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arXiv · 1512.00629

A characterization of probability measure with finite moment and an application to the Boltzmann equation

Yong-Kum Cho·Yoshinori Morimoto·Shuaikun Wang·Tong Yang
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Abstract

We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuity $f(t,v)\in C((0, \infty),L^1_{2k-2 +\alpha})$, where $f(t, v)$ stands for the density of unique measure-valued solution $(F_t)_{t\ge0}$ of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum $F_0$ satisfying \[ \int |v|^{2k-2+\alpha} dF_0(v) < \infty, 0\leq \alpha < 2,k= 2, 3, 4,\cdots \] provided that $F_0$ is not a single Dirac mass.

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Yong-Kum Cho, Yoshinori Morimoto, Shuaikun Wang, Tong Yang. 2015-12-02. A characterization of probability measure with finite moment and an application to the Boltzmann equation. https://arxiv.org/abs/1512.00629

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