Decay properties of the Stokes semigroup for two-phase incompressible viscous flows
Abstract
In this paper, we consider a two-phase problem for two immiscible, viscous, incompressible fluids in the presence of a uniform gravitational field acting vertically downward in the $N$-dimensional Euclidean space $\mathbf{R}^N$, $N\geq 2$. The two fluids are separated from one another by the sharp interface $\Gamma(t)=\{(x’,x_N) : x’\in\mathbf{R}^{N-1}, x_N=\eta(x’,t)\}$ at time $t\geq 0$ and surface tension is included on $\Gamma(t)$. The fluid occupying the region $x_N \gt \eta(x’,t)$ is called the upper fluid, while the other fluid occupying the region $x_N \lt \eta(x’,t)$ is called the lower fluid. It is well-known that the trivial steady state, i.e., the motionless state with the flat interface $x_N=0$, is unstable if the upper fluid is heavier than the lower one due to gravity. This instability is called the Rayleigh–Taylor instability. On the other hand, the present paper treats the following two cases: (i) the lower fluid is heavier than the upper one; (ii) the two fluids have equal density. For these two cases, we prove time decay estimates of $L_p\text{-}L_q$ type for the Stokes semigroup associated with a linearized system of the above two-phase problem. We emphasize that the decay rate of the semigroup generated by the fractional Laplacian appears in the $L_p\text{-}L_q$ time decay estimates of the Stokes semigroup.
