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Vanishing viscosity limit to two interacting shocks from the same family for the compressible Navier-Stokes equations

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Pith's one-line read Two shocks from the same family collide; the vanishing viscosity limit holds through the collision with explicit $L^p$ rates.

desk verdict Genuinely new configuration and a serious proof idea, but the theorem as stated outruns the argument: the proof needs an ε≲δ ordering that the statement never imposes. read the letter →

arxiv 2608.06173 v1 pith:VWWIOM67 submitted 2026-08-06 math.AP

Lin-An Li , Dehua Wang , Yi Wang This is my paper · ORCID
classification math.AP MSC 76N1035Q3535Q3035Q3176N06
keywords VanishingviscositylimitcompressibleNavier-Stokesequationsinteractingshockssamefamilyrarefactionwavesanti-derivativemethodrelativeentropya-contraction
topics Navier-Stokes Regularity
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 establishes that the vanishing viscosity limit can be rigorously justified for the one-dimensional compressible Navier-Stokes equations when the underlying Euler solution contains two shocks from the same characteristic family that collide. This case is more singular than collisions of shocks from different families because the collision time grows like the inverse wave strength and the outgoing wave pattern mixes a shock with a rarefaction wave. The authors prove that, for suitably small wave strengths and viscosity, there are global smooth Navier-Stokes solutions converging to the entropy solution of the Euler equations in every $L^p$ space with $p\in[2,+\infty)$, at rates written explicitly in terms of the wave strength and viscosity. If correct, this supplies a quantitative vanishing-viscosity theorem for an overtaking shock interaction, including the collision point itself.

What carries the argument

Before the approximate collision time $\tau=-A\varepsilon/\delta^2$, the argument uses anti-derivative variables $(\Phi,\Psi)$ for the perturbation around the sum of two shifted viscous shock profiles; the energy estimates close only up to this carefully chosen time, producing exponentially small bounds with a large constant $A$. At that time the solution is transferred to a different approximate pattern: a smooth approximate rarefaction wave superposed with a shifted viscous shock, where the shift $X(\tau)$ is governed by a weighted relative entropy and $a$-contraction relation with weight $a=1+(\lambda/\delta_2)(v^s-v_*)$. The transition interval between the two patterns has width $O(\varepsilon)$, and controlling the error there is what lets the proof pass through the collision.

What would settle it

Take $\delta=\varepsilon^2$ with a fixed large constant $A$. Then $-A\varepsilon/\delta^2=-A/\varepsilon$ lies before the initial time $-t_0$ for small $\varepsilon$, so the proof's pre-collision interval $[-t_0,-A\varepsilon/\delta^2]$ is empty. Checking whether the claimed global smooth solutions and convergence rates still exist in this $\delta\ll\varepsilon$ regime, or finding a counterexample, would settle whether Theorem 1.1 needs the extra condition $\varepsilon\lesssim\delta$.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: there exist positive constants $\varepsilon_0$ and $\delta_0$ such that for any $\varepsilon\in(0,\varepsilon_0)$ and wave strengths $\delta_l\sim\delta_r\le\delta_0$, the Cauchy problem for the compressible Navier-Stokes equations has a family of global smooth solutions $(v^\varepsilon,u^\varepsilon)$ converging as $\varepsilon\to0^+$ to the entropy solution $(V,U)$ of the Euler equations. The convergence holds in $L^p(\mathbb{R})$ for every $p\in[2,+\infty)$ with the explicit bounds $\|(v^\varepsilon-V,u^\varepsilon-U)(t,\cdot)\|_{L^p}\le C(\delta_l+\delta_r)^{1/2}\varepsilon^{1/p}$ for $0\le t\le t_0$, plus $C(\delta_l+\delta_r)(t-t_0)^{1/(2p)}\varepsilon^{1/(2p)}$ for $t\ge t_0$. The proof treats the pre-collision phase, the collision point, and the post-collision shock-rarefaction composite as one connected picture, using an approximate collision time to close uniform energy estimates before merging into a shifted composite wave after the collision.

Load-bearing premise

The proof requires the wave strength to be at least a fixed multiple of the viscosity, because the pre-collision analysis stops at the time $-A\varepsilon/\delta^2$ and this time must lie before the actual collision; when $\delta$ is much smaller than $\varepsilon$, that interval is empty and the constructed initial data cannot start the argument.

Editorial extensions

If this is right

  • For any fixed time before the collision, the $L^p$ error between the viscous solution and the Euler entropy solution is $O((\delta_l+\delta_r)^{1/2}\varepsilon^{1/p})$, so the long $1/\delta$ pre-collision time scale does not ruin the convergence rate.
  • After the collision the error bound contains the additional factor $(\delta_l+\delta_r)(t-t_0)^{1/(2p)}\varepsilon^{1/(2p)}$, giving a polynomial-in-$\varepsilon$ rate for every fixed $p\ge2$ even as time advances.
  • The approximate collision time creates a transition layer of width $O(\varepsilon)$ near the true collision, and the paper shows the approximate composite wave matches the exact entropy solution there with the desired rate.
  • The global-in-time uniform estimates simultaneously imply convergence in every $L^p$ space with $p\in[2,+\infty)$, rather than only in $L^2$.
  • The same framework applies in Eulerian coordinates, as noted in the paper, so the Lagrangian proof is not an obstruction to the physically standard formulation.

Reading between the lines

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

  • The proof's stopping time condition suggests that Theorem 1.1 as stated may hide an implicit ordering between the wave strength and the viscosity: when $\delta$ is much smaller than $\varepsilon$, the interval on which the initial data are constructed is empty, so a separate argument would be needed to cover that regime.
  • A natural extension is to non-isentropic compressible Navier-Stokes equations with two interacting same-family shocks, where the outgoing wave pattern would also include a contact discontinuity; the paper explicitly leaves this as future work.
  • The shift $X(\tau)$ is designed to track how the rarefaction wave pushes the outgoing shock; a numerical test of the viscous $p$-system near the collision could check whether the predicted shift matches the actual shock location.
  • The approximate collision time scales as $\varepsilon/\delta^2$, which matches the expected viscous interaction width divided by the relative shock speed, so the same device may transfer to scalar conservation laws with curved flux near shock coalescence.
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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 / 3 minor

Summary. The paper studies the vanishing viscosity limit for 1D compressible Navier-Stokes equations (1.1) in the regime of two interacting shocks from the same characteristic family. The entropy solution of the Euler system consists of two incoming 2-shocks before a collision time t0, and a composite outgoing 2-shock plus a 1-rarefaction wave after the collision. The authors construct a smooth approximate wave pattern using viscous shock profiles and a smooth rarefaction wave, and they introduce an approximate collision time t0 - Aε/δ² to split the analysis. Before this time, an anti-derivative method yields H²-type energy estimates; after this time, a weighted relative entropy method with time-dependent shifts yields H¹-type estimates. Theorem 1.1 claims global smooth Navier-Stokes solutions for all ε∈(0,ε0) and δ_l∼δ_r≤δ0, converging to the Euler entropy solution in L^p with explicit rates C(δ_l+δ_r)^{1/2} ε^{1/p} before t0 and additional time-dependent terms after t0.

Significance. If the main theorem and its proof are correct, this is a substantial advance in the vanishing viscosity problem for interacting shocks: same-family shock interaction is genuinely more singular than different-family interaction because the collision time grows like 1/δ, and the post-collision wave pattern is a shock-rarefaction composite. The proof is entirely analytic and constructive: the approximate collision time, the shift ODE, and the weight function are chosen with explicit constants rather than fitted to the solution, and the claimed convergence rates are explicit and falsifiable. The reliance on prior work is for standard viscous shock profiles and rarefaction wave properties, not for the interaction estimates. The overall architecture of the proof is coherent and the techniques are potentially adaptable to related problems, provided the parameter-relation gaps identified below are resolved.

major comments (2)
  1. [Theorem 1.1, Section 1.3, Section 2.4, Theorem 3.1] The theorem's quantifier order is not supported by the proof. The proof requires the approximate collision time -Aε/δ² to lie strictly inside the pre-collision interval [-t0, 0] (equivalently t0 - Aε/δ² > 0). Since t0 = 1/(s_l - s_r) ∼ C/δ, this forces Aε/δ² < C/δ, i.e. ε ≤ Cδ for some constant C. Theorem 1.1 states the result for any ε∈(0,ε0) and δ_l∼δ_r≤δ0 with ε0 and δ0 independent, which permits δ≪ε. For example, if δ=ε², then t0∼ε^{-2} while Aε/δ² = Aε^{-3}, so -Aε/δ² < -t0 and the interval [-t0, -Aε/δ²] is empty; the initial data (2.13) cannot serve as a starting point for the anti-derivative estimates, and Theorem 3.1 is vacuous or inapplicable. The same condition is needed for the exponential factors such as e^{-cAδε} in (3.8) to be small. This is a load-bearing gap, though it is likely repairable by adding the condition ε ≤ Cδ to Theorem 1.1 and Theorems 3.1-3.2, or by making ε0 depend on δ.
  2. [Theorem 3.1, Eq. (3.8) and Section 4, Eq. (4.40)] The estimate (3.8), ∥(ϕ,ψ)(τ)∥² + ε∫∥ψ_y∥² ≤ Ce^{-cAδε}, does not imply the smallness used in (4.40). In the proof of Theorem 1.1, the bound ∥v(τ,·)-V(τ,·)∥² ≤ C(δ_l+δ_r)ε is obtained by bounding ∥ϕ∥² by C(δ_l+δ_r)ε, but (3.8) only gives ∥ϕ∥² ≤ Ce^{-cAδε}. For fixed δ as ε→0 (the regime of the theorem), e^{-cAδε} → 1, so this does not tend to zero and cannot be absorbed into the O(δ ε) terms in (4.40). This is not a mere presentation issue: the claimed convergence rate as ε→0 relies on the perturbation L² norm being small, and the stated exponential bound does not provide that smallness for the parameter range of the theorem. The authors should either correct the exponent in (3.8) to a quantity that is actually small in the relevant regime (e.g., a polynomial in δε) or revise the proof of (4.40) to use a different estimate.
minor comments (3)
  1. [Section 1.1] The same symbol (v_*, u_*) is used both for the pre-collision intermediate state in (1.6) and for the post-collision intermediate state in (1.12), despite the text saying these are different states. This causes an apparent contradiction: δ_l := v_* - v_- is the first-order left shock strength, while later δ_1 := v_* - v_- is claimed to be third order. Please use distinct notation for the two intermediate states and clarify the definition of δ_1.
  2. [Sections 1.3 and 4] The transition interval [t0 - Aε/δ², t0] is repeatedly described as having 'width of order O(ε)'. Its actual width is Aε/δ², which is larger than O(ε) when δ is small; for example, if δ=√ε the width is of order one. The estimates in the paper do not appear to rely on this width being O(ε), but the statement is inaccurate and should be corrected to 'width O(ε/δ²)'.
  3. [Throughout] There are several typos and formatting errors, e.g., 'T reatment' in the table of contents, 'Y ield' in the proof of Lemma 6.6, and inconsistent spacing around equations. These do not affect the mathematics but should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper constructs NS solutions matching a specified entropy solution and proves the convergence by explicit energy estimates, with no fitted parameters or load-bearing self-citation.

full rationale

The paper's central claim is an existence-and-convergence construction: for a fixed entropy solution (V, U) it defines explicit approximate wave profiles and proves uniform energy estimates. No parameter is fitted to the target solution; A, M, lambda, and kappa = epsilon are explicit constants chosen to close the estimates (Sections 1.3, 2.4, 3.5). The approximate wave profiles are explicit superpositions of viscous shock profiles (Lemma 2.1, cited to Kawashima-Matsumura) and a smooth rarefaction wave (Lemma 2.2, cited to Xin); these are standard external results, not consequences of the conclusion being proved. The shift ODE (2.18) and weight function (3.39) implement the a-contraction method from Kang-Vasseur-Wang [21]; although [21] shares a co-author with the present paper, it is used only as a technical framework and local-existence reference, while the present estimates are carried out in detail. The convergence rate follows from direct triangle-inequality estimates (4.40)-(4.48) applied to the constructed profiles, not from any fitted input. The only notable concern is a statement-level quantifier gap: the proof requires epsilon <= C delta so that the interval [-t0, -A epsilon / delta^2] is nonempty, but Theorem 1.1 states epsilon and delta vary independently. This is a correctness or precision issue about the hypotheses, not a circularity, because the proof does not assume the conclusion it is trying to establish.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. All parameters (A, M, λ, κ) are constants chosen for the proof, and the axioms are standard results from the cited literature on viscous profiles, rarefaction waves, and the inviscid Riemann problem.

free parameters (4)
  • A = large constant, chosen so that e^{-cA} is small enough to close estimates
    Determines the approximate collision time t_0 - Aε/δ²; introduced in Section 2.3 to make pre-collision energy estimates close.
  • M = M = 5α* / (4(s*)²)
    Constant in the shift ODE (2.18); chosen in Lemma 6.5 to make the a-contraction coefficient negative.
  • lambda = δ₂ much less than λ ≤ C√δ₂
    Weight amplitude in (3.39); chosen small enough for the Poincaré-based contraction estimates.
  • kappa = κ = ε
    Smoothing width of the rarefaction wave in (2.8); fixed to ε throughout.
assumptions (5)
  • domain assumption Existence and exponential decay estimates for viscous shock profiles of (1.1)
    Lemma 2.1, cited to [22]. Used to control interactions of the two viscous shocks.
  • domain assumption Smooth approximate rarefaction wave satisfies the estimates of Lemma 2.2
    Lemma 2.2, cited to [31]. Used for the post-collision composite profile.
  • domain assumption Local well-posedness for the perturbation systems
    Cited to [7,29] before collision and [21] after. Standard semigroup theory.
  • domain assumption The entropy solution of (1.2)-(1.3) for two interacting 2-shocks has the structure (1.7)-(1.12): after collision a 2-shock and a 1-rarefaction with strengths δ₂ = δ_l + δ_r + O(δ_l δ_r (δ_l + δ_r)) and δ₁ = O(δ_l δ_r (δ_l + δ_r))
    Cited to [2,4,24,27]. Underlies the whole wave pattern.
  • standard math Poincaré-type inequality of Lemma 6.2
    Cited to [19,21]. Used in the a-contraction step.

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Pith. "Pith review of Vanishing viscosity limit to two interacting shocks from the same family for the compressible Navier-Stokes equations." pith.science (2026). https://pith.science/paper/VWWIOM67

@misc{pith2026260806173,
  author       = {Pith},
  title        = {Pith review of: Vanishing viscosity limit to two interacting shocks from the same family for the compressible Navier-Stokes equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWWIOM67}},
  note         = {Machine review of arXiv:2608.06173}
}
read the original abstract

We investigate the vanishing viscosity limit for the one-dimensional compressible Navier-Stokes equations in the regime of two interacting shock waves from the same characteristic family of the underlying Euler equations. Unlike the interaction of two shocks from distinct families, which produces two outgoing shocks in their respective original families, the collision of two same-family shocks generates an outgoing shock in the same family together with a rarefaction wave in the other family. This configuration is more singular because the collision occurs on a larger time scale that depends inversely on the wave strength, making it challenging to justify the vanishing viscosity limit not only in the processes before and after the shock collision but, more crucially, at the collision point. Furthermore, the emergence of both shock and rarefaction waves after the collision introduces an additional difficulty. To overcome these obstacles, we first employ the anti-derivative method before the collision, which allows us to fix the locations of both viscous shocks precisely up to the collision point. However, uniform estimates with respect to the viscosity cannot be closed up to the collision time; we therefore introduce a carefully constructed approximate collision time to obtain the required higher-order energy bounds. After the collision, we apply a weighted relative entropy method, combined with time-dependent shifts, to handle the composite wave structure arising from the coexistence of shock and rarefaction waves. Our main result establishes that, for suitably small wave strengths and viscosity coefficients, there exists a family of global smooth solutions to the Navier-Stokes equations that converge to the entropy solution of the Euler equations with an explicit convergence rate. The techniques are expected to be applicable to other related problems in vanishing viscosity theory.

Figures

Figures reproduced from arXiv: 2608.06173 by the authors.

Figure 1
Figure 1. below. Since the shock Sl propagates faster than Sr, the left shock Sl will inevitably overtake the right one Sr and collide at some point Q = (t0, x0) with t0 = 1 sl−sr > 0. After the collision time t0, there is another intermediate state (v ∗ , u∗ ) with (v ∗ , u∗ ) ∈ S2(v+, u+) and (v−, u−) ∈ R1(v ∗ , u∗ ), which means that the overtaking interaction of two incoming 2-shocks Sl and Sr forms an outgoing 2-shock S2… view at source ↗
Figure 2
Figure 2. Approximation wave interactions For the first sub-interval τ ∈ [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

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Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    of Math.161 (2), 223-342 (2005)

    Bianchini, S., Bressan, A.:Vanishing viscosity solutions of nonlinear hyperbolic systems.Ann. of Math.161 (2), 223-342 (2005)

    work page 2005

  2. [2]

    The one-dimensional Cauchy problem.Oxford Univer- sity Press, Oxford, 2000

    Bressan, A.:Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem.Oxford Univer- sity Press, Oxford, 2000

    work page 2000

  3. [3]

    Bressan, A., Caravenna, L., and Shen, W.:Local asymptotic patterns for viscous approximations of conser- vation laws.https://arxiv.org/abs/2605.00189, (2026)

    work page Pith review arXiv 2026

  4. [4]

    and Jenssen, H.:Pairwise wave interactions in ideal polytropic gases.Arch

    Chen, G., Endres, E. and Jenssen, H.:Pairwise wave interactions in ideal polytropic gases.Arch. Ration. Mech. Anal.204, 787-836 (2012)

    work page 2012

  5. [5]

    From Navier-Stokes to BV solutions of the barotropic Euler equations

    Chen, G., Kang, M.-J. and Vasseur, A.:From Navier-Stokes to BV solutions of the barotropic Euler equations. https://arxiv.org/abs/2401.09305, 2024. V ANISHING VISCOSITY LIMIT FOR SAME F AMILY INTERACTING SHOCKS 49

    work page Pith review arXiv 2024

  6. [6]

    Pure Appl

    Chen, G.-Q., Perepelitsa, M.:Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow.Comm. Pure Appl. Math.63, 1469-1504 (2010)

    work page 2010

  7. [7]

    Rational Mech

    Goodman, J., Xin, Z.:Viscous limits for piecewise smooth solutions to systems of conservation laws.Arch. Rational Mech. Anal.121, 235-265 (1992)

    work page 1992

  8. [8]

    Hoff, D., Liu, T.-P.:The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data.Indiana Univ. Math. J.38, 861-915 (1989)

    work page 1989

Show all 35 references
  1. [9]

    Pure Appl

    Hopf, E.:The partial differential equationu t +uu x =µu xx.Comm. Pure Appl. Math.3, 201-230 (1950)

    1950

  2. [10]

    and Wang, Y.:Zero dissipation limit of full compressible Navier-Stokes equations with a Riemann initial data.Commun

    Huang, F., Jiang, S. and Wang, Y.:Zero dissipation limit of full compressible Navier-Stokes equations with a Riemann initial data.Commun. Inf. Syst.13, 211-246 (2013)

    2013

  3. [11]

    and Wang, Y.:Zero dissipation limit to rarefaction wave with vacuum for one-dimensional compressible Navier-Stokes equations.SIAM J

    Huang, F., Li, M. and Wang, Y.:Zero dissipation limit to rarefaction wave with vacuum for one-dimensional compressible Navier-Stokes equations.SIAM J. Math. Anal.44, 1742-1759 (2012)

    2012

  4. [12]

    and Wang, Y.:Uniqueness of composite wave of shock and rarefaction in the inviscid limit of Navier-Stokes equations.SIAM J

    Huang, F., Wang, W., Wang, Y. and Wang, Y.:Uniqueness of composite wave of shock and rarefaction in the inviscid limit of Navier-Stokes equations.SIAM J. Math. Anal.56, 3924-3967 (2024)

    2024

  5. [13]

    and Yang, T.:The limit of the Boltzmann equation to the Euler equations for Riemann problems.SIAM J

    Huang, F., Wang, Y., Wang, Y. and Yang, T.:The limit of the Boltzmann equation to the Euler equations for Riemann problems.SIAM J. Math. Anal.45, 1741-1811 (2013)

    2013

  6. [14]

    and Yang, T.:Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks.Sci

    Huang, F., Wang, Y., Wang, Y. and Yang, T.:Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks.Sci. China Math.58, 653-672 (2015)

    2015

  7. [15]

    and Yang, T.:Fluid dynamic limit to the Riemann solutions of Euler equations: I

    Huang, F., Wang, Y. and Yang, T.:Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity.Kinet. Relat. Models3, 685-728 (2010)

    2010

  8. [16]

    and Yang, T.:Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to a Riemann problem.Arch

    Huang, F., Wang, Y. and Yang, T.:Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to a Riemann problem.Arch. Rational Mech. Anal.203, 379-413 (2012)

    2012

  9. [17]

    and Sun, W.:Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids.SIAM J

    Jiang, S., Ni, G. and Sun, W.:Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids.SIAM J. Math. Anal.38, 368-384 (2006)

    2006

  10. [18]

    Math.224, 55-146 (2021)

    Kang, M.-J., Vasseur, A.:Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of Navier-Stokes systems.Invent. Math.224, 55-146 (2021)

    2021

  11. [19]

    Kang, M.-J., Vasseur, A.:Contraction property for large perturbations of shocks of the barotropic Navier- Stokes system.J. Eur. Math. Soc.23, 585-638 (2021)

    2021

  12. [20]

    Kang, M.-J., Vasseur, A.:Well-posedness of the Riemann problem with two shocks for the isentropic Euler system in a class of vanishing physical viscosity limits.J. Diff. Equa.338, 128-226 (2022)

    2022

  13. [21]

    and Wang, Y.:Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equations.Adv

    Kang, M.-J., Vasseur, A. and Wang, Y.:Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equations.Adv. Math.419, Paper No. 108963, 66 pp. (2023)

    2023

  14. [22]

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

    1985

  15. [23]

    and Wang, Y.:The limit to rarefaction wave with vacuum for 1D compressible fluids with temperature-dependent transport coefficients.Anal

    Li, M., Wang, T. and Wang, Y.:The limit to rarefaction wave with vacuum for 1D compressible fluids with temperature-dependent transport coefficients.Anal. Appl. (Singap.)13, 555-589 (2015)

    2015

  16. [24]

    Liu, T.-P.:Shock waves.Graduate Studies in Mathematics,215, Amer. Math. Soc., Providence, RI, 2021

    2021

  17. [25]

    Ma, S.:Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations. J. Diff. Equa.248, 95-110 (2010)

    2010

  18. [26]

    and Zhang, Y.:Vanishing viscosity for non-isentropic gas dynamics with interacting shocks

    Shi, X., Yong, Y. and Zhang, Y.:Vanishing viscosity for non-isentropic gas dynamics with interacting shocks. Acta Math. Sci. Ser. B (Engl. Ed.),36, 1699-1720 (2016)

    2016

  19. [27]

    Smoller, J.:Shock waves and reaction-diffusion equations.Grundlehren der mathematischen Wissenschaften, 258, Springer-Verlag, New York, second edition, 1994

    1994

  20. [28]

    Vasseur, A., Wang, Y.:The inviscid limit to a contact discontinuity for the compressible Navier-Stokes- Fourier system using the relative entropy method.SIAM J. Math. Anal.47, 4350-4359 (2015)

    2015

  21. [29]

    Wang, H.:Viscous limits for piecewise smooth solutions of the p-system.J. Math. Anal. Appl.299, 411-432 (2004)

    2004

  22. [30]

    Wang, Y.:Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock.Acta Math. Sci. Ser. B.28, 727-748 (2008)

    2008

  23. [31]

    Pure Appl

    Xin, Z.:Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of com- pressible isentropic gases.Comm. Pure Appl. Math.46, 621-665 (1993)

    1993

  24. [32]

    Xin, Z., Zeng, H.:Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations.J. Diff. Equa.249, 827-871 (2010)

    2010

  25. [33]

    Rational Mech

    Yu, S.-H.:Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws.Arch. Rational Mech. Anal.146, 275-370 (1999)

    1999

  26. [34]

    and Tan, Z.:Zero dissipation limit to a Riemann solution consisting of two shock waves for the 1D compressible isentropic Navier-Stokes equations.Sci

    Zhang, Y., Pan, R. and Tan, Z.:Zero dissipation limit to a Riemann solution consisting of two shock waves for the 1D compressible isentropic Navier-Stokes equations.Sci. China Math.56, 2205-2232 (2013). 50 L.-A. LI, D. W ANG, AND Y. W ANG

    2013

  27. [35]

    and Tan, Z.:Zero dissipation limit with two interacting shocks of the 1D non-isentropic Navier-Stokes equations.Indiana Univ

    Zhang, Y., Pan, R., Wang, Y. and Tan, Z.:Zero dissipation limit with two interacting shocks of the 1D non-isentropic Navier-Stokes equations.Indiana Univ. Math. J.62, 249-309 (2013). School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing N...

    2013

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