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

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### Bibtex

@article{20ec175ab5bc48bdb4ffa33704a9408a,

title = "Steady three-dimensional rotational flows: An approach via two stream functions and Nash-Moser iteration",

abstract = "We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region D = (0, L) × ℝ2. 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 ∂D. The Bernoulli equation states that the {"}Bernoulli function{"} H := 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 also prescribe the value of H on ∂D. 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 v = ∇ f × ∇g and deriving a degenerate nonlinear elliptic system for f and g. 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 H to be nonconstant on ∂D, our theory includes three-dimensional flows with nonvanishing vorticity.",

keywords = "Boundary conditions, Incompressible flows, Nash-Moser iteration method, Vorticity",

author = "Boris Buffoni and Erik Wahl{\'e}n",

year = "2019",

doi = "10.2140/apde.2019.12.1225",

language = "English",

volume = "12",

pages = "1225--1258",

journal = "Analysis and PDE",

issn = "2157-5045",

publisher = "Mathematical Sciences Publishers",

number = "5",

}