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Uniqueness of Yudovich's solutions to the 2D incompressible Euler equation despite the presence of sources and sinks

Florent Noisette, Franck Sueur

TL;DR

This work establishes the uniqueness of Yudovich-type weak solutions to the 2D incompressible Euler equations in a smooth, multiply-connected domain with interior sources and sinks (permeable boundary) under bounded vorticity. Building on the Weigant–Papin framework for rectangles, the authors develop a two-pronged energy method: a direct energy estimate for the velocity difference and a second estimate obtained via an auxiliary harmonic test function, augmented by a generalized Lamb-type identity. A careful regularity analysis of the difference stream function and an Osgood-type argument yield a quantitative stability estimate, which implies uniqueness for identical data and provides a robust modulus of continuity with respect to initial and boundary data. The approach highlights the feasibility of handling non-local boundary conditions and multiple interfaces in fluid-structure-like settings, extending the classical Yudovich theory to permeable boundaries with interior sources and sinks. The results have implications for controllability and stability analyses of 2D flows with inflow/outflow, and may inform extensions to related nonlinear PDEs with non-conservative boundary conditions.

Abstract

In $1962$, Yudovich proved the existence and uniqueness of classical solutions to the 2D incompressible Euler equations in the case where the fluid occupies a bounded domain with entering and exiting flows on some parts of the boundary. The normal velocity is prescribed on the whole boundary, as well as the entering vorticity. The uniqueness part of Yudovich's result holds for Hölder vorticity, by contrast with his 1961 result on the case of an impermeable boundary, for which the normal velocity is prescribed as zero on the boundary, and for which the assumption that the initial vorticity is bounded was shown to be sufficient to guarantee uniqueness. Whether or not uniqueness holds as well for bounded vorticities in the case of entering and exiting flows has been left open until $2014$, when Weigant and Papin succeeded to tackle the case where the domain is a rectangle. In this paper we adapt Weigant and Papin's result to the case of a smooth domain with several internal sources and sinks.

Uniqueness of Yudovich's solutions to the 2D incompressible Euler equation despite the presence of sources and sinks

TL;DR

This work establishes the uniqueness of Yudovich-type weak solutions to the 2D incompressible Euler equations in a smooth, multiply-connected domain with interior sources and sinks (permeable boundary) under bounded vorticity. Building on the Weigant–Papin framework for rectangles, the authors develop a two-pronged energy method: a direct energy estimate for the velocity difference and a second estimate obtained via an auxiliary harmonic test function, augmented by a generalized Lamb-type identity. A careful regularity analysis of the difference stream function and an Osgood-type argument yield a quantitative stability estimate, which implies uniqueness for identical data and provides a robust modulus of continuity with respect to initial and boundary data. The approach highlights the feasibility of handling non-local boundary conditions and multiple interfaces in fluid-structure-like settings, extending the classical Yudovich theory to permeable boundaries with interior sources and sinks. The results have implications for controllability and stability analyses of 2D flows with inflow/outflow, and may inform extensions to related nonlinear PDEs with non-conservative boundary conditions.

Abstract

In , Yudovich proved the existence and uniqueness of classical solutions to the 2D incompressible Euler equations in the case where the fluid occupies a bounded domain with entering and exiting flows on some parts of the boundary. The normal velocity is prescribed on the whole boundary, as well as the entering vorticity. The uniqueness part of Yudovich's result holds for Hölder vorticity, by contrast with his 1961 result on the case of an impermeable boundary, for which the normal velocity is prescribed as zero on the boundary, and for which the assumption that the initial vorticity is bounded was shown to be sufficient to guarantee uniqueness. Whether or not uniqueness holds as well for bounded vorticities in the case of entering and exiting flows has been left open until , when Weigant and Papin succeeded to tackle the case where the domain is a rectangle. In this paper we adapt Weigant and Papin's result to the case of a smooth domain with several internal sources and sinks.

Paper Structure

This paper contains 16 sections, 19 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometry of the domain
  3. The equations at stake
  4. An open problem on the uniqueness of solutions with bounded vorticity
  5. Reminder on the definition and existence of solutions with bounded vorticity
  6. Statement of the main result
  7. Statement of the main result and a few remarks
  8. Strategy of the proof of Theorem \ref{['main-thm']} and organization of the rest of the paper.
  9. Equations satisfied by the difference of two solutions
  10. Estimate of the stream function of the difference
  11. Energy estimate of the difference
  12. An auxiliary test function
  13. A generalized Lamb-type lemma
  14. An auxiliary energy-type estimate
  15. Osgood argument and end of the proof
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $(\mathcal{C}_i)_i\in \mathbb{R}^{I^*}$, $g$ in the Hölder space $C^{1,\alpha}(\Gamma)$ for some $\alpha\in (0,1)$, with zero average on $\Gamma$, and $\omega$ in $L^{\infty}(\Omega)$. Then the system ellip admits an unique solution Moreover there is a constant $C>0$ such that, for all $p\geq 2$,

Figures (1)

  • Figure 1: Example of a fluid domain with one source and two sinks.

Theorems & Definitions (46)

  • Proposition 2.1
  • Remark 2.2
  • Definition 2.1
  • Remark 2.3
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 36 more