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

Abstract

We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region $D=\left(0,L\right)\times {ℝ}^2$. 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.