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The vortex blob method as a second-grade non-Newtonian fluid

Marcel Oliver, Steve Shkoller

TL;DR

The paper rigorously connects Chorin's vortex blob regularization in 2D to the second-grade (Euler-$\alpha$) fluid model by recasting the problem as a geodesic flow on the volume-preserving diffeomorphism group with a weak metric. It establishes global existence and uniqueness of weak solutions for measure-valued vorticity, and introduces a weak co-adjoint framework that shows invariance of co-adjoint orbits under the blob flow, including point-vortex data. Kernel estimates ensure quasi-Lipschitz control of the regularized and classical Biot–Savart kernels, enabling a Kato-type well-posedness argument and a robust convergence theory. The convergence result demonstrates that vortex blob solutions converge to Euler solutions as the blob size vanishes, even when starting from measure-valued (e.g., point-vortex) data, provided the initial data are approximated weakly with bounded total variation. Collectively, these results provide a rigorous PDE foundation and convergence theory for grid-based vortex methods used in simulating 2D incompressible flows.

Abstract

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures ${\mathcal M}({\mathbb R}^2)$. The solutions of this regularized PDE, also known as the averaged Euler or Euler-$α$ equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in ${\mathcal M}({\mathbb R}^2)$ such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes of the type that are usually used for vortex methods.

The vortex blob method as a second-grade non-Newtonian fluid

TL;DR

The paper rigorously connects Chorin's vortex blob regularization in 2D to the second-grade (Euler-) fluid model by recasting the problem as a geodesic flow on the volume-preserving diffeomorphism group with a weak metric. It establishes global existence and uniqueness of weak solutions for measure-valued vorticity, and introduces a weak co-adjoint framework that shows invariance of co-adjoint orbits under the blob flow, including point-vortex data. Kernel estimates ensure quasi-Lipschitz control of the regularized and classical Biot–Savart kernels, enabling a Kato-type well-posedness argument and a robust convergence theory. The convergence result demonstrates that vortex blob solutions converge to Euler solutions as the blob size vanishes, even when starting from measure-valued (e.g., point-vortex) data, provided the initial data are approximated weakly with bounded total variation. Collectively, these results provide a rigorous PDE foundation and convergence theory for grid-based vortex methods used in simulating 2D incompressible flows.

Abstract

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures . The solutions of this regularized PDE, also known as the averaged Euler or Euler- equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes of the type that are usually used for vortex methods.

Paper Structure

This paper contains 5 sections, 10 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Kernel estimates
  3. Well-posedness
  4. Weak co-adjoint action and reduction
  5. Convergence

Key Result

Theorem 1

For initial data $q_0\in {\mathcal{M}}({\mathbb R}^2)$, there exists a unique global weak solution to e.alpha-vorticity with where the subscript ${\mathrm {div}}$ denotes divergence-free. As a consequence, the co-Adjoint action $\operatorname{Ad}^*_{\eta}(q)$ and the weak co-adjoint action $\operatorname{w-ad}^*_{u}(q)$ are conserved.

Theorems & Definitions (25)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Lemma 3
  • ...and 15 more