Instability solutions for the Rayleigh–Taylor problem of non-homogeneous viscoelastic fluids in bounded domains

Abstract

It is well-known that the Rayleigh–Taylor (RT) problem of an inhomogeneous viscoelastic fluid defined on a bounded domain is unstable, if the elasticity coefficient κ is less than some threshold κC. In this paper, we rigorously prove the existence of a unique unstable strong solution in the sense of L1-norm for the RT problem in Lagrangian coordinates based on a bootstrap instability method, when κ<κC. Applying an inverse transformation of Lagrangian coordinates to the obtained unstable solution, we can further get a unique unstable solution for the RT problem of inhomogeneous viscoelastic fluids in Eulerian coordinates.