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

Research output: Contribution to journalArticle


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.


External organisations
  • Swiss Federal Institute of Technology
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematics


  • Boundary conditions, Incompressible flows, Nash-Moser iteration method, Vorticity
Original languageEnglish
Pages (from-to)1225-1258
Number of pages34
JournalAnalysis and PDE
Issue number5
Publication statusPublished - 2019
Publication categoryResearch