Dispersive shocks in Quantum Hydrodynamics with viscosity

Corrado Lattanzio; Pierangelo Marcati; Delyan Zhelyazov

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arXiv · 1812.10279

Dispersive shocks in Quantum Hydrodynamics with viscosity

Corrado Lattanzio·Pierangelo Marcati·Delyan Zhelyazov

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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.

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Corrado Lattanzio, Pierangelo Marcati, Delyan Zhelyazov. 2018-12-26. Dispersive shocks in Quantum Hydrodynamics with viscosity. https://arxiv.org/abs/1812.10279

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