upon noticing that $m_1(\Lambda^N)= \lVert {\boldsymbol{y}}\rVert_{\overline Y^N}$, $$\label{T139}
\lVert {\boldsymbol{b}}^N({\boldsymbol{y}})\rVert_{\overline Y^N}=\frac1N \sum_{i=1}^N \lVert b_{\Lambda^N}(y^i)\rVert_{\overline Y} {\leqslant}\frac{M}{N} \sum_{i=1}^N \big(1+\lVert y^i\rVert_{\overline Y}+m_1(\Lambda^N)\big)= M\big(1+ 2\lVert {\boldsymbol{y}}\rVert_{\overline Y^N}\big),$$ Therefore we can apply Corollary \[cor:Brezis\], which proves the statement.

We are now in a position to state the main result of our paper, concerning the mean-field limit as $N\to\infty$ of the solutions $(y_t^1,\ldots,y_t^N)$ to , or equivalently the limiting behavior of the associated empirical measures $\Lambda_t^N$. In order to do so, we first need to recall the concept of Eulerian solution to the continuity equation.

\[T038\] Let $\Lambda\in C^0([0,T];({\mathcal P}_1(Y),{\mathcal W}_1))$ and let $\bar\Lambda\in{\mathcal P}_c(Y)$ be a given initial datum. We say that $\Lambda$ is a Eulerian solution to the initial value problem for the equation $$\label{T140}
\partial_t \Lambda_t+{\operatorname{div}}(b_{\Lambda_t}\Lambda_t)=0$$ starting from $\bar\Lambda$ if and only if $\Lambda_0=\bar\Lambda$ and, for every $\phi\in C^1_b([0,T]\times \overline Y)$, $$\label{T037}
\int_Y \phi(t,y)\,{\mathrm d}\Lambda_t(y)- \int_Y \phi(0,y)\,{\mathrm d}\Lambda_0(y) =\int_0^t\int_Y \big(\partial_t\phi(s,y)+ D\phi(s,y)\cdot b_{\Lambda_s}(y)\big)\,{\mathrm d}\Lambda_s(y){\mathrm d}s,
$$ where $D\phi(s,y)$ is the Fréchet differential of $\phi$ in the $y$ variable.

The main result of our paper is the following theorem, stating the existence of a unique Eulerian solution to and its characterization as the mean-field limit of solutions to the discrete problem .

\[T150\] Let 