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arxiv: 2605.04658 · v1 · submitted 2026-05-06 · 🧮 math.AP

Recent progress in generalized Hamiltonian gradient flow: Singularities

Wei Cheng , Jiahui Hong This is my paper

Pith reviewed 2026-05-08 17:22 UTC · model grok-4.3

classification 🧮 math.AP
keywords generalized Hamiltonian gradient flowsingularitiesMather measuresMañé critical valueweak KAM theoryHamilton-Jacobi equationsminimizing movementsinvariant measures
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The pith

The only invariant measures of the GHGF semi-flow attaining Mañé's critical value c[H] are the projected Mather measures.

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

The paper surveys the generalized Hamiltonian gradient flow framework for Hamilton-Jacobi equations and its handling of singularities, with ties to weak KAM theory, optimal transport, and mean field control. It gives a variational construction of generalized characteristics through a minimizing movement scheme, where weak limits and Young measure compactness establish that the resulting curves obey the generalized characteristic differential inclusion. The main new result lifts the forward dynamics of strict singular characteristics to a semi-flow on trajectory space and identifies its invariant probability measures that reach the critical value c[H] as precisely the projected Mather measures.

Core claim

By lifting the forward dynamics of strict singular characteristics to a semi-flow on the space of trajectories and studying its invariant probability measures, the paper proves that the only invariant measures of the GHGF semi-flow that attain the critical value c[H] are precisely the projected Mather measures, thereby giving a new dynamical characterization of Mather's minimal measures as well as Mañé's critical value.

What carries the argument

The GHGF semi-flow on the space of trajectories, whose invariant measures attaining c[H] are shown to coincide with projected Mather measures.

If this is right

  • Mather's minimal measures receive an equivalent description as the invariant measures of the GHGF semi-flow that attain the critical value.
  • The propagation of singularities in solutions to Hamilton-Jacobi equations becomes linked to the ergodic properties of the semi-flow.
  • The variational minimizing movement construction supplies a way to approximate generalized characteristics while preserving the differential inclusion in the limit.
  • Connections between singular dynamics, optimal transport, and mean field control are made explicit through the analysis of invariant measures.

Where Pith is reading between the lines

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

  • Questions on uniqueness of strict singular characteristics may be approachable by examining rectifiability of the cut loci under the semi-flow.
  • Stability of the characterization under small perturbations of the Hamiltonian would follow if the semi-flow depends continuously on the Hamiltonian in a suitable topology.
  • The vanishing noise limit in related stochastic problems could correspond to measures attaining c[H] in the deterministic GHGF setting.

Load-bearing premise

The Hamiltonian satisfies the standard convexity, superlinearity, and regularity conditions from weak KAM theory that guarantee existence of Mather measures and well-posedness of the GHGF semi-flow.

What would settle it

Discovery of an invariant probability measure for the GHGF semi-flow on trajectories that attains c[H] but is not a projected Mather measure would disprove the claimed characterization.

Figures

Figures reproduced from arXiv: 2605.04658 by Jiahui Hong, Wei Cheng.

Figure 1
Figure 1. Figure 1: Construction of intrinsic singular characteristics satisfying the following differential inclusion is called a generalized charac￾teristic from x ∈ R n : (GC) ( x˙(t) ∈ co Hp(x(t), D+ϕ(x(t))), a.e. t ∈ [0, +∞) x(0) = x, for any ϕ ∈ SCL (M). We call such a curve x a strict generalized character￾istic ([24]) if we rule out the convex hull in (GC). The authors proved ([2]) if ϕ is a viscosity solution of the … view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of Arnaud’s Theorem view at source ↗
Figure 3
Figure 3. Figure 3: minimizing movement from x0. τ∆(t) = inf{τi | τi > t}. Then one can define a vector field W∆(t, x) on [0, ∞) × M as (2.4) W∆(t, x) = Hp(x, DT + τ∆(t)−t ϕ(x)), t ∈ [0, ∞), x ∈ M. Notice that the vector fields W∆ is uniformly bounded for any partition ∆, and W∆ is Lipschitz continuous on each [τi−1, τi−ε]×M for any small ε > 0. We consider the differential equation (2.5) ( γ˙(t) = W∆(t, γ(t)), t ∈ [0, ∞), γ(… view at source ↗
read the original abstract

This paper is a survey of the generalized Hamiltonian gradient flow (GHGF) framework for Hamilton-Jacobi equations, with an emphasis on the propagation of singularities and its connections to weak KAM theory, optimal transport and mean field control. In addition to reviewing the main ideas and known results, we present two new contributions. First, we provide a variational construction of generalized characteristics via a minimizing movement scheme; by taking the weak limit of approximate solutions and using Young measure compactness, we show that the limiting curve satisfies the generalized characteristic differential inclusion. Second, we lift the forward dynamics of strict singular characteristics to a semi-flow on the space of trajectories and study its invariant probability measures. We prove that the only invariant measures of the GHGF semi-flow that attain the critical value \(c[H]\) are precisely the projected Mather measures, thereby giving a new dynamical characterization of Mather's minimal measures as well as Ma\~n\'e's critical value. Finally, we discuss a number of open problems that arise from the GHGF perspective, including questions on uniqueness of strict singular characteristics, rectifiability of cut loci, stability under perturbations, contact Hamiltonian systems, vanishing noise limits, and extensions to non-convex or low-regularity Hamiltonians. These problems highlight the deeper connections between singular dynamics, ergodic theory, optimal transport, and geometric analysis, and indicate directions for future research.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript surveys the generalized Hamiltonian gradient flow (GHGF) framework for Hamilton-Jacobi equations, with emphasis on singularity propagation and connections to weak KAM theory, optimal transport, and mean field control. It reviews known results and presents two new contributions: (i) a variational construction of generalized characteristics via a minimizing movement scheme, taking weak limits of approximate solutions and applying Young measure compactness to obtain curves satisfying the generalized characteristic differential inclusion; (ii) lifting the forward dynamics of strict singular characteristics to a semi-flow on the space of trajectories and proving that the only invariant measures of this semi-flow attaining the critical value c[H] are precisely the projected Mather measures. The paper concludes by listing open problems, including uniqueness of strict singular characteristics.

Significance. If the central claims hold, the second contribution supplies a new dynamical characterization of Mather's minimal measures and Mañé's critical value in terms of invariant measures for the GHGF semi-flow, strengthening links between singular dynamics, ergodic theory, and geometric analysis. The variational construction via minimizing movements and Young measures adds a constructive tool for handling singularities under standard convexity and superlinearity assumptions from weak KAM theory.

major comments (2)
  1. [Abstract (second new contribution)] Abstract (second new contribution): the lifting of forward dynamics of strict singular characteristics to a semi-flow on trajectory space, followed by the identification of its invariant measures attaining c[H] with projected Mather measures, presupposes a well-defined (single-valued) semi-flow. The manuscript itself flags uniqueness of strict singular characteristics as an open problem. In the absence of uniqueness the dynamics are multi-valued, so the definition of invariant probability measures requires an unspecified selection mechanism; the claimed identification may then hold only for particular selections rather than for the full dynamics.
  2. [Abstract (first new contribution)] Abstract (first new contribution): the variational construction via minimizing movement scheme and Young measure compactness is asserted to yield curves satisfying the generalized characteristic differential inclusion, but the abstract supplies no error estimates, boundary-case verification, or explicit compactness constants. The full manuscript must confirm that the weak-limit argument remains valid under the precise regularity and growth conditions stated for the Hamiltonian.
minor comments (1)
  1. Distinguish proved statements from open problems more explicitly in the introduction and conclusion so that readers can immediately separate established results from conjectures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our survey of the generalized Hamiltonian gradient flow framework. We address each major comment below with clarifications and indicate planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract (second new contribution)] Abstract (second new contribution): the lifting of forward dynamics of strict singular characteristics to a semi-flow on trajectory space, followed by the identification of its invariant measures attaining c[H] with projected Mather measures, presupposes a well-defined (single-valued) semi-flow. The manuscript itself flags uniqueness of strict singular characteristics as an open problem. In the absence of uniqueness the dynamics are multi-valued, so the definition of invariant probability measures requires an unspecified selection mechanism; the claimed identification may then hold only for particular selections rather than for the full dynamics.

    Authors: We acknowledge the referee's observation on the open uniqueness question for strict singular characteristics. The semi-flow is constructed on the space of trajectories via the closure of the generalized characteristic differential inclusion, which naturally accommodates set-valued dynamics. Invariant measures are defined as those probability measures on trajectories for which the support is invariant under the inclusion (i.e., there exists a measurable selection of the velocity field consistent with the measure). The proof that any such measure attaining c[H] must be a projected Mather measure relies on the variational characterization of Mather measures via the critical value and does not depend on a particular selection; it holds for the full set-valued dynamics. We will add a clarifying paragraph in the relevant section to make this definition explicit and address the multi-valued case directly. revision: partial

  2. Referee: [Abstract (first new contribution)] Abstract (first new contribution): the variational construction via minimizing movement scheme and Young measure compactness is asserted to yield curves satisfying the generalized characteristic differential inclusion, but the abstract supplies no error estimates, boundary-case verification, or explicit compactness constants. The full manuscript must confirm that the weak-limit argument remains valid under the precise regularity and growth conditions stated for the Hamiltonian.

    Authors: The full manuscript contains the complete proof of the minimizing-movement construction. Under the standard convexity and superlinearity assumptions on the Hamiltonian (as in weak KAM theory), the approximate solutions are uniformly bounded in suitable Sobolev spaces, and Young measure compactness yields a limiting curve satisfying the differential inclusion. The argument is qualitative and does not provide quantitative error estimates or explicit constants, as the goal is existence rather than rates. We confirm that the weak-limit step holds under the precise regularity and growth conditions stated in the paper. No changes to the abstract are necessary, though a brief reference to the assumptions could be added if the referee prefers. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via compactness and prior weak KAM theory

full rationale

The paper's two new contributions—a variational construction of generalized characteristics via minimizing movements and Young-measure compactness, plus the characterization of GHGF semi-flow invariant measures attaining c[H] as projected Mather measures—are established through standard compactness arguments and the existing framework of weak KAM theory. No step reduces a claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the dynamical characterization is independent of the input data used to define c[H] and does not rename or smuggle in prior results by construction. The explicit listing of uniqueness of strict singular characteristics as an open problem further indicates the work does not presuppose its own conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard domain assumptions from weak KAM theory; no new free parameters or invented entities are introduced in the described results.

axioms (1)
  • domain assumption Hamiltonian H is C^2, convex, and superlinear in the momentum variable, as required for existence of Mather measures and well-posedness of GHGF.
    These are the background conditions under which the semi-flow, critical value c[H], and projected Mather measures are defined in the abstract.

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Works this paper leans on

90 extracted references · 90 canonical work pages

  1. [1]

    Structural properties of singularities of semi- concave functions.Ann

    Paolo Albano and Piermarco Cannarsa. Structural properties of singularities of semi- concave functions.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28(4):719–740, 1999

    work page 1999

  2. [2]

    Propagation of singularities for solutions of nonlinear first order partial differential equations.Arch

    Paolo Albano and Piermarco Cannarsa. Propagation of singularities for solutions of nonlinear first order partial differential equations.Arch. Ration. Mech. Anal., 162(1):1–23, 2002

    work page 2002

  3. [3]

    Long time behaviour of generalised gradient flows via occupational measures

    Paolo Albano, Piermarco Cannarsa, Wei Cheng, and Cristian Mendico. Long time behaviour of generalised gradient flows via occupational measures. preprint, arXiv:2410.20943v2, 2024

    work page arXiv 2024

  4. [4]

    Sin- gular gradient flow of the distance function and homotopy equivalence.Math

    Paolo Albano, Piermarco Cannarsa, Khai Tien Nguyen, and Carlo Sinestrari. Sin- gular gradient flow of the distance function and homotopy equivalence.Math. Ann., 356(1):23–43, 2013. 42 WEI CHENG AND JIAHUI HONG

    work page 2013

  5. [5]

    Springer, 2021

    Luigi Ambrosio, Elia Bru´ e, and Daniele Semola.Lectures on optimal transport, volume 130 ofUnitext. Springer, 2021

    work page 2021

  6. [6]

    On the propagation of singularities of semi-convex functions.Ann

    Luigi Ambrosio, Piermarco Cannarsa, and Halil Mete Soner. On the propagation of singularities of semi-convex functions.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 20(4):597–616, 1993

    work page 1993

  7. [7]

    Lectures in Mathematics ETH Z¨ urich

    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¨ urich. Birkh¨ auser Verlag, Basel, second edition, 2008

    work page 2008

  8. [8]

    Fabio Ancona, Piermarco Cannarsa, and Khai T. Nguyen. Compactness estimates for Hamilton-Jacobi equations depending on space.Bull. Inst. Math. Acad. Sin. (N.S.), 11(1):63–113, 2016

    work page 2016

  9. [9]

    Fabio Ancona, Piermarco Cannarsa, and Khai T. Nguyen. Quantitative compactness estimates for Hamilton-Jacobi equations.Arch. Ration. Mech. Anal., 219(2):793–828, 2016

    work page 2016

  10. [10]

    M.-C. Arnaud. Pseudographs and the Lax-Oleinik semi-group: a geometric and dy- namical interpretation.Nonlinearity, 24(1):71–78, 2011

    work page 2011

  11. [11]

    Invariant measures of differential inclusions applied to singular pertur- bations.J

    Zvi Artstein. Invariant measures of differential inclusions applied to singular pertur- bations.J. Differential Equations, 152(2):289–307, 1999

    work page 1999

  12. [12]

    Systems & Control: Foundations & Applications

    Martino Bardi and Italo Capuzzo-Dolcetta.Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems & Control: Foundations & Applications. Birkh¨ auser Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia

    work page 1997

  13. [13]

    Existence ofC 1,1 critical sub-solutions of the Hamilton-Jacobi equa- tion on compact manifolds.Ann

    Patrick Bernard. Existence ofC 1,1 critical sub-solutions of the Hamilton-Jacobi equa- tion on compact manifolds.Ann. Sci. ´Ecole Norm. Sup. (4), 40(3):445–452, 2007

    work page 2007

  14. [14]

    The Lax-Oleinik semi-group: a Hamiltonian point of view.Proc

    Patrick Bernard. The Lax-Oleinik semi-group: a Hamiltonian point of view.Proc. Roy. Soc. Edinburgh Sect. A, 142(6):1131–1177, 2012

    work page 2012

  15. [15]

    The Monge problem for supercritical Ma˜ n´ e po- tentials on compact manifolds.Adv

    Patrick Bernard and Boris Buffoni. The Monge problem for supercritical Ma˜ n´ e po- tentials on compact manifolds.Adv. Math., 207(2):691–706, 2006

    work page 2006

  16. [16]

    Optimal mass transportation and Mather theory

    Patrick Bernard and Boris Buffoni. Optimal mass transportation and Mather theory. J. Eur. Math. Soc. (JEMS), 9(1):85–121, 2007

    work page 2007

  17. [17]

    Weak KAM pairs and Monge-Kantorovich duality

    Patrick Bernard and Boris Buffoni. Weak KAM pairs and Monge-Kantorovich duality. InAsymptotic analysis and singularities—elliptic and parabolic PDEs and related problems, volume 47 ofAdv. Stud. Pure Math., pages 397–420. Math. Soc. Japan, Tokyo, 2007

    work page 2007

  18. [18]

    SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t, x).SIAM J

    Stefano Bianchini and Daniela Tonon. SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t, x).SIAM J. Math. Anal., 44(3):2179–2203, 2012

    work page 2012

  19. [19]

    Perestroikas of shocks and singularities of minimum functions.Phys

    Ilya Aleksandrovich Bogaevsky. Perestroikas of shocks and singularities of minimum functions.Phys. D, 173(1-2):1–28, 2002

    work page 2002

  20. [20]

    Discontinuous gradient differential equations, and trajectories in the calculus of variations.Mat

    Ilya Aleksandrovich Bogaevsky. Discontinuous gradient differential equations, and trajectories in the calculus of variations.Mat. Sb., 197(12):11–42, 2006

    work page 2006

  21. [21]

    Semiconcavity and sensitivity analysis in mean-field optimal control and applications.J

    Benoˆ ıt Bonnet and H´ el` ene Frankowska. Semiconcavity and sensitivity analysis in mean-field optimal control and applications.J. Math. Pures Appl. (9), 157:282–345, 2022

    work page 2022

  22. [22]

    Generalized characteristics and Lax-Oleinik operators: global theory.Calc

    Piermarco Cannarsa and Wei Cheng. Generalized characteristics and Lax-Oleinik operators: global theory.Calc. Var. Partial Differential Equations, 56(5):Art. 125, 31, 2017

    work page 2017

  23. [23]

    On and beyond propagation of singularities of viscosity solutions

    Piermarco Cannarsa and Wei Cheng. On and beyond propagation of singularities of viscosity solutions. InProceedings of the International Consortium of Chinese Math- ematicians 2017, pages 141–157. Int. Press, Boston, MA, 2020

    work page 2017

  24. [24]

    Local singular characteristics onR 2.Boll

    Piermarco Cannarsa and Wei Cheng. Local singular characteristics onR 2.Boll. Unione Mat. Ital., 14(3):483–504, 2021

    work page 2021

  25. [25]

    Singularities of Solutions of Hamilton–Jacobi Equations.Milan J

    Piermarco Cannarsa and Wei Cheng. Singularities of Solutions of Hamilton–Jacobi Equations.Milan J. Math., 89(1):187–215, 2021. GENERALIZED HAMILTONIAN GRADIENT FLOW 43

    work page 2021

  26. [26]

    On the topology of the set of singularities of a solution to the Hamilton-Jacobi equation.C

    Piermarco Cannarsa, Wei Cheng, and Albert Fathi. On the topology of the set of singularities of a solution to the Hamilton-Jacobi equation.C. R. Math. Acad. Sci. Paris, 355(2):176–180, 2017

    work page 2017

  27. [27]

    Singularities of solutions of time dependent Hamilton-Jacobi equations

    Piermarco Cannarsa, Wei Cheng, and Albert Fathi. Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry.Publ. Math. Inst. Hautes ´Etudes Sci., 133(1):327–366, 2021

    work page 2021

  28. [28]

    Topological and control theoretic properties of Hamilton–Jacobi equations via Lax-Oleinik commutators.Nonlinear Anal

    Piermarco Cannarsa, Wei Cheng, and Jiahui Hong. Topological and control theoretic properties of Hamilton–Jacobi equations via Lax-Oleinik commutators.Nonlinear Anal. Real World Appl., 84:Paper No. 104282, 2025

    work page 2025

  29. [29]

    Variational construction of generalized characteristics and maximal slope curve

    Piermarco Cannarsa, Wei Cheng, Jiahui Hong, and Kaizhi Wang. Variational construction of generalized characteristics and maximal slope curve. preprint, arXiv:2409.00961, 2024

    work page arXiv 2024

  30. [30]

    Global propagation of singularities for magnetic mechanical systems

    Piermarco Cannarsa, Wei Cheng, Jiahui Hong, and Wenxue Wei. Global propagation of singularities for magnetic mechanical systems. preprint, arXiv:2509.13433, 2025

    work page arXiv 2025

  31. [31]

    Herglotz’ variational principle and Lax-Oleinik evolution.J

    Piermarco Cannarsa, Wei Cheng, Liang Jin, Kaizhi Wang, and Jun Yan. Herglotz’ variational principle and Lax-Oleinik evolution.J. Math. Pures Appl. (9), 141:99–136, 2020

    work page 2020

  32. [32]

    Singularities and their propagation in optimal transport.Math

    Piermarco Cannarsa, Wei Cheng, Tianqi Shi, and Wenxue Wei. Singularities and their propagation in optimal transport.Math. Ann., 393(2):2095–2155, 2025

    work page 2095

  33. [33]

    Herglotz’ generalized variational principle and contact type Hamilton-Jacobi equations

    Piermarco Cannarsa, Wei Cheng, Kaizhi Wang, and Jun Yan. Herglotz’ generalized variational principle and contact type Hamilton-Jacobi equations. InTrends in control theory and partial differential equations, volume 32 ofSpringer INdAM Ser., pages 39–67. Springer, Cham, 2019

    work page 2019

  34. [34]

    Some characterizations of optimal tra- jectories in control theory.SIAM J

    Piermarco Cannarsa and Halina Frankowska. Some characterizations of optimal tra- jectories in control theory.SIAM J. Control Optim., 29(6):1322–1347, 1991

    work page 1991

  35. [35]

    Birkh¨ auser Boston, Inc., Boston, MA, 2004

    Piermarco Cannarsa and Carlo Sinestrari.Semiconcave functions, Hamilton-Jacobi equations, and optimal control, volume 58 ofProgress in Nonlinear Differential Equa- tions and their Applications. Birkh¨ auser Boston, Inc., Boston, MA, 2004

    work page 2004

  36. [36]

    On the singularities of the viscosity solu- tions to Hamilton-Jacobi-Bellman equations.Indiana Univ

    Piermarco Cannarsa and Halil Mete Soner. On the singularities of the viscosity solu- tions to Hamilton-Jacobi-Bellman equations.Indiana Univ. Math. J., 36(3):501–524, 1987

    work page 1987

  37. [37]

    Generalized one-sided estimates for solu- tions of Hamilton-Jacobi equations and applications.Nonlinear Anal., 13(3):305–323, 1989

    Piermarco Cannarsa and Halil Mete Soner. Generalized one-sided estimates for solu- tions of Hamilton-Jacobi equations and applications.Nonlinear Anal., 13(3):305–323, 1989

    work page 1989

  38. [38]

    Singular dynamics for semiconcave functions.J

    Piermarco Cannarsa and Yifeng Yu. Singular dynamics for semiconcave functions.J. Eur. Math. Soc. (JEMS), 11(5):999–1024, 2009

    work page 2009

  39. [39]

    Lasry-Lions, Lax-Oleinik and generalized characteristics

    Cui Chen and Wei Cheng. Lasry-Lions, Lax-Oleinik and generalized characteristics. Sci. China Math., 59(9):1737–1752, 2016

    work page 2016

  40. [40]

    Local strict singular characteristics: Cauchy problem with smooth initial data.J

    Wei Cheng and Jiahui Hong. Local strict singular characteristics: Cauchy problem with smooth initial data.J. Differential Equations, 328:326–353, 2022

    work page 2022

  41. [41]

    Local strict singular characteristics II: existence for sta- tionary equations onR 2.NoDEA Nonlinear Differential Equations Appl., 30(5):Paper No

    Wei Cheng and Jiahui Hong. Local strict singular characteristics II: existence for sta- tionary equations onR 2.NoDEA Nonlinear Differential Equations Appl., 30(5):Paper No. 64, 2023

    work page 2023

  42. [42]

    Differentiability properties for a class of non-convex functions.Calc

    Giovanni Colombo and Antonio Marigonda. Differentiability properties for a class of non-convex functions.Calc. Var. Partial Differential Equations, 25(1):1–31, 2006

    work page 2006

  43. [43]

    Contreras, A

    G. Contreras, A. Figalli, and L. Rifford. Generic hyperbolicity of Aubry sets on sur- faces.Invent. Math., 200(1):201–261, 2015

    work page 2015

  44. [44]

    [CK04]Chi, D.P.andKwon, D., Sturmian words,β-shifts, and transcendence.Theoret

    Gonzalo Contreras. Proof of theC 2 Ma˜ n´ e’s conjecture on surfaces. preprint, arXiv:2408.01009, 2024

    work page arXiv 2024

  45. [45]

    Lagrangian flows: the dy- namics of globally minimizing orbits

    Gonzalo Contreras, Jorge Delgado, and Renato Iturriaga. Lagrangian flows: the dy- namics of globally minimizing orbits. II.Bol. Soc. Brasil. Mat. (N.S.), 28(2):155–196, 1997. 44 WEI CHENG AND JIAHUI HONG

    work page 1997

  46. [46]

    22o Col´ oquio Brasileiro de Matem´ atica

    Gonzalo Contreras and Renato Iturriaga.Global minimizers of autonomous La- grangians. 22o Col´ oquio Brasileiro de Matem´ atica. [22nd Brazilian Mathematics Col- loquium]. Instituto de Matem´ atica Pura e Aplicada (IMPA), Rio de Janeiro, 1999

    work page 1999

  47. [47]

    Crandall, Lawrence C

    Michael G. Crandall, Lawrence C. Evans, and Pierre-Louis Lions. Some proper- ties of viscosity solutions of Hamilton-Jacobi equations.Trans. Amer. Math. Soc., 282(2):487–502, 1984

    work page 1984

  48. [48]

    Crandall and Pierre-Louis Lions

    Michael G. Crandall and Pierre-Louis Lions. Viscosity solutions of Hamilton-Jacobi equations.Trans. Amer. Math. Soc., 277(1):1–42, 1983

    work page 1983

  49. [49]

    Dafermos

    Constantine M. Dafermos. Generalized characteristics and the structure of solutions of hyperbolic conservation laws.Indiana Univ. Math. J., 26(6):1097–1119, 1977

    work page 1977

  50. [50]

    Conver- gence of the solutions of the discounted Hamilton-Jacobi equation: convergence of the discounted solutions.Invent

    Andrea Davini, Albert Fathi, Renato Iturriaga, and Maxime Zavidovique. Conver- gence of the solutions of the discounted Hamilton-Jacobi equation: convergence of the discounted solutions.Invent. Math., 206(1):29–55, 2016

    work page 2016

  51. [51]

    The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data.Comm

    Avron Douglis. The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data.Comm. Pure Appl. Math., 14:267– 284, 1961

    work page 1961

  52. [52]

    Evans.Weak convergence methods for nonlinear partial differential equa- tions, volume 74 ofCBMS Regional Conference Series in Mathematics

    Lawrence C. Evans.Weak convergence methods for nonlinear partial differential equa- tions, volume 74 ofCBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the Amer- ican Mathematical Society, Providence, RI, 1990

    work page 1990

  53. [53]

    Weak KAM theorem in Lagrangian dynamics

    Albert Fathi. Weak KAM theorem in Lagrangian dynamics. Cambridge University Press, Cambridge (to appear)

    work page

  54. [54]

    Solutions KAM faibles conjugu´ ees et barri` eres de Peierls.C

    Albert Fathi. Solutions KAM faibles conjugu´ ees et barri` eres de Peierls.C. R. Acad. Sci. Paris S´ er. I Math., 325(6):649–652, 1997

    work page 1997

  55. [55]

    Th´ eor` eme KAM faible et th´ eorie de Mather sur les syst` emes lagrangiens

    Albert Fathi. Th´ eor` eme KAM faible et th´ eorie de Mather sur les syst` emes lagrangiens. C. R. Acad. Sci. Paris S´ er. I Math., 324(9):1043–1046, 1997

    work page 1997

  56. [56]

    Orbites h´ et´ eroclines et ensemble de Peierls.C

    Albert Fathi. Orbites h´ et´ eroclines et ensemble de Peierls.C. R. Acad. Sci. Paris S´ er. I Math., 326(10):1213–1216, 1998

    work page 1998

  57. [57]

    Sur la convergence du semi-groupe de Lax-Oleinik.C

    Albert Fathi. Sur la convergence du semi-groupe de Lax-Oleinik.C. R. Acad. Sci. Paris S´ er. I Math., 327(3):267–270, 1998

    work page 1998

  58. [58]

    Fleming and William M

    Wendell H. Fleming and William M. McEneaney. A max-plus-based algorithm for a Hamilton-Jacobi-Bellman equation of nonlinear filtering.SIAM J. Control Optim., 38(3):683–710, 2000

    work page 2000

  59. [59]

    Fleming and Halil Mete Soner.Controlled Markov processes and viscosity solutions, volume 25 ofStochastic Modelling and Applied Probability

    Wendell H. Fleming and Halil Mete Soner.Controlled Markov processes and viscosity solutions, volume 25 ofStochastic Modelling and Applied Probability. Springer, New York, second edition, 2006

    work page 2006

  60. [60]

    Hamilton-Jacobi equations in the Wasserstein space.Methods Appl

    Wilfrid Gangbo, Truyen Nguyen, and Adrian Tudorascu. Hamilton-Jacobi equations in the Wasserstein space.Methods Appl. Anal., 15(2):155–183, 2008

    work page 2008

  61. [61]

    On differentiability in the Wasserstein space and well-posedness for Hamilton-Jacobi equations.J

    Wilfrid Gangbo and Adrian Tudorascu. On differentiability in the Wasserstein space and well-posedness for Hamilton-Jacobi equations.J. Math. Pures Appl. (9), 125:119– 174, 2019

    work page 2019

  62. [62]

    M. M. Hrustal¨ ev. Necessary and sufficient conditions for optimality in the form of the Bellman equation.Dokl. Akad. Nauk SSSR, 242(5):1023–1026, 1978

    work page 1978

  63. [63]

    Particle dynamics inside shocks in Hamilton-Jacobi equations.Philos

    Konstantin Khanin and Andrei Sobolevski. Particle dynamics inside shocks in Hamilton-Jacobi equations.Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 368(1916):1579–1593, 2010

    work page 1916

  64. [64]

    On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations.Arch

    Konstantin Khanin and Andrei Sobolevski. On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations.Arch. Ration. Mech. Anal., 219(2):861–885, 2016

    work page 2016

  65. [65]

    S. N. Kruˇ zkov. Generalized solutions of Hamilton-Jacobi equations of eikonal type. I. Statement of the problems; existence, uniqueness and stability theorems; certain properties of the solutions.Mat. Sb. (N.S.), 98(140)(3(11)):450–493, 496, 1975. GENERALIZED HAMILTONIAN GRADIENT FLOW 45

    work page 1975

  66. [66]

    N. V. Krylov.Nonlinear elliptic and parabolic equations of the second order, volume 7 ofMathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dor- drecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytski˘ ı]

    work page 1987

  67. [67]

    Li and L

    Y. Li and L. Nirenberg. The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations.Comm. Pure Appl. Math., 58(1):85–146, 2005

    work page 2005

  68. [68]

    On the minimizing measures of Lagrangian dynamical systems.Non- linearity, 5(3):623–638, 1992

    Ricardo Ma˜ n´ e. On the minimizing measures of Lagrangian dynamical systems.Non- linearity, 5(3):623–638, 1992

    work page 1992

  69. [69]

    Generic properties and problems of minimizing measures of La- grangian systems.Nonlinearity, 9(2):273–310, 1996

    Ricardo Ma˜ n´ e. Generic properties and problems of minimizing measures of La- grangian systems.Nonlinearity, 9(2):273–310, 1996

    work page 1996

  70. [70]

    Lagrangian flows: the dynamics of globally minimizing orbits.Bol

    Ricardo Ma˜ n´ e. Lagrangian flows: the dynamics of globally minimizing orbits.Bol. Soc. Brasil. Mat. (N.S.), 28(2):141–153, 1997

    work page 1997

  71. [71]

    Hamilton-Jacobi equations and dis- tance functions on Riemannian manifolds.Appl

    Carlo Mantegazza and Andrea Carlo Mennucci. Hamilton-Jacobi equations and dis- tance functions on Riemannian manifolds.Appl. Math. Optim., 47(1):1–25, 2003

    work page 2003

  72. [72]

    Aubry-Mather theory for conformally symplec- tic systems.Comm

    Stefano Mar` o and Alfonso Sorrentino. Aubry-Mather theory for conformally symplec- tic systems.Comm. Math. Phys., 354(2):775–808, 2017

    work page 2017

  73. [73]

    John N. Mather. Action minimizing invariant measures for positive definite La- grangian systems.Math. Z., 207(2):169–207, 1991

    work page 1991

  74. [74]

    John N. Mather. Variational construction of connecting orbits.Ann. Inst. Fourier (Grenoble), 43(5):1349–1386, 1993

    work page 1993

  75. [75]

    Mather and Giovanni Forni

    John N. Mather and Giovanni Forni. Action minimizing orbits in Hamiltonian sys- tems. InTransition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), volume 1589 ofLecture Notes in Math., pages 92–186. Springer, Berlin, 1994

    work page 1991

  76. [76]

    Regularity and variationality of solutions to Hamilton-Jacobi equations

    Andrea Carlo Giuseppe Mennucci. Regularity and variationality of solutions to Hamilton-Jacobi equations. I. Regularity.ESAIM Control Optim. Calc. Var., 10(3):426–451, 2004

    work page 2004

  77. [77]

    Semiconcave functions in Alexandrov’s geometry

    Anton Petrunin. Semiconcave functions in Alexandrov’s geometry. InSurveys in dif- ferential geometry. Vol. XI, volume 11 ofSurv. Differ. Geom., pages 137–201. Int. Press, Somerville, MA, 2007

    work page 2007

  78. [78]

    Existence of Lipschitz and semiconcave control-Lyapunov functions

    Ludovic Rifford. Existence of Lipschitz and semiconcave control-Lyapunov functions. SIAM J. Control Optim., 39(4):1043–1064, 2000

    work page 2000

  79. [79]

    Semiconcave control-Lyapunov functions and stabilizing feedbacks

    Ludovic Rifford. Semiconcave control-Lyapunov functions and stabilizing feedbacks. SIAM J. Control Optim., 41(3):659–681, 2002

    work page 2002

  80. [80]

    Tyrrell Rockafellar

    R. Tyrrell Rockafellar. Favorable classes of Lipschitz-continuous functions in subgra- dient optimization. InProgress in nondifferentiable optimization, volume 8 ofIIASA Collaborative Proc. Ser. CP-82, pages 125–143. Internat. Inst. Appl. Systems Anal., Laxenburg, 1982

    work page 1982

Showing first 80 references.