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.
Keywords
Cite
@article{arxiv.1812.10279,
title = {Dispersive shocks in Quantum Hydrodynamics with viscosity},
author = {Corrado Lattanzio and Pierangelo Marcati and Delyan Zhelyazov},
journal= {arXiv preprint arXiv:1812.10279},
year = {2019}
}