5 Transport Constraints
If \(\Psi {\gt} 0\), \(f {\gt} 0\), \(f\) smooth, and \(D(f(x, \cdot )) = 0\) for all \(x\), then for every \(x\), \(f(x, \cdot )\) is Maxwellian.
For any \(E_0, B_0 \in \mathbb {R}^3\), \(\nabla \cdot _v (E_0 + v \times B_0) = 0\).
Integration by parts on \(\mathbb {R}^3\) in velocity space: \(\int \partial _i g \cdot h = -\int g \cdot \partial _i h\) under suitable decay.
For any Lorentz force \((E_0 + v \times B_0)\), velocity-space IBP combined with \(\nabla \cdot _v(E_0 + v \times B_0) = 0\) gives \(\int (E_0 + v \times B_0) \cdot \nabla _v f \cdot \log f\, dv = 0\).
Under the Vlasov equation, combined with spatial IBP (on the torus) and the force transport lemma: \(\int _X D(f(x, \cdot ))\, dx = 0\).