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Mean-Field Convergence of Point Vortices to the Incompressible Euler Equation with Vorticity in \(L^\infty \)

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Abstract

We consider the classical point vortex model in the mean-field scaling regime, in which the velocity field experienced by a single point vortex is proportional to the average of the velocity fields generated by the remaining point vortices. We show that if at some time the associated sequence of empirical measures converges in a renormalized \({\dot{H}}^{-1}\) sense to a probability measure with density \(\omega ^0\in L^\infty ({\mathbb {R}}^2)\) and having finite energy as the number of point vortices \(N\rightarrow \infty \), then the sequence converges in the weak-* topology for measures to the unique solution \(\omega \) of the 2D incompressible Euler equation with initial datum \(\omega ^0\), locally uniformly in time. In contrast to previous results Schochet (Commun Pure Appl Math 49:911–965, 1996), Jabin and Wang (Invent Math 214:523–591, 2018), Serfaty (Duke Math J 169:2887–2935, 2020), our theorem requires no regularity assumptions on the limiting vorticity \(\omega \), is at the level of conservation laws for the 2D Euler equation, and provides a quantitative rate of convergence. Our proof is based on a combination of the modulated-energy method of Serfaty (J Am Math Soc 30:713–768, 2017) and a novel mollification argument. We contend that our result is a mean-field convergence analogue of the famous theorem of Yudovich (USSR Comput Math Math Phys 3:1407–1456, 1963) for global well-posedness of 2D Euler with vorticity in the scaling-critical function space \(L^\infty ({\mathbb {R}}^2)\).

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Notes

  1. Even if \(\omega \) is compactly supported, the velocity field u need not be Lipschitz.

  2. See [2] for an example, and see [32] and references therein for more discussion on this point.

  3. See Definition 2.21 for the notion of weak solution intended here.

  4. As is perhaps well-known, the mean-field limit of the Lieb-Liniger model is the one-dimensional cubic nonlinear Schrödinger equation.

  5. As is standard notation, a dot superscript indicates a homogeneous semi-norm for a function space in this paper.

  6. The Banach space \((B_{2,\infty }^{-1}({\mathbb {R}}^2),\Vert \cdot \Vert _{B_{2,\infty }^{-1}({\mathbb {R}}^2)})\) is isomorphic to the dual of the Banach space \((B_{2,1}^{1}({\mathbb {R}}^2),\Vert \cdot \Vert _{B_{2,1}^1({\mathbb {R}}^2)})\).

  7. We note that the truncation parameter \({\underline{\eta }}_N\) is allowed to vary in i. This was an important new contribution from [53], which we shall also make use of in our work.

  8. This inequality is precisely where we use that \(v_{\epsilon _2}\) is Lipschitz.

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Acknowledgements

The author thanks Nataša Pavlović for helpful comments on an earlier draft of the manuscript, as well as Sylvia Serfaty for her feedback, which has improved the discussion in Sect. 1.2 and corrected an earlier misattribution, and Fang-Hua Lin for helpful information concerning references. The author also thanks the anonymous reviewers for their suggestions for improving the exposition. The author acknowledges financial support from the Simons Foundation and from The University of Texas at Austin through a Provost Graduate Excellence Fellowship.

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  1. Department of Mathematics, Massachusetts Institute of Technology, Simons Building, Room 106, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Matthew Rosenzweig

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Rosenzweig, M. Mean-Field Convergence of Point Vortices to the Incompressible Euler Equation with Vorticity in \(L^\infty \). Arch Rational Mech Anal 243, 1361–1431 (2022). https://doi.org/10.1007/s00205-021-01735-3

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