# Dispersive shocks in Quantum Hydrodynamics with viscosity

## TL;DR

This paper investigates the existence and stability of shock profiles in a 1-D quantum hydrodynamic model incorporating dispersive quantum effects and viscosity, revealing conditions for monotone and oscillatory shock solutions.

## Contribution

It introduces a comprehensive analysis of shock profiles in quantum hydrodynamics with viscosity, including existence, stability, and spectral properties, especially for large shocks.

## Key findings

- Existence of monotone shock profiles for small shocks.
- Global existence of oscillatory profiles for large shocks.
- Derived stability conditions for the spectral spectrum.

## 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.