Viscous surface-internal waves in a horizontally infinite layer

We investigate the surface-internal wave problem for the two-phase incompressible Navier–Stokes equations with gravity in a horizontally infinite layer. When the upper fluid is less dense than the lower, we establish the global well-posedness and asymptotic stability of the problem in flattening coordinates for the small initial data, which are uniform with respect to the surface tension coefficients. In $ 2D $, owing to the slower decay rate compared to the $ 3D $ case, the initial data is assumed additionally to be small in the negative Besov space $ \dot{B}^{-1/2}_{2, \infty} $.