Finite time singularities in the Landau equation with very hard potentials
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912881949278208 |
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| author | Bedrossian, Jacob Chen, Jiajie Gualdani, Maria Pia Ji, Sehyun Vicol, Vlad Yang, Jincheng |
| author_facet | Bedrossian, Jacob Chen, Jiajie Gualdani, Maria Pia Ji, Sehyun Vicol, Vlad Yang, Jincheng |
| contents | We consider the inhomogeneous Landau equation with $γ\in (\sqrt{3},2]$ and construct smooth, strictly positive initial data that develop a finite time singularity. The $C^α$-norm of the distribution function blows up for every $α>0$, whereas its $L^{\infty}$-norm remains uniformly bounded. In self-similar variables, the solution becomes asymptotically hydrodynamic - the distribution function converges to a local Maxwellian, while the hydrodynamic fields develop an asymptotically self-similar implosion whose profile coincides with a smooth imploding profile of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05981 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite time singularities in the Landau equation with very hard potentials Bedrossian, Jacob Chen, Jiajie Gualdani, Maria Pia Ji, Sehyun Vicol, Vlad Yang, Jincheng Analysis of PDEs We consider the inhomogeneous Landau equation with $γ\in (\sqrt{3},2]$ and construct smooth, strictly positive initial data that develop a finite time singularity. The $C^α$-norm of the distribution function blows up for every $α>0$, whereas its $L^{\infty}$-norm remains uniformly bounded. In self-similar variables, the solution becomes asymptotically hydrodynamic - the distribution function converges to a local Maxwellian, while the hydrodynamic fields develop an asymptotically self-similar implosion whose profile coincides with a smooth imploding profile of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data. |
| title | Finite time singularities in the Landau equation with very hard potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.05981 |