Abstract
We study the homogeneous Boltzmann equation with the fractional Laplacian term. Working on the Fourier side we solve the resulting integral equation, and improve a previous result by Y.-K. Cho. We replace the initial data space with a certain space $\mathcal{M}^{\alpha}$ introduced by Morimoto, Wang, and Yang. This space precisely captures the Fourier image of probability measures with bounded fractional moments, providing a more natural initial condition. We show existence of a unique global solution, in addition to the expected maximal growth estimates and stability estimates. As a consequence we obtain a continuous density solution of the original equation.
Citation
Shota Sakamoto. "Some remarks on the homogeneous Boltzmann equation with the fractional Laplacian term." Osaka J. Math. 53 (3) 621 - 636, July 2016.
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