Solvability of a supercritical free surface flow problem under gravity-capillarity

In this paper, we consider a problem of a supercritical free surface flow over an obstacle lying on the bottom of a channel in 2D. The flow is irrotational, stationary and the fluid is ideal and incompressible. We take into account both the gravity and the effects of the superficial tension. The problem is nonlinear, it is formulated by the Laplace operator and the dynamic condition defined on the free surface of the fluid domain (Bernoulli equation). Using the perturbation stream function, we linearize the problem and we give a priori properties of the solution. These a priori properties allow us to construct a space where we can use the Lax–Milgram’s theorem to prove the existence and the uniqueness of the solution of the problem.