We consider the stationary flow of an inviscid and incompressible fluid of constant density in
the region
.
We are concerned with flows that are periodic in the second and third
variables and that have prescribed flux through each point of the boundary
.
The Bernoulli equation states that the “Bernoulli function”
(where
is the velocity
field and
the
pressure) is constant along stream lines, that is, each particle is associated with a particular value of
. We also prescribe
the value of
on
.
The aim of this work is to develop an existence theory near a given
constant solution. It relies on writing the velocity field in the form
and deriving a degenerate nonlinear elliptic system for
and
. This system
is solved using the Nash–Moser method, as developed for the problem of isometric embeddings
of Riemannian manifolds; see, e.g., the book by Q. Han and J.-X. Hong (2006). Since we can
allow
to be
nonconstant on
,
our theory includes three-dimensional flows with nonvanishing vorticity.