 Mean-field alignment equation

The continuum velocity alignment equation with $\mathbf{L}[f](\mathbf{x},\mathbf{v})=\int K(\mathbf{x},{\mathbf{x}}_{\star })({\mathbf{v}}_{\star }-\mathbf{v})f_{\star }{d}^3{x}_{\star }{d}^3{v}_{\star }$. For symmetric nonnegative $K$, it conserves total mass and momentum while dissipating kinetic energy. Its force is affine in velocity, with a nonzero velocity-space divergence.