Abstract
The authors consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. Three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients are established. The availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis are used. The paper is organized as follows. In Section 2, the three formulations of the problem are introduced and their equivalence is established. In Section 3, the authors first introduce the notion of a weak solution to Dubreil-Jacotin's formulation and establish, by means of a shooting method, the existence of at least one laminar flow solution to this latter formulation. In Section 4, the equations are reformulated as an abstract bifurcation problem by using methods from nonlinear functional analysis.
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