2025/05/14 by Pan, Xinghong, Wang, Xue Ping, Zhu, Lu
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.09270
We use methods from microlocal analysis and quantum scattering to study spectral properties near the threshold zero of the Kramers-Fokker-Planck operator with a decaying potential in \mathbb Rn, n ≥ 4, and deduce the large-time behavior of solutions to the kinetic Kramers-Fokker-Planck equation. For short-range potentials, we establish an optimal time-decay estimate in weighted L2-spaces when n≥ 5 is odd. For potentials decaying like O(|x|-ρ) for some ρ> n-1, we obtain, for all dimensions n ≥ 4, a large-time expansion of the solution with the leading term given by the Maxwell-Boltzmann distribution multiplied by the factor (4πt)-\frac n 2 corresponding to the decay for the heat equation. These results complete those obtained in [16, 22] for dimensions n=1 and 3. The same questions for n=2 are still open.