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",

}