Large-Time Asymptotic Stability of Riemann Shocks of Scalar Balance Laws

Duchêne, Vincent; Rodrigues, Luis Miguel

doi:10.1137/18m1221795

Cited by 11 publications

(29 citation statements)

References 22 publications

(29 reference statements)

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“…A very interesting open problem would be to study existence and stability of hydraulic shocks for this more complicated model, in particular comparing results to Saint-Venant profiles and experiment, On the mathematical side, our main contribution here is the treatment for the first time of nonlinear stability of relaxation profiles containing subshocks, a topic that so far as we know has up to now not been addressed. (Though see [DR1,DR2] for related, contemporary, studies of stability of discontinuous solutions of scalar balance laws.) Indeed, at the outset it is perhaps not clear what is the proper framework in which this problem should be approached, as smooth and discontinuous shocks have been treated in the literature by quite different and at first sight incompatible techniques.…”

Section: 2mentioning

confidence: 99%

Stability of Hydraulic Shock Profiles

Yang

Zumbrun

2019

Arch Rational Mech Anal

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We establish nonlinear H 2 ∩ L 1 → H 2 stability with sharp rates of decay in L p , p ≥ 2, of general hydraulic shock profiles, with or without subshocks, of the inviscid Saint-Venant equations of shallow water flow, under the assumption of Evans-Lopatinsky stability of the associated eigenvalue problem. We verify this assumption numerically for all profiles, giving in particular the first nonlinear stability results for shock profiles with subshocks of a hyperbolic relaxation system.An interesting open problem would be to analyze the second case by the introduction/tracking of this additional shock wave in the nonlinear Ansatz, "relieving" incompatibility at t = 0. More generally, it would be interesting to treat lower regularity perturbations than piecewise H 2 , for example in piecewise Lipshitz class by a paradifferential damping estimate following [Me]. To treat perturbations admitting shocks would also be interesting, but appears to require new ideas. Likewise, in the setting of more general balance laws not admitting a damping estimate, it is not clear how to proceed even for the case of arbitrarily smooth compatible initial perturbations. As noted in [JLW], for example, the time-asymptotic stability of piecewise smooth Zeldovich-von Neumann-Doering (ZND) detonations is an important open problem.Finally, it would be very interesting to attack by techniques like those used here the open problem cited in [JNRYZ] of nonlinear time-asymptotic stability of discontinuous periodic "roll wave" solutions of (1.1) or its 3 × 3 analog (RG) in the hydrodynamically unstable regime F > 2. It would appear that a Bloch wave analog of the linear analysis here would apply also for periodic waves, similar to that of [JZN, JNRZ] in the viscous periodic case; for the requisite Bloch wave framework for discontinuous waves, see [JNRYZ]. 1 A difficulty is the apparent lack of a nonlinear damping estimate given instability of constant states. However, as suggested by L. M. Rodrigues [R], one may hope that an "averaged" energy estimate using "gauge functions", or specially chosen weights generalizing the Goodman-and Kawashima-type estimates here, as used to obtain damping estimates in the viscous case in [RZ] might yield a nonlinear damping estimate here as well.Note: Our numerical conclusions have subsequently been verified analytically by generalized Sturm-Liouville considerations in [SYZ], yielding a complete analytical proof of stability. Hydraulic shock profiles of Saint-Venant equationsWe begin by categorizing the family of hydraulic shock profiles, or piecewise smooth traveling wave solutions of (1.1) with discontinuities consisting of entropy-admissible shocks. For closely related analysis, see the study of periodic "Dressler" waves in [JNRYZ, §2]; as discussed in Remark 2.3, this corresponds to the degenerate case H s = H L , F > 2 in our study here. As the firstorder derivative part of (1.1) comprises the familiar equations of isentropic gas dynamics, entropyadmissble discontinuities are in this case Lax 1-or 2-shocks sa...

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Section: 2mentioning

confidence: 99%

Stability of Hydraulic Shock Profiles

Yang

Zumbrun

2019

Arch Rational Mech Anal

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“…The original hyperbolic result. The purely inviscid result [DR20], that we extend to the slightly viscous regimes, is itself quite recent. More generally, despite the fact that hyperbolic models are largely used for practical purposes and that for such models singularities such as shocks and characteristic points are ubiquitous, the analysis of nonlinear asymptotic stability of singular traveling-waves of hyperbolic systems is still in its infancy.…”

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confidence: 58%

“…From the former point of view, the present contribution may be thought as a global-in-time scalar version of [GX92,GR01,Rou02]. From the latter point of view, though of a very different technical nature, by many respects, it shares similar goals with other vanishing viscosity stability programs -see for instance [BGM17,HR18] -and the present contribution is thought as being to [DR20] what [BMV16] is to [BM15].…”

mentioning

confidence: 92%

“…More generally, despite the fact that hyperbolic models are largely used for practical purposes and that for such models singularities such as shocks and characteristic points are ubiquitous, the analysis of nonlinear asymptotic stability of singular traveling-waves of hyperbolic systems is still in its infancy. The state of the art is essentially reduced to a full classification of waves of scalar equations in any dimension [DR20,DRar] (obtained using some significant insights about characteristic points from [JNR `19]) and the case study of a discontinuous wave without characteristic point for a system of two equations in dimension 1 [SYZ20,YZ20].…”

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confidence: 99%

“…We focus on the most basic shock stability result of [DR20]. Consider a scalar balance law in dimension 1, (1.1) B t u `Bx pf puqq " gpuq with traveling wave solutions R ˆR Ñ R, pt, xq Þ Ñ upx ´pψ 0 `σ0 tqq with initial shock position ψ 0 P R, speed σ 0 P R and wave profile u of Riemann shock type, that is,…”

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confidence: 99%

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Uniform asymptotic stability for convection-reaction-diffusion equations in the inviscid limit towards Riemann shocks

Blochas¹,

Rodrigues²

2022

Preprint

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The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter ε. The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale ε-dependent topology reduces to the classical W 1,8 -topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regularity estimates, that the global-in-time analysis is also spatially multiscale due to the coexistence of nontrivial slow parts with fast shock-layer parts, that the limiting smooth spectral problem (in fast variables) has no spectral gap and that uniformity requires a very precise and unusual design of the phase shift encoding orbital stability. In particular, our analysis builds a phase that somehow interpolates between the hyperbolic shock location prescribed by the Rankine-Hugoniot conditions and the non-uniform shift arising merely from phasing out the non-decaying 0-mode, as in the classical stability analysis for fronts of reactiondiffusion equations.

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