We introduce a fast Fourier spectral method for the spatially homogeneous Boltzmann equation with non-cutoff collision kernels. Such kernels contain non-integrable singularity in the deviation angle which arise in a wide range of interaction potentials (e.g., the inverse power law potentials). Albeit more physical, the non-cutoff kernels bring a lot of difficulties in both analysis and numerics, hence are often cut off in most studies (the well-known Grad's angular cutoff assumption). We demonstrate that the general framework of the fast Fourier spectral method developed in [9], [14] can be extended to handle the non-cutoff kernels, achieving the accuracy/efficiency comparable to the cutoff case. We also show through several numerical examples that the solution to the non-cutoff Boltzmann equation enjoys the smoothing effect, a striking property absent in the cutoff case.

对于具有非截止碰撞核的空间齐次Boltzmann方程，我们引入了一种快速傅立叶谱方法。这样的核在偏离角中包含不可积分的奇异性，其在广泛的相互作用势（例如，逆幂定律势）中出现。非截止核虽然更物理，但在分析和数值上都带来了很多困难，因此在大多数研究中经常被截止（众所周知的Grad的角度截止假设）。我们证明了在[9]，[14]中开发的快速傅里叶频谱方法的通用框架可以扩展为处理非截止核，从而获得与截止情况相当的准确性/效率。我们还通过几个数值示例表明，非截止Boltzmann方程的解具有平滑效果，