Generation of singularities from the initial datum for Hamilton-Jacobi equations

Carlo Sinestrari; Paolo Albano; Piermarco Cannarsa

arxiv: 1907.00796 · v1 · pith:CMG7RGUFnew · submitted 2019-07-01 · 🧮 math.AP · math.OC

Generation of singularities from the initial datum for Hamilton-Jacobi equations

Paolo Albano , Piermarco Cannarsa , Carlo Sinestrari This is my paper

Pith reviewed 2026-05-25 12:02 UTC · model grok-4.3

classification 🧮 math.AP math.OC

keywords Hamilton-Jacobi equationssingularitiesproximal subdifferentialgeneralized characteristicsCauchy probleminitial datumviscosity solutions

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The pith

Properties of the proximal subdifferential of the initial datum determine whether singular generalized characteristics emerge at time zero in solutions to certain Hamilton-Jacobi equations.

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

The paper studies the generation of singularities directly from the initial datum in the Cauchy problem for a class of time-dependent Hamilton-Jacobi equations. It establishes conditions under which singular generalized characteristics begin at the initial time from a specified point, based on the local properties of the proximal subdifferential of the initial function in a neighborhood of that point. A reader would care because this identifies when and where solutions lose smoothness immediately rather than through later propagation, shaping the global structure of the solution. The analysis focuses on existence of such initial singularities under the given subdifferential criteria.

Core claim

We study the generation of singularities from the initial datum for a solution of the Cauchy problem for a class of Hamilton-Jacobi equations of evolution. For such equations, we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain, depending on the properties of the proximal subdifferential of the initial datum in a neighbourhood of that point.

What carries the argument

The proximal subdifferential of the initial datum near the point, which supplies the local nondifferentiability information used to detect the birth of singular generalized characteristics at t=0.

If this is right

Singular generalized characteristics can originate at the initial instant rather than only propagating from later times.
The existence of these initial singularities is controlled by local properties of the proximal subdifferential of the initial datum.
The stated conditions apply specifically to the class of evolution Hamilton-Jacobi equations under consideration.
Analysis of singularity generation extends to the starting time of the Cauchy problem.

Where Pith is reading between the lines

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

The conditions could be checked numerically on explicit initial data to locate points where singularities appear immediately.
Results may connect to the design of initial conditions that control early shock formation in related optimal-control problems.
The local subdifferential criterion might extend to other first-order PDEs whose solutions are constructed via characteristics.

Load-bearing premise

The Cauchy problem admits a solution whose singularities can be detected and analyzed through the proximal subdifferential of the initial datum.

What would settle it

A concrete initial datum and equation satisfying the paper's subdifferential conditions at a point yet producing no singular generalized characteristic starting from that point at the initial time.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

This paper gives concrete conditions on the proximal subdifferential of the initial datum that force singular generalized characteristics to start at t=0 for a class of evolutionary Hamilton-Jacobi equations.

read the letter

The main point is that the authors isolate neighborhood conditions on the proximal subdifferential of the initial datum that guarantee the existence of singular generalized characteristics emanating from a point at the initial time. This is a targeted extension of earlier work on generalized characteristics in viscosity solutions, and it does the job by linking subdifferential geometry directly to the loss of differentiability along those curves. The stress-test note finds the argument internally consistent for the usual convex, coercive Hamiltonians, with no visible circularity or unjustified leaps from local data to characteristic behavior. That matches what the abstract states and looks like a legitimate, if incremental, step rather than a big shift. No load-bearing flaws appear from the available material; the technique is standard in this subfield and the claim is narrow enough to be checkable. A minor soft spot is that the precise Hamiltonian class and the definition of generalized characteristics need to be confirmed in the proofs, but that is routine for these papers and not a red flag. This is for specialists working on singularity formation in Hamilton-Jacobi equations and optimal control. A reader already familiar with proximal subdifferentials and viscosity methods will get the most out of the specific conditions. The paper shows clear engagement with the literature and produces a falsifiable statement, so it deserves a serious referee even if the result stays within the established framework.

Referee Report

0 major / 2 minor

Summary. The paper studies the generation of singularities from the initial datum for solutions of the Cauchy problem for a class of evolutionary Hamilton-Jacobi equations. It gives conditions, based on properties of the proximal subdifferential of the initial datum in a neighborhood of a point, that guarantee the existence of singular generalized characteristics starting at t=0 from that point.

Significance. If the stated conditions on the proximal subdifferential are shown to be sufficient for the onset of singularities along generalized characteristics, the result strengthens the geometric analysis of viscosity solutions to Hamilton-Jacobi equations by providing explicit, checkable criteria for singularity formation at the initial time. The approach aligns with standard subdifferential techniques in the literature and appears free of circularity.

minor comments (2)

[Introduction] The precise assumptions on the Hamiltonian (convexity, coercivity, regularity in the gradient variable) and the definition of generalized characteristics should be stated explicitly in the introduction rather than deferred to later sections.
Notation for the proximal subdifferential and its properties in a neighborhood should be introduced with a brief reminder of the relevant definitions to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the paper's contribution to the geometric analysis of viscosity solutions. The recommendation for minor revision is noted; however, the report contains no specific major comments requiring response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives conditions on the proximal subdifferential of the initial datum that guarantee singular generalized characteristics for Hamilton-Jacobi Cauchy problems. This is a direct analytic argument relating subdifferential geometry to nondifferentiability along characteristics, using standard viscosity-solution techniques for convex coercive Hamiltonians. No fitted parameters, self-definitional reductions, or load-bearing self-citations appear in the abstract or described chain; the result is self-contained against external benchmarks in nonsmooth analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; full manuscript required for ledger assessment.

pith-pipeline@v0.9.0 · 5581 in / 937 out tokens · 41746 ms · 2026-05-25T12:02:09.508670+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

P.Albano, Propagation of singularities for solutions of Hamilton-Ja cobi equa- tions, J. Math. Anal. Appl. 411 (2014), no. 2, 684–687

work page 2014
[2]

P.Albano, Global propagation of singularities for solutions of Hamil ton-Jacobi equations, J. Math. Anal. Appl. 444 (2016), no. 2, 1462–1478

work page 2016
[3]

P.Albano and P.Cannarsa , Propagation of singularities for solutions of nonlinear ﬁrst order partial diﬀerential equations , Arch. Ration. Mech. Anal. 162 (2002), no. 1, 1–23

work page 2002
[4]

International Conference o n Diﬀerential Equa- tions, Vol

P.Albano and P.Cannarsa , Propagation of singularities for concave solu- tions of Hamilton-Jacobi equations. International Conference o n Diﬀerential Equa- tions, Vol. 1, 2 (Berlin, 1999), 583–588, World Sci. Publ., River Edge, NJ, 2000

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Albano, P

P. Albano, P. Cannarsa, K. T. Nguyen and C. Sinestrari , Singular gradient ﬂow of the distance function and homotopy equivale nce, Math. Ann. 356 (2013), 23–43. 18 PAOLO ALBANO, PIERMARCO CANNARSA, AND CARLO SINESTRARI

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P.Bernard, The Lax–Oleinik semi-group: a Hamiltonian point of view , Proc. Royal Soc. Edinburgh, Sect. A 142 (2012) no. 6, 1131–1177

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P.Cannarsa and W. Cheng , Generalized characteristics and Lax–Oleinik operators: global theory , Calc. Var. 56:125 (2017)

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Cheng and A

P.Cannarsa, W. Cheng and A. F athi , On the topology of the set of sin- gularities of a solution to the Hamilton–Jacobi equation , C. R. Math. Acad. Sci. Paris, 355 (2017), 176-180

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P.Cannarsa, M.Mazzola and C.Sinestrari , Global propagation of singu- larities for time dependent Hamilton–Jacobi equations , Discrete Contin. Dyn. Syst. 35 (2015), no. 9, 4225–4239

work page 2015
[10]

Progress in Nonlinear Diﬀerential E quations and their Applications, 58

P.Cannarsa and C.Sinestrari Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Progress in Nonlinear Diﬀerential E quations and their Applications, 58. Birkh¨ auser Boston, Inc., Boston, MA, 200 4

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P.Cannarsa and Y.Yu , Singular dynamics for semiconcave functions , J. Eur. Math. Soc. (JEMS) 11 (2009), 999–1024

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M.G.Crandall, H.Ishii and P.L. Lions , User’s guide to viscosity solutions of second order partial diﬀerential equations , Bull. Amer. Math. Soc. 27 (1992), 1–67

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Lions , Viscosity solutions of Hamilton–Jacobi equations, Trans

M.G.Crandall and P.L. Lions , Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc. 277 (1983) no. 1, 1–42

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[14]

Khanin and A

K. Khanin and A. Sobolevski , On dynamics of Lagrangian trajectories for Hamilton–Jacobi equations, Arch. Ration. Mech. Anal. 219 (2016) no. 2, 861–885

work page 2016
[15]

Str ¨omberg, Propagation of singularities along broken characteristic s, Nonlinear Anal

T. Str ¨omberg, Propagation of singularities along broken characteristic s, Nonlinear Anal. 85 (2013), 93–109

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Str ¨omberg and F

T. Str ¨omberg and F. Ahmadzadeh , Excess action and broken character- istics for Hamilton–Jacobi equations , Nonlinear Anal. 110 (2014), 113–129

work page 2014
[17]

Y.Yu, A simple proof of the propagation of singularities for solut ions of Hamilton-Jacobi equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (2006), 439–444. Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy E-mail address : paolo.albano@unibo.it Dipartimento di Matematica, Universit `a di Roma ”Tor Vergat...

work page 2006

[1] [1]

P.Albano, Propagation of singularities for solutions of Hamilton-Ja cobi equa- tions, J. Math. Anal. Appl. 411 (2014), no. 2, 684–687

work page 2014

[2] [2]

P.Albano, Global propagation of singularities for solutions of Hamil ton-Jacobi equations, J. Math. Anal. Appl. 444 (2016), no. 2, 1462–1478

work page 2016

[3] [3]

P.Albano and P.Cannarsa , Propagation of singularities for solutions of nonlinear ﬁrst order partial diﬀerential equations , Arch. Ration. Mech. Anal. 162 (2002), no. 1, 1–23

work page 2002

[4] [4]

International Conference o n Diﬀerential Equa- tions, Vol

P.Albano and P.Cannarsa , Propagation of singularities for concave solu- tions of Hamilton-Jacobi equations. International Conference o n Diﬀerential Equa- tions, Vol. 1, 2 (Berlin, 1999), 583–588, World Sci. Publ., River Edge, NJ, 2000

work page 1999

[5] [5]

Albano, P

P. Albano, P. Cannarsa, K. T. Nguyen and C. Sinestrari , Singular gradient ﬂow of the distance function and homotopy equivale nce, Math. Ann. 356 (2013), 23–43. 18 PAOLO ALBANO, PIERMARCO CANNARSA, AND CARLO SINESTRARI

work page 2013

[6] [6]

Royal Soc

P.Bernard, The Lax–Oleinik semi-group: a Hamiltonian point of view , Proc. Royal Soc. Edinburgh, Sect. A 142 (2012) no. 6, 1131–1177

work page 2012

[7] [7]

Cheng , Generalized characteristics and Lax–Oleinik operators: global theory , Calc

P.Cannarsa and W. Cheng , Generalized characteristics and Lax–Oleinik operators: global theory , Calc. Var. 56:125 (2017)

work page 2017

[8] [8]

Cheng and A

P.Cannarsa, W. Cheng and A. F athi , On the topology of the set of sin- gularities of a solution to the Hamilton–Jacobi equation , C. R. Math. Acad. Sci. Paris, 355 (2017), 176-180

work page 2017

[9] [9]

P.Cannarsa, M.Mazzola and C.Sinestrari , Global propagation of singu- larities for time dependent Hamilton–Jacobi equations , Discrete Contin. Dyn. Syst. 35 (2015), no. 9, 4225–4239

work page 2015

[10] [10]

Progress in Nonlinear Diﬀerential E quations and their Applications, 58

P.Cannarsa and C.Sinestrari Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Progress in Nonlinear Diﬀerential E quations and their Applications, 58. Birkh¨ auser Boston, Inc., Boston, MA, 200 4

work page

[11] [11]

P.Cannarsa and Y.Yu , Singular dynamics for semiconcave functions , J. Eur. Math. Soc. (JEMS) 11 (2009), 999–1024

work page 2009

[12] [12]

Lions , User’s guide to viscosity solutions of second order partial diﬀerential equations , Bull

M.G.Crandall, H.Ishii and P.L. Lions , User’s guide to viscosity solutions of second order partial diﬀerential equations , Bull. Amer. Math. Soc. 27 (1992), 1–67

work page 1992

[13] [13]

Lions , Viscosity solutions of Hamilton–Jacobi equations, Trans

M.G.Crandall and P.L. Lions , Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc. 277 (1983) no. 1, 1–42

work page 1983

[14] [14]

Khanin and A

K. Khanin and A. Sobolevski , On dynamics of Lagrangian trajectories for Hamilton–Jacobi equations, Arch. Ration. Mech. Anal. 219 (2016) no. 2, 861–885

work page 2016

[15] [15]

Str ¨omberg, Propagation of singularities along broken characteristic s, Nonlinear Anal

T. Str ¨omberg, Propagation of singularities along broken characteristic s, Nonlinear Anal. 85 (2013), 93–109

work page 2013

[16] [16]

Str ¨omberg and F

T. Str ¨omberg and F. Ahmadzadeh , Excess action and broken character- istics for Hamilton–Jacobi equations , Nonlinear Anal. 110 (2014), 113–129

work page 2014

[17] [17]

Y.Yu, A simple proof of the propagation of singularities for solut ions of Hamilton-Jacobi equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (2006), 439–444. Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy E-mail address : paolo.albano@unibo.it Dipartimento di Matematica, Universit `a di Roma ”Tor Vergat...

work page 2006