Nonlinear Rayleigh-Taylor instability in compressible viscoelastic fluids with an upper free boundary

Abstract

In this paper, we study the nonlinear Rayleigh-Taylor instability in the compressible viscoelastic fluid governed by the gravity-driven Oldroyd-B model in a finitely deep and horizontally periodic moving domain. Under the instability condition $\kappa <{\kappa }_c$, we first prove that there exist growing mode solutions to the linearized equations around a viscoelastic equilibrium $\left(\overline{\rho }\right(x_1),0,\overline{u}I)$ for a smooth increasing density profile $\overline{\rho }$. Based on the finding of growing modes, we then show the nonlinear Rayleigh-Taylor instability of the above profile by constructing appropriate initial data for the nonlinear perturbation problem departing from the equilibrium and conducting some instability bootstrap arguments.