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.
"Steady three-dimensional rotational flows: an approach via two stream functions and Nash–Moser iteration." Anal. PDE 12 (5) 1225 - 1258, 2019. https://doi.org/10.2140/apde.2019.12.1225