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

2011/09/26 by Yanjin Wang, Ian Tice, Chanwoo Kim · 84 citations
Mathematics · #Advanced Mathematical Physics Problems #Boundary (topology) #Boundary value problem #Classical mechanics #Compressibility #Exponential decay #Exponential function #Exponential stability #Free surface #Geometric Analysis and Curvature Flows #Geometry #Instability #Mathematical analysis #Mathematics #Mechanics #Navier-Stokes equation solutions #Nonlinear system #Physics #Rayleigh–Taylor instability #Surface (topology) #Surface tension #Thermodynamics #Viscous liquid #math.AP #msc:35B40 #msc:35Q30 #msc:35R35 #msc:76D03 #msc:76D45 #msc:76E17

paper · pdf · doi:10.1007/s00205-013-0700-2

published in Archive for Rational Mechanics and Analysis 212(1), 1-92 (Springer Science+Business Media) · 70 pages; v2: typos and minor errors corrected

arxiv created 2011/09/26 · openalex publication_date 2013/12/17 · arxiv updated 2015/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 three-dimensional 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.

Citations

• Almost Exponential Decay of Periodic Viscous Surface Waves without Surface Tension

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