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We consider the stationary flow of an inviscid and incompressible fluid ofconstant density in the region $D=(0, L)\times \mathbb{R}^2$. We are concernedwith flows that are periodic in the second and third variables and that haveprescribed flux through each point of the boundary $\partial D$. The Bernoulliequation states that the "Bernoulli function" $H:= \frac 1 2 |v|^2+p$ (where$v$ is the velocity field and $p$ the pressure) is constant along stream lines,that is, each particle is associated with a particular value of $H$. We alsoprescribe the value of $H$ on $\partial D$. The aim of this work is to developan existence theory near a given constant solution. It relies on writing thevelocity field in the form $v=\nabla f\times \nabla g$ and deriving adegenerate nonlinear elliptic system for $f$ and $g$. This system is solvedusing the Nash-Moser method, as developed for the problem of isometricembeddings of Riemannian manifolds; see e.g. the book by Q. Han and J.-X. Hong(2006). Since we can allow $H$ to be non-constant on $\partial D$, our theoryincludes three-dimensional flows with non-vanishing vorticity.

Steady three-dimensional rotational flows: an approach via two stream functions and Nash-Moser iteration