Spectral Stability of Inviscid Roll Waves

Johnson, Mathew A.; Noble, Pascal; Rodrigues, Luis Miguel; Yang, Zhao; Zumbrun, Kevin

doi:10.1007/s00220-018-3277-7

Cited by 17 publications

(15 citation statements)

References 52 publications

(79 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Beyond (discontinuous or smooth) fronts and constant solutions, the equation may also support spatially periodic traveling waves. Those are however necessarily discontinuous and, as a consequence of admissibility, each of their smooth part must also contain a sonic point (see [JNR + 18] for details, on a closely related system case).…”

Section: Introductionmentioning

confidence: 99%

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

Duchêne¹,

Rodrigues²

2020

SIAM J. Math. Anal.

View full text Add to dashboard Cite

We prove the large-time asymptotic orbital stability of strictly entropic Riemann shock solutions of first order scalar hyperbolic balance laws, under piecewise regular perturbations provided that the source term is dissipative about endstates of the shock. Moreover the convergence towards a shifted reference state is exponential with a rate predicted by the linearized equations about constant endstates.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Duchêne¹,

Rodrigues²

2020

SIAM J. Math. Anal.

View full text Add to dashboard Cite

show abstract

“…Among other applications, they are commonly used in the hydraulic engineering literature to describe flow in a dam spillway, channel, or etc. ; see, e.g., [BM,Je,Br1,Br2,Dr,JNRYZ] for further discussion. Equations (1.1) form a hyperbolic system of balance laws [La, Bre, Da], with the first equation representing conservation of fluid and the second balance between change of momentum and the opposing forces of gravity (h) and turbulent bottom friction (−h −2 |q|q).…”

mentioning

confidence: 99%

“…, for which H * < H s < H R , the corresponding smooth traveling wave profile does not pass the singular point, and so there exist smooth traveling wave solutions as depicted in Figure 2(c). However, there exist no solutions containing subshocks, as these would necessarily jump below H s < H R , and so the solution could never return past H s , since H ′ < 0 on (H * , H s ) blocks approach by smooth solution, and since any admissible discontinuities can only decrease the value of H. See Figure 3 [JNRYZ,§2] with regard to existence of periodic entropy-admissible piecewise smooth relaxation profiles; indeed, existence of periodic and quasiperiodic profiles follows by essentially the same construction used here to show existence of homoclinic ones.…”

mentioning

confidence: 99%

Stability of Hydraulic Shock Profiles

Yang

Zumbrun

2019

Arch Rational Mech Anal

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…For integrable systems, the hyperbolicity of modulation equations and existence of the Riemann invariants were established, for example, for Korteweg-de Vries equation (KdV equation) [44], for nonlinear Schrödinger equation (NLS equation) [39], for sine-Gordon equation [14,25], for Benjamin-Ono equation [6] (see also the book [28] for further references). For non-integrable systems, one can cite [26] where the Whitham equation was studied and modulational instability for short enough waves was shown, or [33] and [27] where the the regions of modulational and spectral stability for roll waves to the Saint-Venant equations were determined. In general, a "right choice" of unknowns in which the modulation equations are written is necessary to have explicit (or almost explicit) expressions of the corresponding characteristic values.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolicity of the Modulation Equations for the Serre–Green–Naghdi Model

2020

View full text Add to dashboard Cite

Serre-Green-Naghdi equations (SGN equations) is the most simple dispersive model of long water waves having "good" mathematical and physical properties. First, the model is a mathematically justified approximation of the exact water wave problem. Second, the SGN equations are the Euler-Lagrange equations coming from Hamilton's principle of stationary action with a natural approximate Lagrangian. Finally, the equations are Galilean invariant which is necessary for physically relevant mathematical models.We have derived the modulation equations to the SGN model and show that they are strictly hyperbolic for any wave amplitude, i.e., the periodic wave trains are modulationally stable. Numerical tests for the full SGN equations are shown. The results confirm the modulational stability analysis.

show abstract

Spectral Stability of Inviscid Roll Waves

Cited by 17 publications

References 52 publications

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

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

Stability of Hydraulic Shock Profiles

Hyperbolicity of the Modulation Equations for the Serre–Green–Naghdi Model

Contact Info

Product

Resources

About