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

TL;DR

This work analyzes the well-posedness and blow-up behavior of a nonlocal granular kinetic equation featuring velocity-space aggregation with inelastic collisions. It develops a structure-preserving, two-step numerical framework combining a transport step and a Fisher-information regularized JKO collision step, augmented by adaptive mesh refinement to resolve potential blow-up regions. The study establishes rigorous blow-up regimes: infinite-time blow-up for $\gamma>2$, finite-time blow-up for $-1<\gamma<2$, and a critical infinite-time case at $\gamma=2$ with finite-mass blow-up possible in the spatial direction under a threshold; for $\gamma<2$ blow-up occurs in velocity. The numerical experiments, supported by explicit self-similar solutions and conditional blow-up analysis, provide insight into how transport shear interacts with nonlinear focusing, with implications for modeling rapid granular flows and their transition to concentrated states.

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.

Figures (26)

• Figure 1: Demonstration of Mesh refinement method for $f(v) = e^{-30v^2}$ for $v \in [-2, 2]$. The left figure shows the mapping function $\mu(s)$ with fixed $\delta=0.5$ and various $\delta_0=0.2, 0.5, 0.8$. The middle and right one compare function $f(v)$ in uniform grid and after mesh refinement (MR).
• Figure 2: Demonstration of Mesh refinement method for $f(v) = e^{-50(v-2)^2 - 50(v+2)^2}$ for $v \in [-4, 4]$. The left figure shows the mapping function $\mu(s)$ with fixed $\delta=0.5$ and various $\delta_0=0.2, 0.5, 0.8$. The middle and right one compare function $f(v)$ in uniform grid and after mesh refinement(MR).
• Figure 3: Numerical solution with initial condition $g_1(v)= e^{-2v^2}$, $\delta=0.2$, $\delta_0 = 0.5$ and fixed time step $\Delta t=0.01$ with the kernel $W(v) = |v|$.
• Figure 4: Time versus minimum $\Delta v$ for initial condition $g_1(v)$ with fixed $\Delta t$ with the kernel $W(v) = |v|$. Here $N_v=121$ and $\delta=0.5$, the images from top left to bottom right correspond to $\delta_0 = 0.05, 0.1, 0.5, 0.8$ respectively.
• Figure 5: Time versus adaptive time step size $\Delta t$ with the kernel $W(v) = |v|$. Here, we consider the initial condition $g_1(v)$ with $N_v=121$ and $\delta=0.5$. On the left is $\delta_0=0.05$, and on the right is $\delta_0=0.5$.

Theorems & Definitions (5)

• Theorem 1
• Proposition 1
• Remark 1
• Theorem 2
• proof