Gradient flow approach to Landau equation: A cross-product structure

We introduce a cross-product Landau gradient that commutes with Gaussian mollification. For the hard-potential interaction kernels of the form $A(\mathrm{\mid }v-v_{\ast }\mathrm{\mid })\sim ⟨v-v_{\ast }{⟩}^{\gamma }\mathrm{\mid }v-v_{\ast }{\mathrm{\mid }}^2$ with $\gamma \in (-\mathrm{\infty },1]$, this construction yields a variational characterisation of the gradient-flow structure of the spatially homogeneous Landau equation. The cross-product gradient induces the same gradient-flow geometry as that introduced by Carrillo, Delgadino, Desvillettes and Wu (2024), while providing a different representation of the Landau gradient. The main advantage of this formulation is that it requires only weighted $L^1$-control. We also discuss a similar variational characterisation for the GENERIC structure of the fuzzy Landau equations.