The classical point vortex model is the system of \(N\) ordinary differential equations written as: \(\overset{.}{x}_{i}(t)=\sum_{1\leq j\leq N,j\neq i}a_{j}(\nabla ^{\perp }\mathfrak{g})(x_{i}(t)-x_{j}(t))\), \(i=1,\dots,N\), where \(N\) is the number of vortices, \(a_{j}\in \mathbb{R}\setminus \{0\}\), \( j=1,\dots,N\), the intensities of the vortices, \(\mathfrak{g}(x)=-\frac{1}{ 2\pi }\ln \left\vert x\right\vert \) the 2D Coulomb potential, and \(\nabla ^{\perp }=(-\partial _{x_{2}},\partial _{x_{1}})\) the perpendicular gradient.The initial condition\(x_{i}(0)=x_{i}^{0}\) is imposed, where \( x_{1}^{0},\dots,x_{N}^{0}\in \mathbb{R}^{2}\) are pairwise distinct initial positions. This model is an idealization of a 2D incompressible and inviscid fluid flow, which may be described in a vorticity form by the 2D incompressible Euler equation: \(\partial _{t}\omega +u\cdot \nabla \omega =0\) , \(u=(\nabla ^{\perp }\mathfrak{g})\ast \omega \), \((t,x)\in \lbrack 0,\infty )\times \mathbb{R}^{2}\), with the initial condition for the vorticity \( \omega (0,x)=\omega ^{0}(x)\). The first main result proves the existence of an absolute constant \(C>0\) such that a weak solution \(\omega \in L^{\infty }([0,\infty );\mathcal{P}(\mathbb{R}^{2})\cap L^{\infty }(\mathbb{R}^{2}))\) to the Euler equation with initial datum \(\omega ^{0}\) satisfies \(\int_{ \mathbb{R}^{2}}\ln\left\langle x\right\rangle \omega ^{0}(x)dx+\int_{(\mathbb{ R}^{2})^{2}}\ln\left\langle x-y\right\rangle \omega ^{0}(x)\omega ^{0}(y)dxdy<\infty \), \(\mathcal{P}\) denoting the space of probability measures. Let \(N\in \mathbb{N}\), \(\underline{x}_{N}\in C^{\infty }([0,\infty );(\mathbb{R}^{2})^{N}\setminus \Delta _{N})\) be a solution to the point vortex system and \(\mathfrak{F}_{N}^{\mathrm{avg}}:[0,\infty )\rightarrow \mathbb{R}\) be the functional defined as \(\mathfrak{F}_{N}^{\mathrm{avg}}(\underline{x} _{N}(t),\omega (t))=\int_{(\mathbb{R}^{2})^{2}\setminus \Delta _{2}} \mathfrak{g}(x-y)d(\omega _{N}-\omega )(t,x)d(\omega _{N}-\omega )(t,y)\), where \(\omega _{N}\) is the empirical measure defined as \(\omega _{N}(t,x)=\sum_{i=1}^{N}a_{i}\delta _{x_{i}(t)}(x)\) and being a solution to the weak vorticity formulation of the Euler equation. Here \(\Delta _{N}=\{(y_{1},\dots,y_{N})\in (\mathbb{R}^{2})^{N}\): \(\exists 1\leq i=j\leq N\) such that \(y_{i}=y_{j}\}\). If for given \(t>0\), \(N\in \mathbb{N}\) sufficiently large so that \(\frac{Ct(\left\Vert \omega ^{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}^{1/2}+\left\Vert \omega ^{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}^{3/2})\left\vert \ln N\right\vert ^{2}}{N} +\left\vert \mathfrak{F}_{N}^{\mathrm{avg}}(\underline{x}_{N}(0),\omega (0))\right\vert <\exp(-e^{Ct\left( \left\Vert \omega ^{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}^{1/2}+\left\Vert \omega ^{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}^{3/2}\right) }\), then \(\mathfrak{F}_{N}^{\mathrm{avg}}( \underline{x}_{N}(t),\omega (t))\) satisfies the inequality \[ \mathfrak{F} _{N}^{\mathrm{avg}}(\underline{x}_{N}(t),\omega (t))\leq \left( \mathfrak{F} _{N}^{\mathrm{avg}}(\underline{x}_{N}(0),\omega (0))+\frac{Ct(\left\Vert \omega ^{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}^{1/2}+\left\Vert \omega ^{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}^{3/2})\left\vert \ln N\right\vert ^{2}}{N}\right) ^{\exp(-e^{Ct\left( \left\Vert \omega ^{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}^{1/2}+\left\Vert \omega ^{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}^{3/2}\right) }}. \] From this main result, the authors derive an estimate for the norm \(\left\Vert \omega _{N}-\omega \right\Vert _{L^{\infty }([0,T];H^{s}(\mathbb{R}^{2}))}\), whence the convergence \(\omega _{N}\overset{\ast }{\rightarrow }_{N\rightarrow \infty }\omega \) in \(\mathcal{M}(\mathbb{R}^{2})\), locally uniformly in time, if \(\mathfrak{F}_{N}^{\mathrm{avg}}(\underline{x}_{N}(0),\omega (0))\rightarrow _{N\rightarrow \infty }0\). For the proof, the authors apply the modulated-energy method developed by M. Duerinckx and S. Serfaty in their works [SIAM J. Math. Anal. 48, No. 3, 2269–2300 (2016; Zbl 1348.82050); Duke Math. J. 169, No. 15, 2887–2935 (2020; Zbl 1475.35341)]. He recalls properties of the Coulomb potential \(\mathfrak{g}\) and of the modulated energy \(\mathfrak{F}_{N}^{\mathrm{avg}}(.,.)\). He introduces a truncation \(\mathfrak{g}_{\eta }:\mathbb{R}^{2}\rightarrow \mathbb{R}\) for the potential \(g\) through \(\mathfrak{g}_{\eta }(x)=\mathfrak{g}(x)\), \( \left\vert x\right\vert \geq \eta \), \(\mathfrak{g}_{\eta }(x)=\widetilde{ \mathfrak{g}}(x)\), \(\left\vert x\right\vert <\eta \), and the associated smearing procedure for point masses through \(\delta ^{(\eta )}0=-\Delta \mathfrak{g}_{\eta }=\sigma _{\partial B(0,\eta )}\), the uniform probability measure on the sphere \(\partial B(0,\eta )\). He defines the notion of weak solution to the Euler equation through Duhamel’s formula and recalls an existence and uniqueness result and properties of such a weak solution.