Publication 18-CNA-015
The Viscous Surface Wave Problem With Generalized Surface Energies
Antoine Remond-Tiedrez
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
aremondt@andrew.cmu.edu
Ian Tice
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
ian.tice@andrew.cmu.edu
Abstract: We study a three-dimensional incompressible viscous
fluid in a horizontally periodic domain
with finite depth whose free boundary is the graph of a function. The
fluid is subject to gravity and
generalized forces arising from a surface energy. The surface energy incorporates both bending and surface
tension effects. We prove that for initial conditions sufficiently close to equilibrium the problem is globally
well-posed and solutions decay to equilibrium exponentially fast, in an appropriate norm. Our proof is
centered around a nonlinear energy method that is coupled to careful estimates of the fully nonlinear surface
energy.
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