Uniqueness and Zeroth-Order Analysis of Weak Solutions to the Non-cutoff Boltzmann equation
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arXiv
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| Format: | Preprint |
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2026
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| author | Deng, Dingqun Sakamoto, Shota |
| author_facet | Deng, Dingqun Sakamoto, Shota |
| contents | We establish the uniqueness of large solutions to the non-cutoff Boltzmann equation with moderate soft potentials. Specifically, the weak solution $F=μ+μ^{\frac{1}{2}}f$ is unique as long as it has finite energy, in the sense that the norm $\|f\|_{L^\infty_t L^{r}_{x,v}}+\|f\|_{L^\infty_t L^2_{x,v}}$ remains bounded for some sufficiently large $r>0$. As a byproduct, we establish $L^2_{t,x,v}$ stability for initial data $f_0\in L^r_{x,v}\cap L^2_{x,v}$. Our approach employs dilated dyadic decompositions in phase space $(v,ξ,η)$ to capture hypoellipticity and to reduce the fractional derivative structure $(-Δ_v)^{s}$ of the Boltzmann collision operator to zeroth order. The difficulties posed by the large solution are overcome through the negative-order hypoelliptic estimate that gains integrability in $(t,x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniqueness and Zeroth-Order Analysis of Weak Solutions to the Non-cutoff Boltzmann equation Deng, Dingqun Sakamoto, Shota Analysis of PDEs We establish the uniqueness of large solutions to the non-cutoff Boltzmann equation with moderate soft potentials. Specifically, the weak solution $F=μ+μ^{\frac{1}{2}}f$ is unique as long as it has finite energy, in the sense that the norm $\|f\|_{L^\infty_t L^{r}_{x,v}}+\|f\|_{L^\infty_t L^2_{x,v}}$ remains bounded for some sufficiently large $r>0$. As a byproduct, we establish $L^2_{t,x,v}$ stability for initial data $f_0\in L^r_{x,v}\cap L^2_{x,v}$. Our approach employs dilated dyadic decompositions in phase space $(v,ξ,η)$ to capture hypoellipticity and to reduce the fractional derivative structure $(-Δ_v)^{s}$ of the Boltzmann collision operator to zeroth order. The difficulties posed by the large solution are overcome through the negative-order hypoelliptic estimate that gains integrability in $(t,x)$. |
| title | Uniqueness and Zeroth-Order Analysis of Weak Solutions to the Non-cutoff Boltzmann equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.15601 |