Dean-Kawasaki Equation with Biot-Savart and Keller-Segel Interactions: Existence and Large Deviations
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| Format: | Preprint |
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2026
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| _version_ | 1866910216411414528 |
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| author | Ji, Xiaohao Sun, Yue Wu, Zhengyan |
| author_facet | Ji, Xiaohao Sun, Yue Wu, Zhengyan |
| contents | We establish the existence of probabilistically weak, renormalized kinetic solutions to the Dean--Kawasaki equation with singular interaction kernels, including those of Biot--Savart and Keller--Segel type. Under a suitable regularization of the square-root noise coefficient, we further prove a restricted large deviation principle for probabilistically weak solutions to the regularized Dean--Kawasaki equation. The Biot--Savart and Keller--Segel type interactions introduce a scaling criticality within the $L^1$ framework of the Dean--Kawasaki equation and the associated skeleton equation, which gives rise to a significant new challenge. In contrast to [Fehrman, Gess; Invent. Math., 2023], our large deviation analysis relies on a novel exponential tightness argument specifically adapted to the Dean--Kawasaki noise. This approach, combined with a weak-strong uniqueness result for the associated skeleton equation, allows us to partially overcome the criticality induced by the singular interaction kernel. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dean-Kawasaki Equation with Biot-Savart and Keller-Segel Interactions: Existence and Large Deviations Ji, Xiaohao Sun, Yue Wu, Zhengyan Probability Analysis of PDEs We establish the existence of probabilistically weak, renormalized kinetic solutions to the Dean--Kawasaki equation with singular interaction kernels, including those of Biot--Savart and Keller--Segel type. Under a suitable regularization of the square-root noise coefficient, we further prove a restricted large deviation principle for probabilistically weak solutions to the regularized Dean--Kawasaki equation. The Biot--Savart and Keller--Segel type interactions introduce a scaling criticality within the $L^1$ framework of the Dean--Kawasaki equation and the associated skeleton equation, which gives rise to a significant new challenge. In contrast to [Fehrman, Gess; Invent. Math., 2023], our large deviation analysis relies on a novel exponential tightness argument specifically adapted to the Dean--Kawasaki noise. This approach, combined with a weak-strong uniqueness result for the associated skeleton equation, allows us to partially overcome the criticality induced by the singular interaction kernel. |
| title | Dean-Kawasaki Equation with Biot-Savart and Keller-Segel Interactions: Existence and Large Deviations |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2605.13479 |