Stability properties of the steady state for the isentropic compressible Navier–Stokes equations with density dependent viscosity in bounded intervals

We prove existence and asymptotic stability of the stationary solution for the compressible Navier–Stokes equations for isentropic gas dynamics with a density-dependent diffusion in a bounded interval. We present the necessary conditions to be imposed on the boundary data which ensure existence and uniqueness of the steady state, and we subsequently investigate its stability properties by means of the construction of a suitable Lyapunov functional for the system. The Saint–Venant system, modeling the dynamics of a shallow compressible fluid, fits into this general framework.

Keywords

Navier–Stokes equations, parabolic-hyperbolic systems, stationary solutions, stability

2010 Mathematics Subject Classification

35B35, 35B40, 35Q35, 76N10

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Received 29 November 2020

Received revised 21 June 2021

Accepted 21 June 2021

Published 10 December 2021