Maxwell molecule collision operator ($Q(f,f)=\frac{1}{4\pi }\int (f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })d{v}_{\ast }d\sigma$)

For the normalized isotropic cutoff model, collisions redistribute the direction of relative velocity uniformly while keeping its length. With center $C=(v+v_{\ast })/2$ and relative speed $r=|v-v_{\ast }|$, outgoing velocities are $C\pm r\sigma /2$. Momentum and kinetic energy are preserved, and the collision rate has no relative-speed factor. The displayed operator uses unit total angular rate. General Maxwell molecule kernels can have nonconstant angular dependence.

A collision invariant has an unchanged pairwise sum in every allowed elastic collision. The functions $1$, each velocity coordinate, and $|v{|}^2$ encode mass, momentum and energy. Symmetrized weak collision identities then give zero moments of the collision operator against these functions. Under standard regularity assumptions, the invariants form their five-dimensional linear span in three velocity dimensions.

For the Maxwell molecule collision operator and the Fourier transform with exponent $-i\xi \cdot v$, put ${\xi }^{\pm }=(\xi \pm |\xi |\sigma )/2$. The angular exchange for elastic collisions, rotation invariance of spherical surface area and factorization of the two velocity integrals give the displayed formula. This identity converts the collision integral into a spherical average of products at two related frequencies.

For finite nonnegative measures with equal mass and first moment and finite second moments, the constant and linear terms of their Fourier transforms cancel. Taylor's remainder bounds their difference by a constant times $|\xi {|}^2$, making this distance finite. Fourier uniqueness gives definiteness. The origin is excluded from the supremum; the quotient need not have a direction-independent limit there.

Unit-mass solutions of the isotropic Maxwell molecule collision operator with matching first moments obey this comparison. The Bobylev identity, the Fourier bound by one, and $|{\xi }^{+}{|}^2+|{\xi }^{-}{|}^2=|\xi {|}^2$ give a scalar damped differential inequality. Its integral form and the Gronwall inequality yield nonexpansion in the Fourier distance of order two. This does not alone establish strict contraction or equilibrium convergence.

Write incoming relative velocity as $r\omega$ and outgoing relative velocity as $r\sigma$. For isotropic scattering, the product measure $dC{r}^{2}drd\omega d\sigma$ is invariant under exchanging the two angular variables. This gives the pre/post-collision weak identities. Keeping $\sigma$ fixed and sending only the velocities to the outgoing pair is not invertible: that map discards the incoming angular direction.