On a fuzzy Landau Equation: Part I. A variational approach
Manh Hong Duong, Zihui He
TL;DR
This work derives a GENERIC-based variational characterisation for the three-dimensional inhomogeneous fuzzy Landau equation with delocalised collisions. It builds a transport grazing rate framework, coupling entropy, entropy dissipation, and a curve action, and proves a chain-rule-enabled nonnegativity result: $\mathcal J_T(f,U)=\mathcal H(f_T)-\mathcal H(f_0) +\tfrac{1}{2}\int_0^T\mathcal D(f_t)\,dt+\tfrac{1}{2}\int_0^T\mathcal A(f_t,U_t)\,dt \ge 0$, with equality if and only if $f$ is an $\mathcal H$-solution of the fuzzy Landau equation $\partial_t f+v\cdot\nabla_x f=Q_{\sf fuz}(f,f)$. The authors regularise in space, velocity, and time to justify the chain rule and extend the construction to higher dimensions and general kernel classes on TEXT. The results connect nonlocal delocalised interactions with thermodynamic structure, enabling a unified variational view of kinetic models and laying groundwork for Part II (existence, regularity) and numerical schemes. Overall, the paper broadens the gradient-flow/GENERIC landscape to include delocalised kinetic equations and sets up a rigorous framework for thermodynamically consistent evolution with grazing collisions.
Abstract
This article is the first in a series of works on the fuzzy Landau equation, where particles interact through delocalised Coulomb collisions. Here, we establish a variational characterisation that recasts the fuzzy Landau equation within the framework of GENERIC systems (General Equations for Non-Equilibrium Reversible-Irreversible Coupling).