On a fuzzy Landau Equation: Part III. The grazing collision limit
Manh Hong Duong, Boris Golubkov, Zihui He
TL;DR
This work rigorously justifies the grazing collision limit from the non-cutoff fuzzy Boltzmann equation to the fuzzy Landau equation within a GENERIC variational framework. By combining polar-coordinate collision representations, velocity averaging, and compactness arguments, the authors show that non-quadratic dissipation in the fuzzy Boltzmann picture converges to a quadratic dissipation in the fuzzy Landau setting, with sqrt(f^ε) converging to sqrt(f) and the corresponding variational inequalities converging accordingly. The main contributions include a full variational characterization of the fuzzy Landau equation, a non-quadratic-to-quadratic dissipation limit under grazing scaling, and a rigorous passage of dissipation and curve-action functionals through the limit, covering hard, Maxwellian, and soft potential regimes (up to the moderately soft range). The results advance the gradient-flow/GENERIC perspective on kinetic equations and pave the way for further analysis of large-deviation principles and asymptotic behavior in non-cutoff fuzzy kinetic models.
Abstract
In this paper, we study the grazing limit from the non-cutoff fuzzy Boltzmann equations to the fuzzy Landau equation, where particles interact through delocalised collisions. We show the grazing limit through variational formulations that correspond to the GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling) structure of the respective equations. We show that the variational formulation associated with a non-quadratic dual dissipation pair for the fuzzy Boltzmann equations converges to a variational formulation of the fuzzy Landau equation corresponding to a quadratic dissipation pair.