Granular kinetic equations exhibit blow-up behavior under specific conditions.
摘要
The content discusses the blow-up behavior of granular kinetic equations, focusing on the singularity formation in velocity direction and its mitigation by shear. The analysis involves numerical investigations and heuristic arguments to understand potential blow-up scenarios. The study also delves into the derivation of kinetic equations from statistical mechanics, emphasizing inelastic collisions and dissipation of energy in granular gases. Various mathematical models and methods are employed to capture blow-up behavior effectively.
Structure:
Introduction to Granular Flows and Kinetic Equations
Granular flows in nature and rapid granular media.
Derivation of kinetic equations from statistical mechanics.
Mathematical Description of Inelastic Collisions
Postcollisional velocities and energy dissipation.
Kinetic Description of Inelastic Collisions
Formulation of the kinetic equation for inelastic collisions.
Regularized JKO Scheme with Adaptive Mesh Refinement
Detailed explanation of the numerical method for collision step.
Numerical Examples and Blow-Up Verification
Validation of numerical solver for spatially homogeneous case.
Finite Time Blow-Up Verification: Homogeneous Problem
Examination of one-bump initial conditions with fixed time step strategy.
Adaptive Time Stepping Strategy
Utilization of adaptive time steps to approach analytical blow-up time.
To blow-up or not to blow-up for a granular kinetic equation
"We present a preliminary study through a meticulous numerical investigation."
"As opposed to the Boltzmann equation whose equilibrium is the Maxwellian, the equilibrium of (1.1) is a Dirac mass located at the mean velocity."