To blow-up or not to blow-up for a granular kinetic equation

Abstract

A simplified kinetic description of rapid granular media leads to a nonlocal Vlasov-type equation with a convolution integral operator that is of the same form as the continuity equations for aggregation-diffusion macroscopic dynamics. While the singular behavior of these nonlinear continuity equations is well studied in the literature, the extension to the corresponding granular kinetic equation is highly nontrivial. The main question is whether the singularity formed in velocity direction will be enhanced or mitigated by the shear in phase space due to free transport. We present a preliminary study through a meticulous numerical investigation and heuristic arguments. We have numerically developed a structure-preserving method with adaptive mesh refinement that can effectively capture potential blow-up behavior in the solution for granular kinetic equations. We have analytically constructed a finite-time blow-up infinite mass solution and discussed how this can provide insights into the finite mass scenario.