For \(v \in \mathbb {R}^3\), define \(|v|^2 = \sum _{i=1}^{3} v_i^2\).

For \(v \in \mathbb {R}^3\), define \(|v| = \sqrt{|v|^2}\).

Given a kernel function \(\Psi : \mathbb {R}\to \mathbb {R}\), the Landau matrix \(A(z)\) is the \(3 \times 3\) matrix with entries

A function \(f : \mathbb {R}^3 \to \mathbb {R}\) is Maxwellian if there exist \(a_0 \in \mathbb {R}\), \(b \in \mathbb {R}^3\), and \(c_0 {\lt} 0\) such that

for all \(v \in \mathbb {R}^3\).

For \(\rho _{\mathrm{ion}} {\gt} 0\) and \(T {\gt} 0\), the equilibrium Maxwellian is

The Landau collision operator is

The entropy dissipation functional is

A flat 3-torus is a compact, nonempty, first-countable measure space \(X\) equipped with spatial gradient, divergence, and curl operators, a differentiability predicate IsSpatiallyDiff, and axioms encoding: operator linearity, closure of IsSpatiallyDiff under const/add/smul/log/grad, integration by parts, curl-integral vanishing, harmonic \(\Rightarrow \) constant, maximum principle, Killing/curl-free/div-free \(\Rightarrow \) harmonic, and Fubini for spatial–velocity double integrals.

The VelocityDecayConditions structure bundles 17 integrability/Fubini/IBP conditions on \(f\), \(E\), \(B\) that justify the analytical manipulations (integration by parts on \(\mathbb {R}^3\), Fubini exchange, Leibniz differentiation under the integral, entropy dissipation continuity, etc.).