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2019 Steady three-dimensional rotational flows: an approach via two stream functions and Nash–Moser iteration
Boris Buffoni, Erik Wahlén
Anal. PDE 12(5): 1225-1258 (2019). DOI: 10.2140/apde.2019.12.1225


We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region D=(0,L)×2. 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:=12|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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Boris Buffoni. Erik Wahlén. "Steady three-dimensional rotational flows: an approach via two stream functions and Nash–Moser iteration." Anal. PDE 12 (5) 1225 - 1258, 2019.


Received: 18 September 2017; Revised: 26 July 2018; Accepted: 18 October 2018; Published: 2019
First available in Project Euclid: 5 January 2019

zbMATH: 07006760
MathSciNet: MR3892402
Digital Object Identifier: 10.2140/apde.2019.12.1225

Primary: 35G60, 35Q31, 58C15, 76B03, 76B47

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.12 • No. 5 • 2019
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