Asymptotic stability of the linearised Euler equations with long-memory impedance boundary condition

Abstract

This work focuses on the well-posedness and stability of the linearised Euler equations (1) with impedance boundary condition (2,3). The first part covers the acoustical case ($u_0 = 0$), where the complexity lies solely in the chosen impedance model. The existence of an asymptotically stable $C_0$-semigroup of contractions is shown when the passive impedance admits a dissipative realisation; the only source of instability is the time-delay $\tau$. The second part discusses the more challenging aeroacoustical case($u_0 \neq 0$), which is the subject of ongoing research. A discontinuous Galerkin discretisation is used to investigate both cases.