On exponential stability of gravity driven viscoelastic flows

We investigate stability of an equilibrium state to a nonhomogeneous incompressible viscoelastic fluid driven by gravity in a bounded domain Ω⊂R3Ω⊂R3 of class C3C3. First, we establish a critical number κCκC, which depends on the equilibrium density and the gravitational constant, and is a threshold of the elasticity coefficient κ   for instability and stability of the linearized perturbation problem around the equilibrium state. Then we prove that the equilibrium state is exponential stability provided that κ>κCκ>κC and the initial disturbance quantities around the equilibrium state satisfy some relations. In particular, if the equilibrium density ρ¯ is a Rayleigh–Taylor (RT) type and ρ¯′ is a constant, our result strictly shows that the sufficiently large elasticity coefficient can prevent the RT instability from occurrence.