  It is challenging to perform a multiscale analysis of mesoscopic systems
exhibiting singularities at the macroscopic scale. In this paper, we study the
hydrodynamic limit of the Boltzmann equations $$\mathrm{St} \partial_t F +
v\cdot \nabla_x F = \frac{1}{\mathrm{Kn}} Q(F ,F ) $$ toward the singular
solutions of 2D incompressible Euler equations whose vorticity is unbounded
$$\partial_t u + u \cdot \nabla_x u + \nabla_x p = 0,\text{div }u =0.$$ We
obtain a microscopic description of the singularity through the so-called
kinetic vorticity and understand its behavior in the vicinity of the
macroscopic singularity. As a consequence of our new analysis, we settle
affirmatively an open problem of the hydrodynamic limit toward Lagrangian
solutions of the 2D incompressible Euler equation whose vorticity is unbounded
($\omega \in L^\mathfrak{p}$ for any fixed $1 \leq \mathfrak{p} < \infty$).
Moreover, we prove the convergence of kinetic vorticities toward the vorticity
of the Lagrangian solution of the Euler equation. In particular, we obtain the
rate of convergence when the vorticity blows up moderately in $L^\mathfrak{p}$
as $\mathfrak{p} \rightarrow \infty$ (localized Yudovich class).
