On classical solutions of Rayleigh–Taylor instability in inhomogeneous viscoelastic fluids

We study the nonlinear Rayleigh–Taylor (RT) instability of an inhomogeneous incompressible viscoelastic fluid in a bounded domain. It is well known that there exist strong solutions of RT instability in H 2 -norm in inhomogeneous incompressible viscoelastic fluids, when the elasticity coefficient κ is less than some threshold κ C . In this paper, we prove the existence of classical solutions of RT instability in L 1 -norm in Lagrangian coordinates based on a bootstrap instability method with finer analysis, if κ < κ C . Moreover, we also get classical solutions of RT instability in L 1 -norm in Eulerian coordinates by further applying an inverse transformation of Lagrangian coordinates.