Stability of Hydraulic Shock Profiles

Abstract

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...

A Sturm–Liouville theorem for quadratic operator pencils

“…where h denotes fluid height; q = hu total flow, with u fluid velocity; and F > 0 the Froude number, a nondimensional parameter depending on reference height/velocity and inclination. Following [29], we here focus on the hydrodynamically stable case 0 < F < 2, and associated hydraulic shock profile solutions (5.2) These are piecwise smooth traveling-wave solutions satisfying the Rankine-Hugoniot jump and Lax entropy conditions at any discontinuities. Their existence theory reduces to the study of an explicitly solvable scalar ODE with polynomial coefficients [29] We now turn to the discussion of stability.…”

“…Following [29], we here focus on the hydrodynamically stable case 0 < F < 2, and associated hydraulic shock profile solutions (5.2) These are piecwise smooth traveling-wave solutions satisfying the Rankine-Hugoniot jump and Lax entropy conditions at any discontinuities. Their existence theory reduces to the study of an explicitly solvable scalar ODE with polynomial coefficients [29] We now turn to the discussion of stability. Linearizing (5.1) about a smooth profile (H, Q) following [18,26], we obtain eigenvalue equations…”

“…It is shown in [29] that essential spectrum of L := −A∂ x − ∂ x A + E is confined to {λ : λ < 0} ∪ {0}, with an embedded eigenvalue at λ = 0. Moreover, it is shown that the embedded eigenvalue at λ = 0 is of multiplicity one in a generalized sense defined in terms of an associated Evans function defined as in [1,18].…”

“…where W := (H, Q) T and [h] := h(0 + ) − h(0 − ) denotes jump in h across x = 0; see [29] for further details. Similarly as in the smooth case, it is shown in [29] that essential spectrum of L with boundary condition (5.5) is confined to {λ : λ < 0} ∪ {0}, with an embedded eigenvalue at λ = 0, of multiplicity one in a generalized sense defined by an associated Evans-Lopatinsky function.…”

“…where W := (H, Q) T and [h] := h(0 + ) − h(0 − ) denotes jump in h across x = 0; see [29] for further details. Similarly as in the smooth case, it is shown in [29] that essential spectrum of L with boundary condition (5.5) is confined to {λ : λ < 0} ∪ {0}, with an embedded eigenvalue at λ = 0, of multiplicity one in a generalized sense defined by an associated Evans-Lopatinsky function. It follows by the general theory of [29] that discontinuous hydraulic shock profiles are nonlinearly orbitally stable so long as they are weakly spectrally stable in the sense that there exist no decaying solutions of (5.3)-(5.5) on {λ : λ ≥ 0} \ {0}.…”