Interplay of hypoellipticity and shock formation for more singular kernels

Determine whether, for interaction kernels g(x) ∼ |x|^{−s} with (d − 2)_+ < s < d, the hypoellipticity of the corresponding nonlocal model suppresses or dominates shock formation, and characterize the resulting qualitative behavior of solutions.

Background

The paper highlights a class of more singular interactions where the model becomes hypoelliptic. Understanding whether the smoothing effect induced by hypoellipticity outweighs the nonlinear mechanisms leading to shocks is essential for the qualitative theory and for designing appropriate analytical and numerical techniques.

This question connects to recent work on nonlocal pressure models and singular limit regimes, where regularization and nonlinearity compete in subtle ways.