On gelation for the Smoluchowski equation
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866929663838781440 |
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| author | Fournier, Nicolas |
| author_facet | Fournier, Nicolas |
| contents | Motivated by the recent results of Andreis-Iyer-Magnanini (2023), we provide a short proof, revisiting the one of Escobedo-Mischler-Perthame (2002), that for a large class of coagulation kernels, any weak solution to the Smoluchowski equation looses mass in finite time. The class of kernels we consider is essentially the same as the one of Andreis-Iyer-Magnanini (2023): homogeneous kernels of degree $γ>1$ not vanishing on the diagonal, or homogeneous kernels of degree $γ=1$ not vanishing on the diagonal with some additional logarithmic factor. We also show that when $γ=1$, the power of the logarithmic factor ensuring gelation may depend on the shape of the kernel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On gelation for the Smoluchowski equation Fournier, Nicolas Analysis of PDEs 45K05, 45M99 Motivated by the recent results of Andreis-Iyer-Magnanini (2023), we provide a short proof, revisiting the one of Escobedo-Mischler-Perthame (2002), that for a large class of coagulation kernels, any weak solution to the Smoluchowski equation looses mass in finite time. The class of kernels we consider is essentially the same as the one of Andreis-Iyer-Magnanini (2023): homogeneous kernels of degree $γ>1$ not vanishing on the diagonal, or homogeneous kernels of degree $γ=1$ not vanishing on the diagonal with some additional logarithmic factor. We also show that when $γ=1$, the power of the logarithmic factor ensuring gelation may depend on the shape of the kernel. |
| title | On gelation for the Smoluchowski equation |
| topic | Analysis of PDEs 45K05, 45M99 |
| url | https://arxiv.org/abs/2501.03867 |