Global renormalized solutions for hard potential non-cutoff Boltzmann equation without defect measure
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910021029199872 |
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| author | Luo, Yi-Long Nie, Jing-Xin |
| author_facet | Luo, Yi-Long Nie, Jing-Xin |
| contents | The existence of global renormalized solutions to the Boltzmann equation with long-range interactions without angular cutoff was first established by Alexandre and Villani [Comm. Pure Appl. Math., 55(1), 30-70, 2002]. Their result relies on a definition of renormalized solutions involving a non-negative defect measure. In this paper, we address this issue for the inverse power law model in the case of hard potentials ($0 \leq γ\leq 1$). By exploiting the stronger coercivity estimates provided by hard potentials, we prove that the defect measure actually vanishes. Consequently, we establish the global existence of renormalized solutions for the non-cutoff Boltzmann equation with hard potentials in the standard sense, without any defect measure. Finally, we construct a counterexample showing that the approach developed for the hard potential case fails for soft potential model ($-3 < γ< 0$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12511 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global renormalized solutions for hard potential non-cutoff Boltzmann equation without defect measure Luo, Yi-Long Nie, Jing-Xin Analysis of PDEs The existence of global renormalized solutions to the Boltzmann equation with long-range interactions without angular cutoff was first established by Alexandre and Villani [Comm. Pure Appl. Math., 55(1), 30-70, 2002]. Their result relies on a definition of renormalized solutions involving a non-negative defect measure. In this paper, we address this issue for the inverse power law model in the case of hard potentials ($0 \leq γ\leq 1$). By exploiting the stronger coercivity estimates provided by hard potentials, we prove that the defect measure actually vanishes. Consequently, we establish the global existence of renormalized solutions for the non-cutoff Boltzmann equation with hard potentials in the standard sense, without any defect measure. Finally, we construct a counterexample showing that the approach developed for the hard potential case fails for soft potential model ($-3 < γ< 0$). |
| title | Global renormalized solutions for hard potential non-cutoff Boltzmann equation without defect measure |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.12511 |