The Landau equation is a fundamental collisional kinetic model arising in plasma physics. In the spatially homogeneous setting, global well-posedness is known, and solutions exhibit strong regularizing effects. This naturally raises the question: can singularities arise due to spatial inhomogeneity?

 

In this talk, I will present a construction of smooth, strictly positive initial data for the inhomogeneous Landau equation with \(\gamma \in (\sqrt{3}, 2]\) that develops a finite-time implosion singularity. Interestingly, the singularity manifests at the hydrodynamic level: macroscopic quantities such as density and temperature blow up, while the \(L^{\infty}\)-norm of the distribution function remains uniformly bounded. Working in self-similar variables, our blow-up profile is asymptotically self-similar (Type II). The distribution function converges to a local Maxwellian, and the associated macroscopic fields approach to a smooth imploding solution of the compressible Euler equations.

 

To our knowledge, this provides the first example of a collisional kinetic equation that is globally well-posed in the homogeneous setting but exhibits finite-time singularity formation for smooth inhomogeneous data. This is joint work with Jacob Bedrossian, Jiajie Chen, Maria Gualdani, Vlad Vicol, and Jincheng Yang.