Steady three-dimensional rotational flows: existence via Kato's approach to locally coercive problems

TL;DR

This paper proves the existence of steady three-dimensional rotational flows in a specific domain using Kato's approach, improving upon previous methods by reducing regularity requirements and establishing solutions as local minimizers.

Contribution

It adapts Kato's method to locally coercive problems for steady flows, providing more precise regularity conditions and a variational characterization of solutions.

Findings

Existence of solutions with minimal Sobolev regularity

Solutions are local minimizers of an associated functional

Method applies to a simplified PDE with weaker ellipticity

Abstract

Stationary flows of an inviscid and incompressible fluid of constant density in the region $D=(0, L)\times \mathbb R^2$, periodic in the second and third variables, are considered. The flux and the Bernoulli function are prescribed at each point of the boundary $\partial D$. The previous existence proof relying on the Nash-Moser iteration scheme is replaced by an adaptation of Kato's approach to locally coercive problems, allowing a more precise statement: the regularity required in Sobolev spaces is the one needed to ensure a basic local coercivity property, and there is a loss of control of only two derivatives in the obtained solutions. The underlying variational structure gives an additional property: the obtained solutions are local minimizers of an integral functional. The strategy of proof is first developed for a simpler nonlinear partial differential equation in two variables which satisfies a weaker form of ellipticity.