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$L^2$-contraction and asymptotic stability of large shock for scalar viscous conservation laws

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Pith's one-line read For the scalar viscous conservation law u_t+(u^p)_x=u_xx with 2≤p≤4, every viscous shock of arbitrary amplitude is L2-contracting and asymptotically stable under small H1 perturbations, and L1 perturbations decay at the rate t^{-1/4}.

desk verdict This paper proves L2 contraction and asymptotic stability for arbitrarily large viscous shocks for polynomial fluxes u^p, 2≤p≤4, and the main result is new and credible, but the proof of the key dissipation lemma has a repairable gap at p=2. read the letter →

arxiv 2509.02965 v1 pith:HHRW43BD submitted 2025-09-03 math.AP math-phmath.MP

Alexis F. Vasseur , Yi Wang , Jian Zhang This is my paper · ORCID
classification math.APmath-phmath.MP MSC 35L6535B3535B4035L67
keywords L2contractionviscousshockwavelarge-amplitudea-contractionmethodtime-dependentshiftscalarconservationlawpolynomialfluxasymptoticstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper settles a question left open by a recent counterexample: whether the a-contraction property, known for arbitrary shock amplitudes at the inviscid level and for small viscous shocks, also holds for large viscous shocks. For the strictly convex polynomial flux f(u)=u^p with 2≤p≤4, the authors prove that for any end states 0

What carries the argument

The proof is carried by a weighted relative-entropy functional rather than by the usual anti-derivative or spectral method. The weight is a(U)= (f(U)-f(u_-))/(U-u_-) - (f(U)-f(u_+))/(U-u_+), positive and bounded on the shock layer, and the shift X(t) is chosen by the ODE (2.5) so that the transport term created by shifting becomes a square with favorable sign. The decisive algebraic step is Lemma 3.2, which asserts that the effective diffusion coefficient g(U) in (3.14) is strictly negative on [u_+,u_-]; combined with the weighted Poincaré inequality of Lemma 3.1, this negativity converts the weighted energy identity into pure dissipation plus small cubic terms controlled by the H1 smallness

What would settle it

For fixed p∈[2,4] and end states 0<u_+<u_-, evaluate m(U) in (3.16) on a dense grid of U∈[u_+,u_-]. If m(U)≤0 at any interior U, then g(U)≥0 and the dissipation term in (3.28) loses its sign, so the weighted contraction estimate cannot hold as stated; a concrete numerical search would settle the lemma's validity, especially at the endpoint p=2.

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Extended reading notes

Core claim

The central claim of Theorem 1.1 is that the viscous shock profile U of u_t+(u^p)_x=u_xx is nonlinearly stable in H1 for arbitrary shock strength when p∈[2,4]. More precisely, there is a critical ϵ* so that if the initial H1 distance to the profile is below ϵ*, then a time-dependent shift X(t) and a positive weight a(U) exist with d/dt ∫ a(U(x-st-X(t))) |u(t,x)-U(x-st-X(t))|^2 dx ≤ 0 for all t>0. The shift obeys an ODE that compensates the longitudinal drift of the perturbation; the weight, which depends on the shock strength, is trivial exactly for Burgers flux p=2. As a corollary of the monotonicity and the H1 a priori estimates, the solution converges to the shifted shock in L∞, the shift

Load-bearing premise

The argument's load-bearing premise is that a certain algebraic expression built from the flux and the shock profile—the effective dissipation coefficient g(U)—is strictly negative on the whole interval [u_+,u_-]; the proof of this lemma as written uses a strict inequality that is actually zero at p=2, though retaining a discarded quadratic term appears to restore the needed negativity.

Editorial extensions

If this is right

  • Global existence and uniform H1 bounds hold for small H1 perturbations of arbitrarily large shocks in (1.1) for p∈[2,4].
  • The weighted L2 distance to the shifted profile is a Lyapunov function, so contraction holds for all time, not only asymptotically.
  • L∞ convergence of the solution to the shifted shock holds, and the shift velocity tends to zero, so the shift grows at most sublinearly.
  • If the initial perturbation is in L1, the L2 error decays as C||φ0||/(1+C t^{1/4}||φ0||).
  • The theorem answers the open question raised by a recent counterexample to a-contraction for large viscous shocks, at least for polynomial fluxes of degree 2 through 4.

Reading between the lines

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

  • One endpoint issue in the paper as written: Eq. (3.18) uses the assertion l1(u_-,u_+)<0 for all p∈[2,4], but this quantity is 0 at p=2; the strict negativity needed for dissipation appears recoverable by retaining a quadratic term the proof drops, so the p=2 case likely survives with a small correction.
  • Because ϵ* and the constants in the estimates depend on u± and p, “arbitrarily large shock” means arbitrary amplitude with small perturbation; extending the contraction to genuinely large perturbations remains open in general.
  • The same shift-and-weight mechanism may transfer to planar multidimensional viscous shocks or to degenerate Oleinik shocks, with the algebra of the negativity lemma as the main obstacle to overcome.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper studies the scalar viscous conservation law u_t+(u^p)_x=u_{xx} with 2≤p≤4 and end states 0<u_+<u_-. It constructs an explicit time-dependent shift X(t) and a shock-strength-dependent weight a(U) so that, for H^1-small perturbations of any (possibly arbitrarily large) viscous shock, the weighted L^2 norm of the perturbation is non-increasing in time. From this contraction property the authors derive L∞ convergence of the solution to the shifted shock profile, and, when the initial perturbation is in L1, a t^{-1/4} L2 decay rate. The proof follows the a-contraction framework: a weighted Poincaré inequality on the shock layer, a time-dependent shift chosen to remove a nonlocal term, and a priori H1 estimates. The central algebraic step is Lemma 3.2, which asserts that a certain effective diffusion coefficient g(U) is uniformly negative on [u_+,u_-], yielding the dissipative term in the weighted energy identity.

Significance. If the identified gaps are repaired, this is a meaningful advance: it extends the a-contraction approach for large viscous shocks from the Burgers case to the family of polynomial fluxes f(u)=u^p for 2≤p≤4, thereby addressing the open question raised by Blochas-Cheng about possible viscous destabilization for large shocks. The construction of the shift and weight is explicit and parameter-free, and the stability/decay statements are concrete and falsifiable. The paper uses prior results (L1-contraction, the weighted Poincaré inequality) as black boxes rather than fitting anything to the target conclusion, which is a genuine strength. The main concern is not the overall strategy but the correctness of several displayed estimates inside Lemma 3.2 and the passage from the weighted energy inequality to the final contraction estimate.

major comments (2)
  1. [Lemma 3.2, Eq. (3.18)] The assertion l1(u_-,u_+)<0 for all p∈[2,4] is false at p=2. Direct substitution gives l1=(2-p)u_+(u_- -u_+)u_-^{p-2}+u_+^2(u_-^{p-2}-u_+^{p-2})=0 when p=2. Hence the displayed chain in (3.18) yields only g(u_-)≤0 at p=2, not the strict negativity needed to define β. The uniform negativity of g is exactly what converts the weighted Poincaré estimate into the dissipative bound (3.28); without it the contraction proof fails as written. The gap is repairable—for p=2 one can retain the discarded quadratic term, which equals -1/2 in the bracket and gives g(u_-)≤-a(u_-)<0, or treat p=2 separately using the known Burgers result—but this repair is not present in the manuscript.
  2. [Section 3, Eqs. (3.26)-(3.28)] In passing from (3.26) to (3.28), the cubic remainder ∫ O(1)(φ^X)^3 Uξ dξ is dropped without justification. The estimate (3.27) controls only the shift term 1/2 Ẋ∫(φ^X)^2 aξ, not the cubic term. The cubic term can be absorbed by the H^1 smallness: it is bounded by C∥φ^X∥_{L∞}∫(φ^X)^2|Uξ| ≤ C N(T)∫(φ^X)^2|Uξ|, and this contribution should be included in the coefficient β−Cϵ1. Since (3.28) is the core contraction inequality, the absorption of both the shift term and the cubic remainder should be displayed explicitly.
minor comments (6)
  1. [Eq. (3.21)] The statement h'''(Ubar)>0 for all p∈[2,4] is false at p=2, where h'''=0. The inequality in (3.21) remains valid with h'''≥0; please adjust the wording to distinguish the equality case.
  2. [Section 2.5, Eq. (2.28)] The inequality d/dt ∫(Ca+1)(φ^X)^2 ≤ -2∫(φξ^X)^2 is asserted without proof. It can be obtained by taking C large and combining (3.28) with (3.34), but the argument should be shown, especially the choice of the large constant that dominates the positive right-hand side of (3.34).
  3. [Introduction and references] The citation 'Kang [20]' in the introduction does not match the bibliography, where [20] is Kružkov; the relevant reference appears to be [10]. Please correct the citation/numbering.
  4. [Eq. (2.24)] The second term on the right-hand side should be C0∥φ0∥²_{H1} to be consistent with (2.23). If the weaker form C0∥φ0∥_{H1} is intended, this should be stated explicitly.
  5. [Proposition 2.1 and Appendix] For non-integer p, f(u)=u^p is not a polynomial on R and is not smooth at u=0. The local existence statement for arbitrary M>0 needs a smallness/positivity hypothesis, or p should be specified as an integer. Since the continuity argument only uses small M, this does not affect the main result, but the proposition as stated is overbroad.
  6. [General presentation] There are several minor typos: 'intepolation' for 'interpolation', 'satisfing' for 'satisfying', an incomplete sentence at the end of Section 3 ('complete the proof of Proposition 2.2 by .'), and the symbol β is used for two different constants in Lemma 3.2 and Section 2.5.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the contraction and stability estimates are proved from explicit constructions and independent prior results; a technical gap in Lemma 3.2 is a correctness issue, not circularity.

full rationale

The paper's central claim is not obtained by fitting or by restating an input. The weight function a(U) in (2.4) and the shift X(t) in (2.5) are explicit, and the relative-entropy inequality is derived from the equation by weighted energy estimates. The key algebraic estimate on g(U) (Lemma 3.2) is proved directly from the flux and profile, not imported as an assumption equivalent to the theorem. The cited tools — L1-contraction [27], the weighted Poincaré inequality [14], and the a-contraction framework [13] — are independent published results with proofs whose assumptions do not include the target contraction inequality. I found no step where a 'prediction' reduces to its inputs, no parameter fitted to data and renamed as a prediction, and no load-bearing self-citation chain. I did note a technical, non-circular defect: in Eq. (3.18) the assertion l1(u_-,u_+) < 0 for all p in [2,4] fails at p=2, where l1 = 0, so the displayed argument gives only g(u_-) <= 0 rather than strict negativity. This affects the proof of the uniform dissipation constant β, but it is a correctness gap, not a circularity; it appears repairable by retaining the discarded quadratic term.

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

The theorem is a pure PDE proof. No fitted parameters or invented objects appear. It relies on standard tools (weighted Poincaré inequality, L1-contraction, Gagliardo-Nirenberg interpolation), a bootstrap smallness assumption, and one unproved algebraic inequality that is central to Lemma 3.2 but fixable.

assumptions (6)
  • standard math Weighted Poincaré inequality (3.3) with constant 1/2 (Lemma 3.1, cited from [14])
    Used in (3.9)-(3.11) to convert the time-dependent shift term into positive dissipation; accepted as known but not proved in this paper.
  • standard math L1-contraction for scalar viscous conservation laws (Serre [27])
    Used in (2.21) to bound I1(t) in L1 and derive a uniform bound on the shift; external theorem.
  • standard math Gagliardo-Nirenberg interpolation inequality (2.27)
    Used to turn L1 and H1 bounds into L2 decay; standard.
  • domain assumption Existence, uniqueness, and monotonicity of the viscous shock profile U(xi) for (1.3)
    Background for the problem, stated in Section 1 and standard for strictly convex flux.
  • domain assumption Small H1 perturbation bootstrap N(T) <= epsilon_0 in Proposition 2.2
    The a priori estimates are proven under this smallness assumption and then closed by continuity; this defines the small-perturbation scope of the theorem.
  • ad hoc to paper Algebraic inequality l1(u-,u+) < 0 asserted in (3.18)
    Unproved in the text and false at p=2, where l1=0; needed for the strict negativity of g at the u- endpoint, but the conclusion can be recovered from the discarded term.

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Pith. "Pith review of $L^2$-contraction and asymptotic stability of large shock for scalar viscous conservation laws." pith.science (2026). https://pith.science/paper/HHRW43BD

@misc{pith2026250902965,
  author       = {Pith},
  title        = {Pith review of: $L^2$-contraction and asymptotic stability of large shock for scalar viscous conservation laws},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HHRW43BD}},
  note         = {Machine review of arXiv:2509.02965}
}
abstract

We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the strictly convex flux $f(u)=u^p $ with $2\leq p \leq 4$, we can prove $L^2$-contraction and time-asymptotic stability of arbitrarily large viscous shock profile in $H^1$-framework by using $a$-contraction method with time-dependent shift and suitable weight function. Additionally, if the initial perturbation belongs to $L^1$, then $L^2$ time-asymptotic decay rate $t^{-\frac{1}{4}}$ can be obtained.

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Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Viscous Destabilization for Large Shocks of Conservation Laws

    P. Blochas and J. Cheng. Viscous Destabilization for Large Shocks of Conservation Laws, arXiv preprint: 2501.01537, 2025. 28 V ASSEUR, W ANG, AND ZHANG

    work page Pith review arXiv 2025

  2. [2]

    Choi, M.-J

    K. Choi, M.-J. Kang, Y. Kwon and A. Vasseur. Contraction for large perturbations of trav- eling waves in a hyperbolic-parabolic system arising from a chemotaxis model. Math. Models Methods Appl. Sci, 30(2): 387-437, 2020

    work page 2020

  3. [3]

    Freist¨ uhler and D

    H. Freist¨ uhler and D. Serre. L1-stability of shock waves in scalar viscous conservation laws, Comm. Pure Appl. Math., 51: 291-301, 1998

    work page 1998

  4. [4]

    J. Goodman. Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Archive for Rational Mechanics and Analysis, 95: 325-344, 1986

    work page 1986

  5. [5]

    J. Goodman. Stability of viscous scalar shock fronts in several dimensions. Trans. Am. Math. Soc., 311: 683-695, 1989

    work page 1989

  6. [6]

    E. Hopf. The partial differential equations ut + uux = µuxx. Comm. Pure. Appl. Math., 3: 201-230, 1950

    work page 1950

  7. [7]

    F. M. Huang, Y. Wang and J. Zhang. Time-asymptotic stability of composite waves of degen- erate Oleinik shock and rarefaction for non-convex conservation laws. Math. Ann., 392: 1-46, 2025

    work page 2025

  8. [8]

    Humpherys, G

    J. Humpherys, G. Lyng and K. Zumbrun, Multidimensional stability of large-amplitude Navier- Stokes shocks. Arch. Ration. Mech. Anal., 226: 923-973, 2017

    work page 2017

Show all 30 references
  1. [9]

    A. M. Il’in and O. A. Oleinik. Asymptotic behavior of solution of the Cauchy problem for some quasilinear equations for large values of the time. Mat. Sb. (N.S.), 51 (93): 191-216, 1960

    1960

  2. [10]

    M.-J. Kang. L2 -type contraction for shocks of scalar viscous conservation laws with strictly convex flux. J. Math. Pure Appl., 145: 1-43, 2021

    2021

  3. [11]

    Kang and H

    M.-J. Kang and H. Oh. L2 decay for large perturbations of viscous shocks for multi-d Burgers equation. arXiv preprint: 2430.08445, 2024

    2024

  4. [12]

    Kang and A

    M.-J. Kang and A. Vasseur. Criteria on contractions for entropic discontinuities of systems of conservation laws. Arch. Ration. Mech. Anal., 222(1): 343–391, 2016

    2016

  5. [13]

    Kang and A

    M.-J. Kang and A. Vasseur. L2-contraction for shock waves of scalar viscous conservation laws. Ann. l’Institut Henri Poincare C, Analyse non lineaire, 34 (1): 139-156, 2017

    2017

  6. [14]

    Kang and A

    M.-J. Kang and A. Vasseur. Contraction property for large perturbations of shocks of the barotropic Navier-Stokes system. J. Eur. Math. Soc., 23 (2): 585-638, 2020

    2020

  7. [15]

    M.-J. Kang, A. Vasseur and Y. Wang. L2-contraction of large planar shock waves for multi- dimensional scalar viscous conservation laws. Journal of Differential Equations, 267 (5): 2737- 2791, 2019

    2019

  8. [16]

    M.-J. Kang, A. Vasseur and Y. Wang. Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equations. Adv. Math., 419: 108963, 2023

    2023

  9. [17]

    M.-J. Kang, A. Vasseur and Y. Wang. Time-Asymptotic stability of generic Riemann solutions for compressible Navier-Stokes-Fourier equations. Arch. Rational Mech. Anal., 249 (42), 2025. https://doi.org/10.1007/s00205-025-02116-w

    2025 doi

  10. [18]

    Kawashima and A

    S. Kawashima and A. Matsumura. Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Comm. Math. Phys., 101: 97-127, 1985

    1985

  11. [19]

    C. E. Kenig, and F. Merle. Asymptotic stability and Liouville theorem for scalar viscous conservation laws in cylinders. Comm. Pure. Appl. Math., 59: 769-796, 2006

    2006

  12. [20]

    S. N. Kruˇ zkov. First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81(123): 228-255, 1970. L2-CONTRACTION OF LARGE SHOCK 29

    1970

  13. [21]

    T.-P. Liu. Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc., 56: 233-236, 1985

    1985

  14. [22]

    Liu and Y

    T.-P. Liu and Y. N. Zeng. Shock waves in conservation laws with physical viscosity. Mem. Amer. Math. Soc., 234 (1105): vi+168, 2015

    2015

  15. [23]

    Mascia and K

    C. Mascia and K. Zumbrun. Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems. Comm. Pure Appl. Math., 57: 841-876, 2004

    2004

  16. [24]

    Matsumura and K

    A. Matsumura and K. Nishihara. On the stability of traveling wave solutions of a one- dimensional model system for compressible viscous gas, Jpn. J. Appl. Math., 2: 17-25, 1985

    1985

  17. [25]

    Nishihara

    K. Nishihara. A note on the stability of traveling wave solutions of the Burgers’ equation. Jpn. J. Appl. Math., 2 (1): 27-35, 1985

    1985

  18. [26]

    Sattinger

    D. Sattinger. On the stability of waves of nonlinear parabolic systems. Adv. Math., 22 (3): 312-355, 1976

    1976

  19. [27]

    D. Serre. L1-stability of nonlinear waves in scalar conservation laws. Handbook of differential equations: Evolutionary equations. Vol. 1. North-Holland, 473-553, 2002

    2002

  20. [28]

    Szepessy and Z.-P

    A. Szepessy and Z.-P. Xin. Nonlinear stability of viscous shock waves. Arch. Ration. Mech. Anal., 122: 53-104, 1993

    1993

  21. [29]

    Wang and Y

    T. Wang and Y. Wang. Nonlinear stability of planar viscous shock wave to three-dimensional compressible Navier-Stokes equations. arXiv preprint:2204.09428, To appear in JEMS, 2026

    2026 arXiv

  22. [30]

    K. Zumbrun. Planar stability criteria for viscous shock waves of systems with real viscosity. In: Hyperbolic systems of balance laws, Lecture Notes in Math. 1911, Springer, Berlin, 229-326 2007. (Alexis F. Vasseur) Department of Mathematics, The University of Texas at Austin, ...

    1911

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