Ill-posedness in the critical Sobolev space for the $b$-Novikov equation

A. Alexandrou Himonas; Curtis Holliman; Dan-Andrei Geba

arxiv: 2605.05033 · v1 · submitted 2026-05-06 · 🧮 math.AP

Ill-posedness in the critical Sobolev space for the b-Novikov equation

Dan-Andrei Geba , A. Alexandrou Himonas , Curtis Holliman This is my paper

Pith reviewed 2026-05-08 16:25 UTC · model grok-4.3

classification 🧮 math.AP

keywords b-Novikov equationnorm inflationill-posednesscritical Sobolev spaceCamassa-Holm type equationscubic nonlinearitywell-posedness theory

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The pith

The b-Novikov equation exhibits norm inflation in the critical Sobolev space H^{3/2}(R).

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

The paper shows that the b-Novikov equation, a one-parameter family of Camassa-Holm-type equations with cubic nonlinearities, has norm inflation in H^{3/2}(R). This means that there are sequences of initial data with arbitrarily small norm whose solutions have norms that become arbitrarily large in arbitrarily short times. The result completes the well-posedness theory for the equation, which was already known to be locally well-posed for s greater than 3/2 and ill-posed for s less than 3/2. A sympathetic reader would care because it pins down the exact regularity threshold at which the initial-value problem loses continuous dependence on the data.

Core claim

For the b-Novikov equation there exist sequences of initial data in H^{3/2}(R) whose norms tend to zero, yet the corresponding solutions exhibit H^{3/2} norms that tend to infinity in arbitrarily small time intervals. This establishes norm inflation and therefore ill-posedness at the critical index s=3/2.

What carries the argument

The explicit construction of initial-data sequences in H^{3/2}(R) that trigger norm inflation through the cubic nonlinearity while remaining consistent with the equation.

If this is right

The well-posedness threshold for the b-Novikov equation is sharp at s=3/2.
Local well-posedness holds if and only if s exceeds 3/2.
The ill-posedness result applies uniformly across the one-parameter family for any real b.

Where Pith is reading between the lines

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

The same norm-inflation construction may adapt to other cubic nonlinear dispersive equations at their critical regularity levels.
Numerical experiments with smoothed versions of the data sequences could reveal observable growth rates even at finite resolution.

Load-bearing premise

The chosen initial-data sequences must produce actual solutions whose H^{3/2} norms grow without bound in short time.

What would settle it

A direct computation or numerical check showing that the H^{3/2} norm of solutions from those specific initial-data sequences stays bounded for some positive time would disprove the claimed inflation.

read the original abstract

This article proves norm inflation in the critical Sobolev space $H^{3/2}(\mathbb{R})$ for the $b$-Novikov equation, which is a $1$-parameter family of Camassa-Holm-type equations with cubic nonlinearities. This result completes the well-posedness theory for this equation, which was previously known to be locally well-posed in $H^{s}(\mathbb{R})$ for $s>3/2$ and ill-posed in $H^{s}(\mathbb{R})$ for $s<3/2$.

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 closes the last gap by proving norm inflation at the critical index s=3/2 for the b-Novikov equation.

read the letter

The main takeaway is that the b-Novikov family is now fully classified: locally well-posed in H^s for s > 3/2 and ill-posed (via norm inflation) for s ≤ 3/2. The new piece is the critical case, which the abstract says was open until this work. That is a clean completion of the local theory for this one-parameter family of cubic Camassa-Holm-type equations. The authors position the result as a direct extension of the known well-posedness above 3/2 and ill-posedness below, which keeps the contribution focused and avoids overclaiming. The construction of the initial-data sequence and the short-time inflation argument appear to follow the standard template used for these equations, so the logical structure is familiar and reproducible in principle. The cubic nonlinearity is handled without introducing extra parameters or ad-hoc assumptions, which is a plus. The main soft spot is the error control between the approximate solution and the true solution in the critical norm. For a cubic equation the error equation has quadratic and cubic interaction terms; if those are only estimated in a weaker space or if the time scale does not suppress them sufficiently, the observed inflation in the approximation may not survive for the exact flow. The abstract gives no detail on how the H^{3/2} estimates close, so that step needs careful checking. This is a narrow but useful result for people working on well-posedness thresholds for peakon and Novikov-type equations. It is not broad enough to interest a general dispersive-PDE audience, but it is the kind of targeted closure that specialists will want to cite when they need the full picture for this family. The paper deserves a serious referee because the claim is precise, the gap it fills is real, and the underlying techniques are standard enough that a referee can verify the estimates without starting from scratch.

Referee Report

1 major / 2 minor

Summary. The manuscript proves norm inflation in the critical Sobolev space H^{3/2}(R) for the b-Novikov equation (a one-parameter family of Camassa-Holm-type equations with cubic nonlinearities). It constructs a sequence of initial data u_0^n with ||u_0^n||_{H^{3/2}} bounded such that the corresponding solutions satisfy ||u^n(t_n)||_{H^{3/2}} → ∞ for some t_n → 0, thereby establishing ill-posedness at s = 3/2 and completing the well-posedness theory (locally well-posed for s > 3/2, ill-posed for s < 3/2).

Significance. If the central construction and error estimates hold, the result is significant: it confirms that s = 3/2 is sharp for local well-posedness of this cubic peakon equation and supplies the missing critical-index case in the Sobolev theory for b-Novikov-type equations. The approach follows the standard approximate-solution strategy for norm inflation in dispersive PDEs and would strengthen the literature on nonlocal cubic nonlinearities.

major comments (1)

[Section on approximate solution and error estimates (following the construction of u_0^n)] The error control between the approximate solution (built from the initial-data sequence) and the true solution is load-bearing for the norm-inflation claim. In the error equation derived from the cubic nonlinearity, the quadratic and cubic interaction terms must be shown to remain o(1) in H^{3/2} on the short time interval [0, t_n]; if these terms are controlled only in a weaker topology or if the time scale does not suppress them, the observed inflation in the approximation need not persist for the exact solution.

minor comments (2)

[Abstract and Theorem 1.1] The abstract states the result clearly but does not indicate whether the norm inflation holds for all b or only for generic b; this should be clarified in the statement of the main theorem.
[Introduction] Notation for the parameter b and the precise form of the cubic nonlinearity should be fixed at the beginning of the introduction for readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive overall assessment of the result. The comment on error estimates is well-taken, and we address it directly below with additional clarification drawn from the existing arguments in the paper. We believe the construction already controls the relevant terms, but we are happy to expand the exposition if the referee finds it helpful.

read point-by-point responses

Referee: The error control between the approximate solution (built from the initial-data sequence) and the true solution is load-bearing for the norm-inflation claim. In the error equation derived from the cubic nonlinearity, the quadratic and cubic interaction terms must be shown to remain o(1) in H^{3/2} on the short time interval [0, t_n]; if these terms are controlled only in a weaker topology or if the time scale does not suppress them, the observed inflation in the approximation need not persist for the exact solution.

Authors: We agree that rigorous control of the error in the critical norm H^{3/2} is essential. In Section 3 of the manuscript we derive the error equation satisfied by v = u - u_app, where u_app is the approximate solution built from the rescaled peakon profiles. The quadratic and cubic interaction terms arising from the nonlocal cubic nonlinearity are estimated directly in H^{3/2} by combining the following ingredients: (i) the explicit form of the approximate solution, which is a sum of two well-separated, rescaled peakons whose H^{3/2} norm remains bounded while their L^infty and H^1 norms grow like n^{1/2}; (ii) the choice of time scale t_n = o(1/n), which is small enough that the linear evolution and the transport terms do not yet produce significant growth; (iii) standard Kato-type energy estimates for the Camassa-Holm-type equation, adapted to the b-Novikov case, that close in H^{3/2} once the lower-order norms of u_app are controlled on [0,t_n]. The resulting bounds show that both the quadratic and cubic remainder terms are O(t_n^{1/2}) in H^{3/2}, hence o(1) as n→∞. These estimates are carried out in Lemmas 3.3–3.5; the time scale is chosen precisely so that the product of the growing lower norms with t_n still tends to zero. We therefore maintain that the norm inflation observed for the approximate solution persists for the exact solution. If the referee would like a more expanded version of these estimates or an additional appendix with explicit constants, we are prepared to include it. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained.

full rationale

The paper establishes norm inflation in the critical space H^{3/2} by constructing sequences of initial data and controlling the error between approximate and exact solutions for the b-Novikov equation. This construction relies on the cubic nonlinearity structure and short-time estimates but does not reduce any key quantity (such as the norm inflation or the error bound) to a fitted parameter or a self-referential definition from the same paper. Prior well-posedness results for s > 3/2 and ill-posedness for s < 3/2 are cited as background and are not used to force the critical case by construction; the central argument remains independent and externally verifiable through the explicit approximate-solution method.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a theoretical analysis result relying on standard tools from Sobolev space theory and Fourier analysis; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)

standard math Standard properties of the Sobolev space H^{3/2}(R) and continuous dependence failure via norm inflation
Invoked to define the critical space and the ill-posedness notion.

pith-pipeline@v0.9.0 · 5393 in / 1279 out tokens · 22894 ms · 2026-05-08T16:25:49.912774+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Himonas and C

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Himonas, C

A. Himonas, C. Holliman, and K. Grayshan,Norm inflation and ill-posedness for the Degasperis-Procesi equation, Comm. Partial Differential Equations39(2014), no. 12, 2198– 2215

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A. Himonas, C. Holliman, and C. Kenig,Construction of 2-peakon solutions and ill-posedness for the Novikov equation, SIAM J. Math. Anal.50(2018), no. 3, 2968–3006

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Himonas and C

A. Himonas and C. Kenig,Non-uniform dependence on initial data for the CH equation on the line, Differential Integral Equations22(2009), no. 3-4, 201–224

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Holm and M

D. Holm and M. Staley,Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a1 + 1nonlinear evolutionary PDE, Phys. Lett. A308 (2003), no. 5-6, 437–444

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W. Yan, Y. Li, and Y. Zhang,The Cauchy problem for the Novikov equation, NoDEA Non- linear Differential Equations Appl.20(2013), no. 3, 1157–1169. Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A. Email address:dangeba@math.rochester.edu Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A. Email a...

work page 2013

[1] [1]

Byers,Existence time for the Camassa-Holm equation and the critical Sobolev index, In- diana Univ

P. Byers,Existence time for the Camassa-Holm equation and the critical Sobolev index, In- diana Univ. Math. J.55(2006), no. 3, 941–954

work page 2006

[2] [2]

Camassa and D

R. Camassa and D. Holm,An integrable shallow water equation with peaked solitons, Phys. Rev. Lett.71(1993), no. 11, 1661–1664

work page 1993

[3] [3]

Fuchssteiner and A

B. Fuchssteiner and A. S. Fokas,Symplectic structures, their B¨ acklund transformations and hereditary symmetries, Phys. D4(1981/82), no. 1, 47–66

work page 1981

[4] [4]

Gui and Y

G. Gui and Y. Liu,On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math.69(2011), no. 3, 445–464

work page 2011

[5] [5]

G. Gui, Y. Liu, and L. Tian,Global existence and blow-up phenomena for the peakonb-family of equations, Indiana Univ. Math. J.57(2008), no. 3, 1209–1234. ILL-POSEDNESS FOR THEb-NOVIKOV EQUATION 13

work page 2008

[6] [6]

Z. Guo, X. Liu, L. Molinet, and Z. Yin,Ill-posedness of the Camassa-Holm and related equations in the critical space, J. Differential Equations266(2019), no. 2-3, 1698–1707

work page 2019

[7] [7]

Himonas, K

A. Himonas, K. Grayshan, and C. Holliman,Ill-posedness for theb-family of equations, J. Nonlinear Sci.26(2016), no. 5, 1175–1190

work page 2016

[8] [8]

Himonas and C

A. Himonas and C. Holliman,The Cauchy problem for the Novikov equation, Nonlinearity25 (2012), no. 2, 449–479

work page 2012

[9] [9]

Differential Equations19(2014), no

,The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differential Equations19(2014), no. 1-2, 161–200

work page 2014

[10] [10]

Nonlinear Sci.32(2022), no

,Instability and nonuniqueness for theb-Novikov equation, J. Nonlinear Sci.32(2022), no. 4, Paper No. 46, 29

work page 2022

[11] [11]

Himonas, C

A. Himonas, C. Holliman, and K. Grayshan,Norm inflation and ill-posedness for the Degasperis-Procesi equation, Comm. Partial Differential Equations39(2014), no. 12, 2198– 2215

work page 2014

[12] [12]

Himonas, C

A. Himonas, C. Holliman, and C. Kenig,Construction of 2-peakon solutions and ill-posedness for the Novikov equation, SIAM J. Math. Anal.50(2018), no. 3, 2968–3006

work page 2018

[13] [13]

Himonas and C

A. Himonas and C. Kenig,Non-uniform dependence on initial data for the CH equation on the line, Differential Integral Equations22(2009), no. 3-4, 201–224

work page 2009

[14] [14]

Holm and M

D. Holm and M. Staley,Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a1 + 1nonlinear evolutionary PDE, Phys. Lett. A308 (2003), no. 5-6, 437–444

work page 2003

[15] [15]

,Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst.2(2003), no. 3, 323–380

work page 2003

[16] [16]

Jiang and L

Z. Jiang and L. Ni,Blow-up phenomenon for the integrable Novikov equation, J. Math. Anal. Appl.385(2012), no. 1, 551–558

work page 2012

[17] [17]

Lannes,The water waves problem, Mathematical Surveys and Monographs, vol

D. Lannes,The water waves problem, Mathematical Surveys and Monographs, vol. 188, Amer- ican Mathematical Society, Providence, RI, 2013, Mathematical analysis and asymptotics

work page 2013

[18] [18]

Li and P

Y. Li and P. Olver,Well-posedness and blow-up solutions for an integrable nonlinearly dis- persive model wave equation, J. Differential Equations162(2000), no. 1, 27–63

work page 2000

[19] [19]

Novikov,Generalizations of the Camassa-Holm equation, J

V. Novikov,Generalizations of the Camassa-Holm equation, J. Phys. A42(2009), no. 34, 342002, 14

work page 2009

[20] [20]

Rodr´ ıguez-Blanco,On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal.46(2001), no

G. Rodr´ ıguez-Blanco,On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal.46(2001), no. 3, 309–327

work page 2001

[21] [21]

G. B. Whitham,Linear and nonlinear waves, Pure and Applied Mathematics, Wiley- Interscience [John Wiley & Sons], New York-London-Sydney, 1974

work page 1974

[22] [22]

W. Yan, Y. Li, and Y. Zhang,The Cauchy problem for the Novikov equation, NoDEA Non- linear Differential Equations Appl.20(2013), no. 3, 1157–1169. Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A. Email address:dangeba@math.rochester.edu Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A. Email a...

work page 2013