Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux

HyeonSeop Oh; Moon-Jin Kang

REVIEW 4 major objections 6 minor 18 references

Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Large perturbations of multi-D viscous shocks decay at rate $t^{-1/4}$.

desk verdict A credible multi-D extension of the a-contraction framework giving a new t^{-1/4} decay for large perturbations of planar viscous shocks; worth refereeing despite the imported lemmas. read the letter →

arxiv 2501.04311 v2 pith:CUMK7KR6 submitted 2025-01-08 math.AP

Moon-Jin Kang , HyeonSeop Oh This is my paper · ORCID

classification math.AP MSC 35L6535L6735B3535B40

keywords viscousconservationlawplanarshocklargeperturbationcontractionrelativeentropyPoincaréinequalitytimedecaya-contractionwithshifts

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 proves that large, bounded perturbations of a planar viscous shock in a multi-dimensional scalar viscous conservation law—on $\mathbb{R}\times\mathbb{T}^2$ with a strictly convex flux—contract under the flow to the shifted shock profile, and that such perturbations decay in $L^2$ at the rate $(1+t)^{-1/4}$ provided they also have finite $L^1$ mass. The contraction is measured by a weighted relative entropy and holds up to a time-dependent shift that solves a feedback ODE. This is, the authors claim, the first quantitative convergence result to a planar shock for large (not small) perturbations, extending the one-dimensional theory of [8] to the multi-dimensional setting. If correct, it reduces a stability question for arbitrarily large deviations to a single nonlinear Poincaré inequality and the construction of a suitable entropy.

What carries the argument

The argument uses the method of $a$-contraction with shifts: a weight function $a(\xi)=1+\lambda(u_--\tilde u(\xi))/\varepsilon$ makes the entropy production positive, and a shift $X(t)$ solves the feedback ODE $\dot X=\Phi_\varepsilon(Y)(2|B|+1)$ that enforces the contraction. The key new estimate is the multi-dimensional nonlinear Poincaré-type inequality (Proposition 2.1) on $[0,1]\times\mathbb{T}^2$: it bounds the cubic error $\int W^3$ by the weighted dissipation $\int z(1-z)|\partial_z W|^2 + \varepsilon^{-3/2}\int |\partial_{x'}W|^2$ for all $W$ with $\|W\|_{L^\infty}\le 1/\varepsilon$ and $\int W^2\le M$. This inequality, together with a truncation of the large values of $|\eta'(u)-\eta'(\tilde u)|$, yields the contraction and then the decay through a Gagliardo-Nirenberg interpolation.

What would settle it

Take the Burgers flux $f(u)=u^2/2$ in (1.1) on $\mathbb{R}\times\mathbb{T}^2$, choose a large initial perturbation $u_0-\tilde u \in L^1\cap L^\infty$, and compute numerically the shifted $L^2$ distance $\|u(t,\cdot+X(t),\cdot)-\tilde u(\cdot-\sigma t)\|_{L^2}$ over time; observing a decay rate slower than $t^{-1/4}$ or a shift $X(t)$ that grows without bound would disprove Theorem 1.2.

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Extended reading notes

Core claim

The central result is Theorem 1.2: for any initial datum $u_0$ with $u_0 - \tilde u \in L^1\cap L^\infty(\Omega)$, the solution $u(t,\cdot)$ of (1.1) satisfies $\|u(t,\cdot+X(t),\cdot)-\tilde u(\cdot-\sigma t)\|_{L^2(\Omega)} \le C/(1+t^{1/4})$ for all $t>0$, with a shift $X(t)$ that is absolutely continuous and uniformly bounded in time. Theorem 1.1 provides the underlying contraction: the weighted relative entropy $\int a(\xi)\,\eta(u^X|\tilde u)\,d\xi dx'$ is non-increasing in time once the shift is chosen by the ODE (3.2). Together they give the first quantitative large-perturbation decay estimate for a planar viscous shock in multi-dimensions.

Load-bearing premise

The whole proof leans on a multi-dimensional nonlinear Poincaré inequality (Proposition 2.1) that holds only for deviations $W$ with $\|W\|_{L^\infty}\le 1/\varepsilon$ and under smallness constraints on the shock strength $\varepsilon$ and the truncation parameter; if that inequality fails for the required class of perturbations, the contraction estimate and the decay theorem collapse.

Editorial extensions

If this is right

The contraction inequality (1.5) gives a global-in-time $L^2$ bound for the shifted perturbation, so planar viscous shocks are stable to arbitrarily large $L^2$ perturbations in $\mathbb{R}\times\mathbb{T}^2$.
If the initial perturbation is also in $L^1$, the shifted solution converges to the shock profile in $L^2$ with the explicit rate $t^{-1/4}$, while the shift $X(t)$ remains uniformly bounded.
The same decay rate holds in $\mathbb{R}\times\mathbb{T}^{n-1}$ for every $n\ge2$, as noted in Remark 1.3.
The uniform bound on the shift follows from the $L^1$ contraction of the scalar equation, so the decay statement is about the shape of the perturbation rather than about a shift drifting to infinity.

Reading between the lines

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

The same $a$-contraction machinery may yield decay for planar shocks in physical systems with a single convex entropy (for example, Navier-Stokes type systems), provided a multi-dimensional Poincaré inequality analogous to Proposition 2.1 is available.
The $t^{-1/4}$ rate appears to be an artifact of the Gagliardo-Nirenberg interpolation with $L^1$; with stronger integrability or a better lower bound on the diffusion, the method might yield a $t^{-1/2}$ rate.
A numerical experiment with the Burgers flux and a large rectangular bump perturbation could directly test whether the weighted relative entropy is monotone and whether the shift remains bounded, probing the sharpness of Proposition 2.1.
The periodic transverse directions are used to obtain explicit Poincaré constants; on a fully unbounded domain $\mathbb{R}^3$ the cubic term would require a different control, so the proof does not automatically transfer outside the periodic setting.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Main take: this is a serious, technically demanding extension of the a-contraction with shifts program to multi-D scalar viscous conservation laws on R x T^2. The genuinely new result is Theorem 1.2: for strictly convex flux, large L1 intersect L∞ perturbations of a small planar viscous shock converge in L2 at rate t^{-1/4}, up to a uniformly bounded shift. That rate is not in the cited literature; the Burgers analogue is the authors' companion paper and the 1D analogue is Kang's 2021 paper. The proof machinery is the established framework, so the novelty is incremental but real.

What the paper does well: the central reduction is coherent. Theorem 1.1 is reduced to Proposition 3.1, which is split into a Taylor expansion regime and a truncation regime. Proposition 2.1, the multi-D nonlinear Poincare inequality, is proved in detail and is the genuine technical heart: the change of variables to z, Lemma 2.4's pointwise estimate, and the control of the cubic term by diffusion look sound. Appendix A supplies the needed algebra and Appendix B sketches the Gagliardo-Nirenberg interpolation; the overall structure is checkable.

Soft spots, in proportion: the paper is not self-contained at several load-bearing points. Lemma 2.2, the weighted relative entropy time-derivative identity, is imported without proof from [8]. In R x T^2 it involves transversal flux terms and the full gradient in the diffusion term, so this is not a trivial transcription; the stress-test worry about a missing transversal boundary term is reasonable, and if the identity fails or a sign is off, (3.3) fails at the first step. I could not find an actual error, and simple constant-W checks do not produce a counterexample. Lemma 4.2 and Lemma 4.3 are likewise deferred with 'essentially same as in [8]'. There is a minor notational inconsistency in Proposition 2.1 (epsilon^2 versus epsilon^{3/2} in the transversal diffusion denominator), and the reader's sign typo in that proposition matches a discrepancy in the proof; both appear fixable. The relation to [14] (Kang-Vasseur-Wang) should be clarified: the contraction part of Theorem 1.1 may overlap with prior work, and the authors should state precisely what is new. The L1 contraction is used as a black box, which is fine for scalar laws, but the case split in Appendix B.2 is sketched rather than fully justified; that is minor.

Who this is for: researchers in shock stability, especially users of a-contraction methods. It deserves a serious referee: the main theorem is new, the proof is plausible and largely checkable, and the gaps are concentrated and likely fillable. I would not desk reject; I would send it to a knowledgeable PDE analyst with instructions to verify Lemma 2.2 and Proposition 2.1 carefully.

Referee Report

4 major / 6 minor

Summary. The paper studies multi-dimensional scalar viscous conservation laws with strictly convex flux on R x T^2, around a planar viscous shock of small strength. The main results are Theorem 1.1, a contraction property for large L^2 perturbations up to a dynamical shift, and Theorem 1.2, a quantitative t^{-1/4} decay rate in L^2 when the initial perturbation is also in L^1. The proof reduces Theorem 1.1 to Proposition 3.1 via a shift ODE, then splits the analysis into a tail/truncation regime and a small-|η'(u)-η'(ũ)| regime. The latter uses a multi-dimensional nonlinear Poincaré inequality (Proposition 2.1). Theorem 1.2 is obtained in Appendix B by combining the contraction estimate with L^1 contraction and a Gagliardo-Nirenberg interpolation inequality. The manuscript is clearly written and the overall architecture is coherent, but several load-bearing ingredients are imported from earlier papers without full verification, and there is a gap between the assumptions of Theorem 1.2 and the quantities used in its proof.

Significance. If the main results are correct, this is the first quantitative large-perturbation decay estimate for planar viscous shocks in a multi-dimensional scalar viscous conservation law, extending the one-dimensional results of Kang [8] and the multi-D Burgers result in [9]. The proof introduces a genuinely new ingredient: a multi-dimensional nonlinear Poincaré inequality with L^∞ constraint on R x T^2, used to control cubic terms by the diffusion along both the shock and transversal directions. The paper also provides a detailed proof of the algebraic lemma behind that inequality (Appendix A). These are substantial contributions. However, the manuscript currently depends critically on unproved or only sketched imported lemmas, so the significance will be fully realized only after those gaps are closed.

major comments (4)

[§2.2, Lemma 2.2] Lemma 2.2 is the central relative-entropy identity on which Proposition 3.1 and Theorem 1.1 rest, but its proof is omitted with only the statement 'the proof is essentially the same as in [8, Lemma 2.2]'. This identity contains the transversal flux components q2 and q3 and the full gradient in the diffusion term, so it is not a purely one-dimensional computation. The time derivative of ∫ a η(uX|ũ) must be verified in detail for the R x T^2 setting; if any transversal boundary term survives or a sign is incorrect, the key estimate (3.3) fails at the first step. This is a load-bearing gap and should be addressed before the contraction claim is accepted.
[Theorem 1.2 and Appendix B] Theorem 1.2 assumes only u0 - ũ ∈ L^1 ∩ L^∞(Ω), but the proof in Appendix B explicitly uses ‖u0 - ũ‖_{L^2(Ω)} and the L^2 contraction estimate (1.7), which requires the L^2 hypothesis of Theorem 1.1. On R x T^2, L^1 ∩ L^∞ does not imply L^2, so the stated class of initial data is insufficient. Either Theorem 1.2 should include u0 - ũ ∈ L^2, or the proof must establish finite relative entropy and finite L^2 size directly from L^1 ∩ L^∞; the latter seems unlikely without additional assumptions. This is a load-bearing issue for the theorem as stated.
[§2.5, Proposition 2.1 and Lemma 2.4] There are notational inconsistencies in the statement and proof of the central nonlinear Poincaré inequality. In Lemma 2.4 the domain is written as (0,1) x T^2, but the estimate is stated at (z,y) with y ∈ T; the proof then integrates over T^2. In the proof of Proposition 2.1, the line applying Lemma 2.4 writes √(L(z)-L(1-z)) although Lemma 2.4 gives √(L(z)+L(1-z)). Also, the transversal diffusion coefficient appears as (1-δ)/ε^{3/2} in the statement, as (1-δ)/ε^2 in one displayed formula inside the proof, and later is compared with ε^{3/2}. These inconsistencies are likely fixable, but because Proposition 2.1 is a key new result, the final version must give a clean, self-consistent statement and proof.
[§4.2, Lemmas 4.2 and 4.3; §4.2, proof of Proposition 4.2] Several steps that control the region |η'(u)-η'(ũ)| ≥ δ1 are deferred: Lemma 4.2 is stated with 'we omit the proof', Lemma 4.3 is said to follow 'easily' with details in [8,12], and the proof of Proposition 4.2 repeatedly refers to [8, Proposition 4.5] for the main estimates, including (4.8) and (4.10). These lemmas are not cosmetic; they provide the uniform bound on Y and the comparison between the truncated and original functionals that are essential for Case I and Case II of the conclusion. The manuscript should either include complete proofs or give a detailed verification that each one-dimensional argument extends to R x T^2, including the transversal diffusion and the periodic directions.

minor comments (6)

[§1.1, Remark 1.2] There is a typographical error: 'Theroem' should be 'Theorem'.
[§2.2, Lemma 2.2] In the display for B(u), the term `(η'(u)-η(ũ))` should presumably be `(η'(u)-η'(ũ))`; the current text is inconsistent with the surrounding factors.
[§2.5, proof of Proposition 2.1] The notation `T2` is sometimes used for T^2, and the proof of Lemma 2.4 uses variables (z,y) with y ∈ T while integrating over T^2; this should be harmonized for readability.
[§4.2, after Lemma 4.2] The word 'Propotision' appears in 'Propotision 4.5'; it should be 'Proposition'.
[Appendix B, proof of Theorem 1.2] In the interpolation step, the exponent `B^{2/4}` is written where `B^{1/2}` is meant; also the constants `C0` and `C∗` are conflated in several inequalities, which makes the dependence of constants hard to track.
[§2.5, Lemma 2.4] The estimate in the statement is only proved for f ∈ C^1, but it is applied to W that is merely L^2 with √(z(1-z))∂zW ∈ L^2; the approximation step should be mentioned explicitly so the reader knows the lemma applies to the relevant functions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the decay rate and contraction bound are derived, not fitted or assumed.

full rationale

The paper's central claims are Theorem 1.1 (L2 contraction with shift) and Theorem 1.2 (t^{-1/4} decay for L1∩L∞ perturbations). Neither claim is presupposed by the proof setup. The contraction follows from Proposition 3.1, whose proof is carried out in Section 4; Proposition 4.1 reduces the relevant functional to the multi-dimensional nonlinear Poincaré-type inequality Proposition 2.1. Proposition 2.1 is not taken as an input equal to the desired conclusion: it is proved in Section 2.5 using Lemma 2.4 (proved in-text), Lemma 2.5 (proved in Appendix A), and the external Poincaré-type inequality Lemma 2.6 cited from [18]. The final decay in Theorem 1.2 is obtained in Appendix B by combining the contraction differential inequality (B.3), the standard L1 contraction Lemma B.1, and the Gagliardo-Nirenberg interpolation Lemma B.2; the bound then follows by solving d/dt E ≤ -C E^3. No fitting of the rate t^{-1/4} occurs, and no identity in the chain reduces by construction to the target estimate. Several auxiliary statements are imported from the same group's prior papers, especially Lemma 2.2 from [8] and parts of Lemma 4.2, Lemma 4.3, and Proposition 4.2 from [8,9]; these are omitted or sketched proofs of algebraic/localization estimates, not assumptions of the conclusion, and they concern established technique rather than the theorem being proved. The weight function and shift are constructed objects, not disguised versions of the final decay. Thus the derivation is self-contained in the sense required for circularity analysis, with the caveat that some routine but nontrivial computations are deferred to earlier works.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard PDE background and on technical lemmas imported from prior works by the same authors. No empirical parameters are fitted; the proof constants are smallness thresholds. The only explicit free design parameter is lambda. No new physical entities are introduced; the shift X(t) and weight a are mathematical constructs of the method.

free parameters (1)

lambda (weight asymptotic jump) = any value in (delta0^{-1} epsilon, delta0)
The weight function a(xi) = 1 + lambda (u_minus - u_tilde(xi))/epsilon has endpoint jump lambda. The theorem holds for every such lambda, so this is a design parameter of the method rather than a fitted quantity.

assumptions (7)

domain assumption Flux f1 is strictly convex and grows at most exponentially: |f1(u)| <= a e^{b|u|} (condition (1.4)).
Used in Theorem 1.1 and Theorem 1.2; it guarantees the entropy constructed in [8, Appendix A] controls large values of the flux.
domain assumption There exists an entropy eta satisfying hypotheses (A1) and (A2), for instance eta(u) = e^{bu} + e^{-bu} + u^4 + u^2.
The contraction argument is formulated for such an entropy; existence is cited to [8, Appendix A] and not proved in this paper.
domain assumption Weak shock profile bounds of Lemma 2.1, imported from [8, Lemma 2.1]: -C^{-1} epsilon^2 e^{-C1 epsilon |xi|} <= u_tilde'(xi) <= -C epsilon^2 e^{-C2 epsilon |xi|}, plus |u_tilde''| <= C epsilon |u_tilde'|.
Used for the weight function, for the change of variables to z, and for smallness estimates; proof omitted, cited to [8].
domain assumption Weighted relative entropy identity in Lemma 2.2: d/dt integral a eta(u^X|u_tilde) = X_dot Y + B - G; proof omitted, stated as in [8, Lemma 2.2].
This identity is the backbone of the contraction estimate; this paper does not re-derive it.
standard math L1 contraction for scalar viscous conservation law (Lemma B.1, cited to Dafermos [2, Theorem 6.3.2]).
Needed for the L1 bound of the perturbation and the uniform bound of the shift in Appendix B.
standard math Gagliardo-Nirenberg inequality on R x T^2 from Huang and Yuan [7, Theorem 1.4] (Lemma B.2).
Used to convert the differential inequality on the relative entropy into the t^{-1/4} L2 decay.
domain assumption Truncation comparison identities in Lemma 4.3: 0 <= G0(u) - G0(u_trunc) <= G0(u), and Di(u) - Di(u_trunc) = integral a mu(u) |partial_i(eta'(u) - eta'(u_trunc))|^2, proof omitted.
The proof is described as easily following from identity (4.7), with details in [8, 12].

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Pith. "Pith review of Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux." pith.science (2026). https://pith.science/paper/CUMK7KR6

@misc{pith2026250104311,
  author       = {Pith},
  title        = {Pith review of: Decay of large solutions around shocks to multi-D viscous conservation law with strictly convex flux},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUMK7KR6}},
  note         = {Machine review of arXiv:2501.04311}
}

abstract

We consider a planar viscous shock for a scalar viscous conservation law with a strictly convex flux in multi-dimensional setting, where the transversal direction is periodic. We first show the contraction property for any solutions evolving from a large bounded initial perturbation in $L^2$ of the viscous shock. The contraction holds up to a dynamical shift, and it is measured by a weighted relative entropy. This result for the contraction extends the existing result in 1D \cite{Kang19} to the multi-dimensional case. As a consequence, if the large bounded initial $L^2$-perturbation is also in $L^1$, then the large perturbation decays of rate $t^{-1/4}$ in $L^2$, up to a dynamical shift that is uniformly bounded in time. This is the first result for the quantitative estimate converging to a planar shock under large perturbations.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

18 extracted references · 17 canonical work pages

[8]

L2-type contraction for shocks of scalar viscous conservatio n laws with strictly convex ﬂux

Kang, M.-J. L2-type contraction for shocks of scalar viscous conservatio n laws with strictly convex ﬂux. J. Math. Pures Appl. (9) 145 (2021), 1–43

work page 2021
[14]

F., and W ang, Y

Kang, M.-J., V asseur, A. F., and W ang, Y. L2-contraction of large planar shock waves for multi- dimensional scalar viscous conservation laws. J. Diﬀerential Equations 267 , 5 (2019), 2737–2791

work page 2019
[9]

$L^2$ decay for large perturbations of viscous shocks for multi-D Burgers equation

Kang, M.-J., and Oh, H. l2 decay for large perturbations of viscous shocks for multi-d burgers equation. arXiv:2403.08445, To appear in Anal. Appl. (2024)

work page Pith review arXiv 2024
[1]

Hyperbolic systems of conservation laws: the one-dimensio nal Cauchy problem

Bressan, A. Hyperbolic systems of conservation laws: the one-dimensio nal Cauchy problem . Oxford University Press, Oxford, 2000

work page 2000
[2]

Hyperbolic conservation laws in continuum physics , vol

Dafermos, C. Hyperbolic conservation laws in continuum physics , vol. 3. Springer, 2005

work page 2005
[3]

L 1 stability of shock waves in scalar viscous conservation law s

Freist¨uhler, H., and Serre, D. L 1 stability of shock waves in scalar viscous conservation law s. Comm. Pure Appl. Math. 51 , 3 (1998), 291–301

work page 1998
[4]

Stability of viscous scalar shock fronts in several dimensi ons

Goodman, J. Stability of viscous scalar shock fronts in several dimensi ons. Trans. Amer. Math. Soc. 311, 2 (1989), 683–695

work page 1989
[5]

Asymptotic behavior of multidimensional scalar viscous sh ock fronts

Hoff, D., and Zumbrun, K. Asymptotic behavior of multidimensional scalar viscous sh ock fronts. Indiana Univ. Math. J. 49 , 2 (2000), 427–474

work page 2000

Show all 18 references

[6]

Pointwise Green’s function bounds for multidimensional sc alar viscous shock fronts

Hoff, D., and Zumbrun, K. Pointwise Green’s function bounds for multidimensional sc alar viscous shock fronts. J. Diﬀerential Equations 183 , 2 (2002), 368–408. 35

2002
[7]

Stability of planar rarefaction waves for scalar viscous co nservation law under periodic perturbations

Huang, F., and Yuan, Q. Stability of planar rarefaction waves for scalar viscous co nservation law under periodic perturbations. Methods Appl. Anal. 28 , 3 (2021), 337–353

2021
[10]

Kang, M.-J., and V asseur, A. F. Criteria on contractions for entropic discontinuities of s ystems of conservation laws. Arch. Ration. Mech. Anal. 222 , 1 (2016), 343–391

2016
[11]

Kang, M.-J., and V asseur, A. F. L2-contraction for shock waves of scalar viscous conservatio n laws. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire 34, 1 (2017), 139–156

2017
[12]

Kang, M.-J., and V asseur, A. F. Contraction property for large perturbations of shocks of t he barotropic Navier-Stokes system. J. Eur. Math. Soc. 23 , 2 (2021), 585–638

2021
[13]

Kang, M.-J., and V asseur, A. F. Uniqueness and stability of entropy shocks to the isentropi c Euler system in a class of inviscid limits from a large family of Nav ier-Stokes systems. Invent. Math. 224 , 1 (2021), 55–146

2021
[15]

Kruˇzkov, S. N. First order quasilinear equations with several independen t variables. Mat. Sb. (N.S.) 81 (123) (1970), 228–255

1970
[16]

Shock Waves

Liu, T.-P. Shock Waves . Graduate Studies in Mathematics. Volume: 215. American Ma thematical Society, 2021

2021
[17]

V asseur, A. F. Relative entropy and contraction for extremal shocks of con servation laws up to a shift. In Recent advances in partial diﬀerential equations and applic ations, vol. 666 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2016, pp. 385–404

2016
[18]

Nonlinear stability of planar viscous shock wave to three-d imensional compressible Navier-Stokes equations

W ang, T., and W ang, Y. Nonlinear stability of planar viscous shock wave to three-d imensional compressible Navier-Stokes equations. arXiv:2204.09428, To appear in JEMS (2024). (Moon-Jin Kang) Department of Mathematical Sciences, Korea Advanced Institute of Mathematical Scien...

2024 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [8]

L2-type contraction for shocks of scalar viscous conservatio n laws with strictly convex ﬂux

Kang, M.-J. L2-type contraction for shocks of scalar viscous conservatio n laws with strictly convex ﬂux. J. Math. Pures Appl. (9) 145 (2021), 1–43

work page 2021

[2] [14]

F., and W ang, Y

Kang, M.-J., V asseur, A. F., and W ang, Y. L2-contraction of large planar shock waves for multi- dimensional scalar viscous conservation laws. J. Diﬀerential Equations 267 , 5 (2019), 2737–2791

work page 2019

[3] [9]

$L^2$ decay for large perturbations of viscous shocks for multi-D Burgers equation

Kang, M.-J., and Oh, H. l2 decay for large perturbations of viscous shocks for multi-d burgers equation. arXiv:2403.08445, To appear in Anal. Appl. (2024)

work page Pith review arXiv 2024

[4] [1]

Hyperbolic systems of conservation laws: the one-dimensio nal Cauchy problem

Bressan, A. Hyperbolic systems of conservation laws: the one-dimensio nal Cauchy problem . Oxford University Press, Oxford, 2000

work page 2000

[5] [2]

Hyperbolic conservation laws in continuum physics , vol

Dafermos, C. Hyperbolic conservation laws in continuum physics , vol. 3. Springer, 2005

work page 2005

[6] [3]

L 1 stability of shock waves in scalar viscous conservation law s

Freist¨uhler, H., and Serre, D. L 1 stability of shock waves in scalar viscous conservation law s. Comm. Pure Appl. Math. 51 , 3 (1998), 291–301

work page 1998

[7] [4]

Stability of viscous scalar shock fronts in several dimensi ons

Goodman, J. Stability of viscous scalar shock fronts in several dimensi ons. Trans. Amer. Math. Soc. 311, 2 (1989), 683–695

work page 1989

[8] [5]

Asymptotic behavior of multidimensional scalar viscous sh ock fronts

Hoff, D., and Zumbrun, K. Asymptotic behavior of multidimensional scalar viscous sh ock fronts. Indiana Univ. Math. J. 49 , 2 (2000), 427–474

work page 2000