supplies uniform tightness. This implies that if our constructed solutions have finite energy at some positive time, then they converge to the target narrowly and strongly in negative-order Sobolev spaces.

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 6 linked inside Pith

  1. [1]

    Uniqueness of signed measures solving the continuity equation for O sgood vector fields

    Luigi Ambrosio and Patrick Bernard. Uniqueness of signed measures solving the continuity equation for O sgood vector fields. Rend. Lincei Mat. Appl. , 19(3):237--245, 2008

    2008

  2. [2]

    Functions of Bounded Variation and Free Discontinuity Problems

    Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of Bounded Variation and Free Discontinuity Problems . Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000

    2000

  3. [3]

    u rich. Birkh \

    Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar \'e . Gradient Flows: In Metric Spaces and in the Space of Probability Measures . Lectures in Mathematics ETH Z \"u rich. Birkh \"a user, Basel, 2008

    2008

  4. [4]

    Neural Wasserstein gradient flows for discrepancies with R iesz kernels

    Fabian Altekr \"u ger, Johannes Hertrich, and Gabriele Steidl. Neural Wasserstein gradient flows for discrepancies with R iesz kernels. In Proceedings of the 40th International Conference on Machine Learning , volume 202 of Proceedings of Machine Learning Research , pages 664--690. PMLR, 2023

    2023

  5. [5]

    Maximum Mean Discrepancy gradient flow

    Michael Arbel, Anna Korba, Adil Salim, and Arthur Gretton. Maximum Mean Discrepancy gradient flow. In Advances in Neural Information Processing Systems , volume 32, pages 6484--6494, 2019

    2019

  6. [6]

    Ponce, and R \'e my Rodiac

    Luigi Ambrosio, Augusto C. Ponce, and R \'e my Rodiac. Critical weak- L^p differentiability of singular integrals. Rev. Mat. Iberoam. , 36(7):2033--2072, 2020

    2033

  7. [7]

    Bertozzi, Thomas Laurent, and Flavien L \'e ger

    Andrea L. Bertozzi, Thomas Laurent, and Flavien L \'e ger. Aggregation and spreading via the N ewtonian potential: the dynamics of patch solutions. Math. Models Methods Appl. Sci. , 22(suppl. 1):1140005, 2012

    2012

  8. [8]

    On the global convergence of Wasserstein gradient flow of the C oulomb discrepancy

    Siwan Boufad \`e ne and Fran c ois-Xavier Vialard. On the global convergence of Wasserstein gradient flow of the C oulomb discrepancy. SIAM J. Math. Anal. , 57(4):4556--4587, 2025

    2025

  9. [9]

    On the global convergence of gradient descent for over-parameterized models using optimal transport

    L \'e na \"i c Chizat and Francis Bach. On the global convergence of gradient descent for over-parameterized models using optimal transport. In Advances in Neural Information Processing Systems , volume 31, pages 3036--3046, 2018

    2018

  10. [10]

    Quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancies , 2026

    L \'e na \"i c Chizat, Maria Colombo, Roberto Colombo, and Xavier Fern \'a ndez-Real. Quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancies , 2026. arXiv:2603.01977

    arXiv 2026

  11. [11]

    Propagation of chaos: A review of models, methods and applications

    Louis-Pierre Chaintron and Antoine Diez. Propagation of chaos: A review of models, methods and applications. I . Models and methods. Kinet. Relat. Models , 15(6):895--1015, 2022

    2022

  12. [12]

    The gradient flows of sub- Coulomb Riesz energies

    Antonin Chodron de Courcel and Matthew Rosenzweig. The gradient flows of sub- Coulomb Riesz energies. Manuscript in preparation, 2026

    2026

  13. [13]

    Sharp uniform-in-time mean-field convergence for singular periodic R iesz flows

    Antonin Chodron de Courcel, Matthew Rosenzweig, and Sylvia Serfaty. Sharp uniform-in-time mean-field convergence for singular periodic R iesz flows. Ann. Inst. H. Poincar \'e C Anal. Non Lin \'e aire , 42(2):391--472, 2025

    2025

  14. [14]

    Sharp rates of MMD empirical estimation with power kernels, 2026

    Francesco Colasanto, Matteo Focardi, Massimo Fornasier, and Francesco Mattesini. Sharp rates of MMD empirical estimation with power kernels, 2026. arXiv:2605.18497

    Pith/arXiv arXiv 2026

  15. [15]

    Carrillo, Rishabh S

    Jos \'e A. Carrillo, Rishabh S. Gvalani, and Jeremy S.-H. Wu. An invariance principle for gradient flows in the space of probability measures. J. Differential Equations , 345:233--284, 2023

    2023

  16. [16]

    Generative modeling via Drifting , 2026

    Mingyang Deng, He Li, Tianhong Li, Yilun Du, and Kaiming He. Generative modeling via Drifting , 2026. arXiv:2602.04770

    Pith/arXiv arXiv 2026

  17. [17]

    Wasserstein gradient flows of MMD functionals with distance kernel and Cauchy problems on quantile functions

    Richard Duong, Viktor Stein, Robert Beinert, Johannes Hertrich, and Gabriele Steidl. Wasserstein gradient flows of MMD functionals with distance kernel and Cauchy problems on quantile functions. ESAIM Control Optim. Calc. Var. , 32:Paper No. 10, 37, 2026

    2026

  18. [18]

    Evans and Ronald F

    Lawrence C. Evans and Ronald F. Gariepy. Measure Theory and Fine Properties of Functions . Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015

    2015

  19. [19]

    An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows

    Alessio Figalli and Federico Glaudo. An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows . EMS Textbooks in Mathematics. EMS Press, Berlin, second edition, 2023

    2023

  20. [20]

    Borgwardt, Malte J

    Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Sch \"o lkopf, and Alexander J. Smola. A kernel two-sample test. J. Mach. Learn. Res. , 13:723--773, 2012

    2012

  21. [21]

    Foundations of Quantization for Probability Distributions , volume 1730 of Lecture Notes in Mathematics

    Siegfried Graf and Harald Luschgy. Foundations of Quantization for Probability Distributions , volume 1730 of Lecture Notes in Mathematics . Springer, Berlin, 2000

    2000

  22. [22]

    Mean- Field Limits in Statistical Dynamics

    François Golse. Mean- Field Limits in Statistical Dynamics . arXiv:2201.02005, 2022

    Pith/arXiv arXiv 2022

  23. [23]

    On the Wasserstein gradient flow interpretation of Drifting Models , 2026

    Arthur Gretton, Li Kevin Wenliang, Alexandre Galashov, James Thornton, Valentin De Bortoli, and Arnaud Doucet. On the Wasserstein gradient flow interpretation of Drifting Models , 2026. arXiv:2605.05118

    Pith/arXiv arXiv 2026

  24. [24]

    Optimal quantization for Riesz MMDs

    Elias Hess-Childs, Matthew Rosenzweig, and Sylvia Serfaty. Optimal quantization for Riesz MMDs . Manuscript in preparation, 2026

    2026

  25. [25]

    Wasserstein steepest descent flows of discrepancies with R iesz kernels

    Johannes Hertrich, Manuel Gr \"a f, Robert Beinert, and Gabriele Steidl. Wasserstein steepest descent flows of discrepancies with R iesz kernels. J. Math. Anal. Appl. , 531(1):127829, 2024

    2024

  26. [26]

    Jiaqi Han, Puheng Li, Qiushan Guo, Renyuan Xu, Stefano Ermon, and Emmanuel J. Candès. One-step generative modeling via Wasserstein gradient flows, 2026. arXiv:2605.11755

    Pith/arXiv arXiv 2026

  27. [27]

    On the hydrodynamic limit of Ginzburg-Landau vortices

    Fanghua Lin and Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete Contin. Dynam. Systems , 6(1):121--142, 2000

    2000

  28. [28]

    A mean field view of the landscape of two-layer neural networks

    Song Mei, Andrea Montanari, and Phan-Minh Nguyen. A mean field view of the landscape of two-layer neural networks. Proc. Natl. Acad. Sci. USA , 115(33):E7665--E7671, 2018

    2018

  29. [29]

    Diffeomorphic measure matching with kernels for generative modeling

    Biraj Pandey, Bamdad Hosseini, Pau Batlle, and Houman Owhadi. Diffeomorphic measure matching with kernels for generative modeling. SIAM J. Math. Data Sci. , 7(3):937--964, 2025

    2025

  30. [30]

    Sharp commutator estimates of all order for C oulomb and R iesz modulated energies

    Matthew Rosenzweig and Sylvia Serfaty. Sharp commutator estimates of all order for C oulomb and R iesz modulated energies. Comm. Pure Appl. Math. , 79(2):207--292, 2026

    2026

  31. [31]

    Wasserstein gradient flow of Maximum Mean Discrepancy with energy kernels

    Matthew Rosenzweig, Dejan Slep c ev, and Lihan Wang. Wasserstein gradient flow of Maximum Mean Discrepancy with energy kernels. Manuscript in preparation, 2026

    2026

  32. [32]

    Optimal Transport for Applied Mathematicians

    Filippo Santambrogio. Optimal Transport for Applied Mathematicians . Birkh \"a user, Cham, 2015

    2015

  33. [33]

    Lectures on C oulomb and R iesz Gases , volume 70 of Colloquium Publications

    Sylvia Serfaty. Lectures on C oulomb and R iesz Gases , volume 70 of Colloquium Publications . American Mathematical Society, Providence, RI, 2026. Forthcoming; arXiv:2407.21194

    Pith/arXiv arXiv 2026

  34. [34]

    Compact sets in the space L^p(0,T;B)

    Jacques Simon. Compact sets in the space L^p(0,T;B) . Ann. Mat. Pura Appl. (4) , 146:65--96, 1987

    1987

  35. [35]

    A mean field equation as limit of nonlinear diffusions with fractional L aplacian operators

    Sylvia Serfaty and Juan Luis V \'a zquez. A mean field equation as limit of nonlinear diffusions with fractional L aplacian operators. Calc. Var. Partial Differential Equations , 49(3-4):1091--1120, 2014

    2014

  36. [36]

    V. I. Yudovich. Non-stationary flow of an ideal incompressible liquid. USSR Computational Mathematics and Mathematical Physics , 3(6):1407--1456, 1963

    1963