The Landau equation, introduced by Lev Landau in 1936, is one of the central equations in kinetic theory. We consider the Landau equation with very hard potentials $\gamma \in (\sqrt{3},2]$, which is known to admit global smooth solutions for homogeneous data. Inspired by hydrodynamic limits from kinetic equations to fluid equations, we construct smooth, strictly positive initial data that develop a finite-time singularity by lifting imploding singularities from the compressible Euler equations. In self‑similar variables, the solution becomes asymptotically hydrodynamic—the distribution function converges to a local Maxwellian, while the hydrodynamic fields develop an asymptotically self‑similar implosion whose profile coincides with a smooth imploding profile of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data.

This is joint work with Jacob Bedrossian (UCLA), Maria Gualdani (UT Austin), Sehyun Ji (UChicago), Vlad Vicol (NYU), and Jincheng Yang (JHU).