2026/07/22 by Ling-bing He, Jie Ji, Wu-wei Li
#math.AP
We study the Boltzmann equation in the whole space under the radial anharmonic confining potentials Φ(x)=|x|p/p for p>2 and Φ(x)=⟨ x⟩p/p for 1<p<2. We first classify all positive finite-mass-and-energy entropy-invariant, equivalently zero-entropy-production, solutions. For p>2, the nonlinear equilibrium manifold is parametrized by mass, temperature, and the three components of angular momentum; for 1<p<2, integrability excludes rotating equilibria and only mass and energy remain as equilibrium parameters. We then identify the five-dimensional stationary space of the equation linearized about an arbitrary equilibrium and construct an explicit projection determined by the conserved moments. After normalization, the collision term takes the form CMe-\widetildeΦ(x)\mathsf L, so its microscopic coercivity degenerates at spatial infinity. A far-field weight-transfer estimate compensates for this degeneracy. After subtracting the stationary projection, the corresponding semigroup solution converges algebraically in exponentially weighted L2 spaces. The rate is governed by the growth exponent p and the gap between the two weights, up to an arbitrarily small loss. For 1<p<2, the mismatch between the two-dimensional nonlinear equilibrium manifold and the five-dimensional linear stationary space yields a conditional obstruction to nonlinear asymptotic attraction for perturbations carrying nonzero angular momentum.