Vol. 12, No. 5, 2019

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

Boris Buffoni and Erik Wahlén

Vol. 12 (2019), No. 5, 1225–1258
Abstract

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 := 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
incompressible flows, vorticity, boundary conditions, Nash–Moser iteration method
Mathematical Subject Classification 2010
Primary: 35Q31, 76B03, 76B47, 35G60, 58C15
Milestones
Received: 18 September 2017
Revised: 26 July 2018
Accepted: 18 October 2018
Published: 15 December 2018
Authors
Boris Buffoni
Institut de mathématiques
Station 8
Ecole Polytechnique Fédérale de Lausanne
Lausanne
Switzerland
Erik Wahlén
Centre for Mathematical Sciences
Lund University
Lund
Sweden