Example of blowup for inhomogeneous Landau (or Boltzmann)

The problem asks whether spatial variation can cause finite-time blowup for Landau or Boltzmann even when the corresponding homogeneous equation is globally well-posed, including the isotropic Landau case. No source reports an example or a resolution.

Known results

• Desvillettes, He, and Jiang (2017) gave local existence and continuation criteria for inhomogeneous Landau, while noting that even homogeneous a priori $L^\infty$ control was open.
• A 2021 study ruled out a specified approximate Type I self-similar blowup mechanism, without ruling out blowup generally.
• A 2025 study gave necessary divergence conditions for finite-time singularity in the Landau-Coulomb setting, but no example.
• A 2026 result established continuation under time-integrable weighted $L^\infty$ bounds, not the requested contrast.

May 2026 continuation criteria

A May 2026 paper proves pointwise bounds and continuation criteria for inhomogeneous Landau and non-cutoff Boltzmann equations. It shows that a finite maximal time $T^*$ requires divergence of $\int_0^{T^*}\|f(t)\|_{L^\infty_{3+\gamma}}\,dt$, but constructs neither blowup nor the proposed homogeneous-versus-inhomogeneous separation.

Current status (as of September 2026): The requested example and the isotropic Landau answer remain open; existing work supplies exclusions and continuation criteria, but no blowup construction.