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

Boris Buffoni, Eric Séré

TL;DR

The work develops a streamlined existence theory for steady three-dimensional rotational Euler flows by replacing the Nash–Moser scheme with Kato’s degree-theory approach to locally coercive problems. It introduces an admissible-pair formulation $(f,g)$ with $v=\nabla f\times\nabla g$ and a corresponding variational integral whose critical points yield Euler flows, and it proves a local coercivity inequality with a loss of two derivatives. The analysis proceeds via a two-stage strategy: first a two-dimensional toy problem to illustrate the method, then the hydrodynamic problem in three dimensions with a similar variational structure, yielding a local solution in $H^s_{loc}$ for suitable $s$ under small data. The results also deliver a local minimization principle and uniqueness in a small neighborhood, illustrating the practical impact of Kato’s approach for nonlinear, non-elliptic PDEs in fluid dynamics.

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.

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

TL;DR

The work develops a streamlined existence theory for steady three-dimensional rotational Euler flows by replacing the Nash–Moser scheme with Kato’s degree-theory approach to locally coercive problems. It introduces an admissible-pair formulation with and a corresponding variational integral whose critical points yield Euler flows, and it proves a local coercivity inequality with a loss of two derivatives. The analysis proceeds via a two-stage strategy: first a two-dimensional toy problem to illustrate the method, then the hydrodynamic problem in three dimensions with a similar variational structure, yielding a local solution in for suitable under small data. The results also deliver a local minimization principle and uniqueness in a small neighborhood, illustrating the practical impact of Kato’s approach for nonlinear, non-elliptic PDEs in fluid dynamics.

Abstract

Stationary flows of an inviscid and incompressible fluid of constant density in the region , periodic in the second and third variables, are considered. The flux and the Bernoulli function are prescribed at each point of the boundary . 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.
Paper Structure (23 sections, 15 theorems, 234 equations)

This paper contains 23 sections, 15 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. On the stationary Euler equation in dimension $3$
  3. A simpler problem in dimension $2$
  4. The toy problem
  5. Proof of Theorem \ref{['thm: main']}
  6. Proof of the local coercivity inequality \ref{['eq: conclusion']}
  7. The integral $I_1$
  8. The integrals $I_2$ and $I_3$ (inspired by Kohn-Nirenberg KoNi)
  9. The integral $I_4$
  10. The integral $I_5$
  11. The integrals $I_6$ and $I_7$
  12. The integral $I_8$
  13. End of the proof of \ref{['eq: conclusion']}
  14. Appendix
  15. The hydrodynamic problem
  16. ...and 8 more sections

Key Result

Theorem 1

Let us fix $L,P_y,P_z>0$ and affine functions $\overline f$ and $\overline g$ on $D$ such that Consider a pair $(\widetilde{f}_0,\widetilde{g}_0)\in H^7_{loc}(D)$ such that $\widetilde{f}_0-\overline f$ and $\widetilde{g}_0-\overline g$ are $(P_y,P_z)$-periodic in $y$ and $z$, and a function $H:\mathbb{R}^2\rightarrow \mathbb{R}$ of class $C^6$ such that $\nabla H$ admits the vectorial period Ass

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Kato Kato
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • ...and 17 more