Sharp $H_{x}^{s}$ Ill-posedness of the Hard-sphere Boltzmann Equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913177676021760 |
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| author | Chen, Xuwen Shen, Shunlin Zhang, Zhifei |
| author_facet | Chen, Xuwen Shen, Shunlin Zhang, Zhifei |
| contents | We investigate the ill-posedness mechanism of the hard-sphere Boltzmann equation in $H_{x}^{s}$ Sobolev space. Via a direct construction, we prove a strong-weak type ill-posedness result in the low-regularity regime $s<1$, establishing a sharp threshold in connection to the local $s>1$ well-posedness result [11]. Instead of originating from the large-velocity growth of the collision kernel, this illposedness is generated by the loss term and dispersive effects. Consequently, we prove a dispersion-driven nonlinear instability mechanism for the hard-sphere Boltzmann equation, and provide a capstone of the ill-posedness series [18,20]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01331 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp $H_{x}^{s}$ Ill-posedness of the Hard-sphere Boltzmann Equation Chen, Xuwen Shen, Shunlin Zhang, Zhifei Analysis of PDEs Mathematical Physics We investigate the ill-posedness mechanism of the hard-sphere Boltzmann equation in $H_{x}^{s}$ Sobolev space. Via a direct construction, we prove a strong-weak type ill-posedness result in the low-regularity regime $s<1$, establishing a sharp threshold in connection to the local $s>1$ well-posedness result [11]. Instead of originating from the large-velocity growth of the collision kernel, this illposedness is generated by the loss term and dispersive effects. Consequently, we prove a dispersion-driven nonlinear instability mechanism for the hard-sphere Boltzmann equation, and provide a capstone of the ill-posedness series [18,20]. |
| title | Sharp $H_{x}^{s}$ Ill-posedness of the Hard-sphere Boltzmann Equation |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2606.01331 |