Existence, scaling, and spectral gap for traveling fronts in the 2D renormalized Allen--Cahn equation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918249820585984 |
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| author | Chiusole, Gideon Kuehn, Christian |
| author_facet | Chiusole, Gideon Kuehn, Christian |
| contents | We study the deterministic skeleton of the renormalized stochastic Allen--Cahn equation in spatial dimension $2$. For all sufficiently small regularization parameters $δ>0$, we construct monotone traveling wave front solutions connecting the renormalized equilibria, derive a small-$δ$ asymptotic description of their profile and speed, and identify the leading-order contributions. Linearizing about the wave and working in a naturally chosen weighted space, we show that there exists a spectral gap between the symmetry induced eigenvalue $0$ and the rest of the spectrum. The spectral gap grows linearly in the renormalization constant as $δ\downarrow 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence, scaling, and spectral gap for traveling fronts in the 2D renormalized Allen--Cahn equation Chiusole, Gideon Kuehn, Christian Analysis of PDEs Mathematical Physics 34L15, 37L15, 35C07, 47A56, 47A53 We study the deterministic skeleton of the renormalized stochastic Allen--Cahn equation in spatial dimension $2$. For all sufficiently small regularization parameters $δ>0$, we construct monotone traveling wave front solutions connecting the renormalized equilibria, derive a small-$δ$ asymptotic description of their profile and speed, and identify the leading-order contributions. Linearizing about the wave and working in a naturally chosen weighted space, we show that there exists a spectral gap between the symmetry induced eigenvalue $0$ and the rest of the spectrum. The spectral gap grows linearly in the renormalization constant as $δ\downarrow 0$. |
| title | Existence, scaling, and spectral gap for traveling fronts in the 2D renormalized Allen--Cahn equation |
| topic | Analysis of PDEs Mathematical Physics 34L15, 37L15, 35C07, 47A56, 47A53 |
| url | https://arxiv.org/abs/2512.14245 |