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Well/Ill-posedness of the Boltzmann Equation with Soft Potential

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arxiv 2310.05042 v1 pith:RLYKPJCD submitted 2023-10-08 math.AP math-phmath.MP

Xuwen Chen , Shunlin Shen , Zhifei Zhang This is my paper · ORCID
classification math.APmath-phmath.MP
keywords fracill-posednessboltzmannequationpotentialseparationsoftwell
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abstract

We consider the Boltzmann equation with the soft potential and angular cutoff. Inspired by the methods from dispersive PDEs, we establish its sharp local well-posedness and ill-posedness in $H^{s}$ Sobolev space. We find the well/ill-posedness separation at regularity $s=\frac{d-1}{2}$, strictly $\frac{1}{2}$-derivative higher than the scaling-invariant index $s=\frac{d-2}{2}$, the usually expected separation point.

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  1. Local well-posedness for the periodic Boltzmann equation with constant collision kernel

    math.AP 2024-11 conditional novelty 6.0 of 10

    The periodic Boltzmann equation with constant collision kernel is locally well-posed in L^{2,r}_v H^s_x for s > d/2 − 1/4 and r > d/2.

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