On the nonlinear Rayleigh-Taylor instability of nonhomogeneous incompressible viscoelastic fluids under L2-norm

We investigate the nonlinear instability of a viscoelastic Rayleigh-Taylor equilibrium state of a nonhomogeneous incompressible viscoelastic fluid in the presence of a uniform gravitational field. It's well-known that the viscoelastic Rayleigh-Taylor equilibrium state is unstable in H2(Ω)-norm, if the elasticity coefficient κ is less than a critical number κc, where Ω denotes the domain occupied by the viscoelastic fluid. In this article, by choosing a new energy functional in Gronwall-type inequality, we show that, for the horizontal periodic domain with infinite height, the viscoelastic Rayleigh-Taylor equilibrium state is unstable in L2(Ω)-norm based on a bootstrap instability method.