Dispersive shocks in quantum hydrodynamics with viscosity

Abstract

In this paper we study existence and stability of shock profiles for a 1-D compressible Euler system in the context of Quantum Hydrodynamic models. The dispersive term is originated by the quantum effects described through the Bohm potential; moreover we introduce a (linear) viscosity to analyze its interplay with the former while proving existence, monotonicity and stability of travelling waves connecting a Lax shock for the underlying Euler system. The existence of monotone profiles is proved for sufficiently small shocks; while the case of large shocks leads to the (global) existence for an oscillatory profile, where dispersion plays a significant role. The spectral analysis of the linearized problem about a profile is also provided. In particular, we derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of the eigenvalues in the unstable plane, using a careful analysis of the Evans function.

Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

“…as follows from the Rankine-Hugoniot condition (2.3). Substituting (2.7) into (2.6) and proceeding as in [28] (see section 2), we finally conclude that the profile P satisfies…”

Existence of standing and traveling waves in quantum hydrodynamics with viscosity

“…In this section we shall prove existence of traveling waves for the system (1.1) in the case of large shocks without restriction on the viscosity and dispersion parameters, thus improving the global existence Lemma from [20] which requires that the viscosity is sufficiently strong. This improvement is due to the presence of a better Lyapunov function (see (5.13) below) than the one used in [20]. This new Lyapunov function is suggested by the computation in Section 4.2 about standing waves in the (ρ, m) variables.…”

“…The first attempt to analyze the spectral theory of the linearized operator around dispersive shocks for the p−system with real viscosity and linear capillarity has been discussed in [18]. Moreover, the case of the Euler formulation (1.2), but with linear viscosity, is investigated in [20], and the related Evans function computations are presented in [21]. The spectrum of the linearization of (1.3) around dispersive shocks is considered in [23].…”

“…In [20] global existence of traveling waves for the system (1.1) was proved in the case of sufficiently strong viscosity. We remove this condition thanks to the presence of a Lyapunov function (see (5.13) below) and obtain a proof of global existence of profiles for arbitrary viscosity and dispersion parameters.…”

“…Moreover, global existence of traveling waves is proved for the former without restrictions on the viscosity and dispersion parameters, thanks to a suitable Lyapunov function. This is an improvement with respect to the global existence result in [20], where it was required that the viscosity is sufficiently strong.…”