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arXiv · 2608.22579

Uniqueness for the spatially homogeneous Boltzmann equation in critical Sobolev spaces

Tianhao Dong·Shuchen Guo·Jie Ji
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Abstract

We study the spatially homogeneous Boltzmann equation without angular cutoff for very soft potentials satisfying the inverse power law relation $\gamma+4s=1$. Our main results establish the existence, uniqueness, stability and regularization estimates of solutions in the critical Sobolev space $ H^{-(\gamma + 2s + \frac{3}{2})} $ with a logarithmic correction. Combined with the recently established monotonicity of the Fisher information, the solutions extend globally in time. Our primary tools are energy estimates based on a simultaneous dyadic localization in the phase and frequency variables, together with sharp commutator estimates between the collision operator and the localization operators.

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Tianhao Dong, Shuchen Guo, Jie Ji. 2026-08-23. Uniqueness for the spatially homogeneous Boltzmann equation in critical Sobolev spaces. https://arxiv.org/abs/2608.22579

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