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Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks

Marco Bravin, Franck Sueur

TL;DR

This work extends the classical 2D Euler theory by incorporating boundary sources and sinks, prescribing the entering vorticity and part of the boundary velocity. It develops a vorticity-based formulation, employing a div-curl velocity reconstruction and a hydrodynamical Biot–Savart operator to handle the nontrivial topology from multiple holes. Using smooth viscous approximations and compactness arguments, the authors prove the existence of weak solutions in different senses: distributional for $p>\tfrac{4}{3}$, renormalized for $p>1$, and symmetrized for $p=1$, all obtained via vanishing viscosity limits. The results illuminate the circulation dynamics around sources/sinks, establish duality-based arguments for renormalized solutions, and adapt symmetrization techniques to nontrivial geometries, thereby broadening the scope of Yudovich-type theory to permeable boundaries with injections and withdrawals.

Abstract

A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed. In this paper we investigate the existence of weak solutions to this system by relying on \textit{a priori} bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a $L^p$ integrability in space, with $p $ in $[1,+\infty]$, and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where $p >\frac43$, in the renormalized sense in the case where $p >1$, and in a symmetrized sense in the case where $p =1$.

Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks

TL;DR

This work extends the classical 2D Euler theory by incorporating boundary sources and sinks, prescribing the entering vorticity and part of the boundary velocity. It develops a vorticity-based formulation, employing a div-curl velocity reconstruction and a hydrodynamical Biot–Savart operator to handle the nontrivial topology from multiple holes. Using smooth viscous approximations and compactness arguments, the authors prove the existence of weak solutions in different senses: distributional for , renormalized for , and symmetrized for , all obtained via vanishing viscosity limits. The results illuminate the circulation dynamics around sources/sinks, establish duality-based arguments for renormalized solutions, and adapt symmetrization techniques to nontrivial geometries, thereby broadening the scope of Yudovich-type theory to permeable boundaries with injections and withdrawals.

Abstract

A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed. In this paper we investigate the existence of weak solutions to this system by relying on \textit{a priori} bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a integrability in space, with in , and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where , in the renormalized sense in the case where , and in a symmetrized sense in the case where .

Paper Structure

This paper contains 43 sections, 18 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Transport of vorticity
  3. Velocity field as solution of an elliptic problem
  4. Dynamics of the circulations around the sources and sinks
  5. Decomposition of the velocity
  6. Formal vorticity formulation
  7. A first glance on the main results
  8. Precise existence results of distributional and renormalized solutions
  9. Existence of distributional solutions
  10. Existence of renormalized solutions
  11. Precise existence results of symmetrized solutions
  12. Case of a full plane
  13. Case of a general domain
  14. Symmetrized formulation with $L^{1}$ vorticity
  15. Remainder on the transport equation with given non-tangential velocity
  16. ...and 28 more sections

Key Result

Theorem 1

Let $p$ in $[1,+\infty]$ and $(\omega^{in} , \omega^{+} )$ a couple of input vorticities (CIV) in $L^p$. Then there is $(\omega , \omega^{-} )$ a continuous couple (CC) with values in $L^p$ solution of the system vort-ss2. This solution have to be understood in different ways depending on the range In each of these cases, these solutions can be obtained as vanishing viscosity limits.

Figures (1)

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

Theorems & Definitions (44)

  • Definition 1: Source-sink compatible
  • Remark 1
  • Remark 2
  • Definition 2: Couple of input vorticities
  • Definition 3: Continuous and weakly continuous couple
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • ...and 34 more