Steady three-dimensional rotational flows : an approach via two stream functions and Nash-Moser iteration. (English) Zbl 1406.35244

Summary: We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region \(D=(0, L)\times \mathbb{R}^2\). 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 [Isometric embedding of Riemannian manifolds in Euclidean spaces. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1113.53002)]. Since we can allow \(H\) to be nonconstant on \(\partial D\), our theory includes three-dimensional flows with nonvanishing vorticity.


35Q31 Euler equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B47 Vortex flows for incompressible inviscid fluids
35G60 Boundary value problems for systems of nonlinear higher-order PDEs
58C15 Implicit function theorems; global Newton methods on manifolds
35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations


Zbl 1113.53002
Full Text: DOI arXiv


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