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.

翻译：快速颗粒介质的简化动力学描述导出了一个非局部Vlasov型方程，该方程包含一个与聚集-扩散宏观动力学连续性方程形式相同的卷积积分算子。尽管这些非线性连续性方程的奇异行为在文献中已有充分研究，但将其推广到相应的颗粒动力学方程却极具挑战性。核心问题在于：自由输运引起的相空间剪切是否会增强或缓解速度方向形成的奇异性。我们通过细致的数值实验和启发性论证开展了初步研究。数值上，我们发展了一种具有自适应网格加密的结构保持方法，能够有效捕捉颗粒动力学方程解中潜在的爆破行为。解析上，我们构建了一个有限时间爆破的无限质量解，并讨论了该结果如何为有限质量情形提供启示。