Abstract
The aim of this paper is to show time-decay estimates of solutions to linearized two-phase Navier-Stokes equations with surface tension and gravity. The original two-phase Navier-Stokes equations describe the two-phase incompressible viscous flow with a sharp interface that is close to the hyperplane xN=0 in the N-dimensional Euclidean space, N≥2. It is well-known that the Rayleigh–Taylor instability occurs when the upper fluid is heavier than the lower one, while this paper assumes that the lower fluid is heavier than the upper one and proves time-decay estimates of Lp-Lq type for the linearized equations. Our approach is based on solution formulas for a resolvent problem associated with the linearized equations.
Highlights
Let us consider the motion of two immiscible, viscous, incompressible capillary fluids, fluid+ and fluid−, in the N-dimensional Euclidean space R N for N ≥ 2
Rayleigh–Taylor instability occurs when the upper fluid is heavier than the lower one, while this paper assumes that the lower fluid is heavier than the upper one and proves time-decay estimates of
Our approach is based on solution formulas for a resolvent problem associated with the linearized equations
Summary
Let us consider the motion of two immiscible, viscous, incompressible capillary fluids, fluid+ and fluid− , in the N-dimensional Euclidean space R N for N ≥ 2. Navier-Stokes equations with surface tension, but gravity is not taken into account In this case, they proved the zeros λ± (|ξ 0 |) of the boundary symbol satisfy λ± (|ξ 0 |) = ±ic5 |ξ 0 |3/2 − c6 (1 ± i )|ξ 0 |7/4 + o (|ξ 0 |7/4 ). B q1 (Ṙ N ), which satisfies for any λ ∈ Σε , (8) admits a unique solution (v, q) ∈ Hq2 (Ṙ N ) N × H k(λv, λ1/2 ∇v, ∇2 v, ∇q)k Lq (Ṙ N ) ≤ CN,q,ε kfk Lq (Ṙ N ) These results hold for any ρ± > 0 and play a key role in proving time-decay estimates of solutions for (1) in the present paper.
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