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

Buffoni, Boris and Wahlén, Erik LU (2019) In Analysis and PDE 12(5). p.1225-1258
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... (More)

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.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Boundary conditions, Incompressible flows, Nash-Moser iteration method, Vorticity
in
Analysis and PDE
volume
12
issue
5
pages
34 pages
publisher
Mathematical Sciences Publishers
external identifiers
  • scopus:85058886417
ISSN
2157-5045
DOI
10.2140/apde.2019.12.1225
language
English
LU publication?
yes
id
20ec175a-b5bc-48bd-b4ff-a33704a9408a
date added to LUP
2019-01-02 14:43:27
date last changed
2019-02-20 11:41:14
@article{20ec175a-b5bc-48bd-b4ff-a33704a9408a,
  abstract     = {<p>We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region D = (0, L) × ℝ<sup>2</sup>. 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|<sup>2</sup> + 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.</p>},
  author       = {Buffoni, Boris and Wahlén, Erik},
  issn         = {2157-5045},
  keyword      = {Boundary conditions,Incompressible flows,Nash-Moser iteration method,Vorticity},
  language     = {eng},
  number       = {5},
  pages        = {1225--1258},
  publisher    = {Mathematical Sciences Publishers},
  series       = {Analysis and PDE},
  title        = {Steady three-dimensional rotational flows : An approach via two stream functions and Nash-Moser iteration},
  url          = {http://dx.doi.org/10.2140/apde.2019.12.1225},
  volume       = {12},
  year         = {2019},
}