Local well-posedness for Boltzmann's equation and the Boltzmann hierarchy via Wigner transform
英文摘要：
We use the dispersive properties of the linear Schrödinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$. The proofs are based on the use of the (inverse) Wigner transform along with the spacetime Fourier transform. The norms for the initial data $f_0$ are weighted versions of the Sobolev spaces $L^2_v H^α_x$ with $α\in \left( \frac{d-1}{2},\infty\right)$. Our main results are local well-posedness for the Boltzmann equation for cutoff Maxwell molecules and hard spheres, as well as local well-posedness for the Boltzmann hierarchy for cutoff Maxwell molecules (but not hard spheres); the latter result holds without any factorization assumption for the initial data.
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