Bernoulli free-boundary problems in strip-like domains and a property of permanent waves on water of finite depth

Abstract

We study weak solutions for a class of free boundary problems which includes as a special case the classical problem of traveling waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function.

On a Variational Approach for Water Waves

“…In particular, we show that local minimizers are both "weak solutions" and "variational solutions" of (1.6) in the sense of Varvaruca and Weiss [30] (see Definition 3.1 and 3.2 in [30], see also the work of Shargorodsky and Toland [25] and Varvaruca [29]). In particular, the monotonicity formula established in Theorem 3.5 of [30] applies to local minimizers.…”

New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

“…All the equations in this paper are for non-overturning waves, so η is a continuous function of x. For pure gravity waves, there are no overhanging travelling waves of permanent form [34,43]. The situation is different in presence of surface tensions where overhanging solutions exist [27,42].…”

Pressure beneath a Stokes wave

“…Several assumptions encompassed in the definition of a Stokes wave are not restrictive requirements. The free surface must always be a graph (not an overhanging wave) [32,33]. Moreover, assuming the free boundary to be a continuously differentiable curve, the boundary must be a real-analytic curve with a parametrization y D Á.x/, and the velocity components must have harmonic extensions across it [19,29].…”