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

Buffoni, Boris; Wahlén, Erik

doi:10.2140/apde.2019.12.1225

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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

DOI: 10.2140/apde.2019.12.1225

Abstract

We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region $D = (0, L) \times ℝ^{2}$ . 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 $\partial D$ . The Bernoulli equation states that the “Bernoulli function” $H : = \frac{1}{2} | v |^{2} + p$ (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 $\partial D$ . 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 $v = \nabla f \times \nabla g$ 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 $\partial D$ , 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

Vol. 12, No. 5, 2019