The Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay

Abstract

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a threedimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh-Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.Notice that J = det ∇Θ is the Jacobian of the coordinate transformation. We also denote W = ∂ t Θ 3 K N = (−∂ 1 η, −∂ 2 η, 1)T i = e i + ∂ i ηe 3 for i = 1, 2 θ ij = ∂ j Θ i for i, j = 1, 2, 3.(1.23)

The nonlinear stability regime of the viscous Faraday wave problem

“…Such an omission is justified by the abundance of similar local existence results based on the corresponding a priori estimates. We refer, for instance, to the works [10,13,23,27,29].…”

Wave propagation in a three-layer semi-infinite hydrodynamic system with a rigid lid

“…In paper [1], the problem of oscillations of the contact surface of two non-mixed viscous non-compressible liquids over the hard bottom in the gravitational field was explored. Correctness of the problem both with taking into account surface tension and without it was proved.…”

On the stability of Rayleigh–Taylor problem for stratified rotating viscoelastic fluids

“…Proof Xu et al [39] have established the local and global well-posedness results of the onelayer viscoelastic fluid model with upper free boundary. Exploiting the regularity theory for the stratified viscous flows in [38], we can easily extend the well-posedness results in [39] to the transformed SRVRT problem by a standard iteration method, and thus obtain Proposition 5.1.…”

“…From the physical point of view, the velocity of two viscous fluids meeting at a free boundary is continuous across the interface and the jump in the normal stress is proportional to the mean curvature of the surface multiplied by the normal to the surface (see [38]). Thus, we have the jump conditions tvu = 0 on (t) and…”

“…In the last decades, this phenomenon has been extensively investigated from both physical and numerical aspects, see [3,8,17,19,23,36] for examples. It has been also widely investigated how the RT instability evolves under the effects of other physical factors, internal surface tension [9,38], magnetic fields [3,4,18,21,22,24,25], and so on.…”