Continuous Data Assimilation for Semilinear Parabolic Equations with Multiplicative Observation Noise
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917480023195648 |
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| author | Bröcker, Jochen Del Sarto, Gianmarco Hieber, Matthias Palma, Filippo Zöchling, Tarek |
| author_facet | Bröcker, Jochen Del Sarto, Gianmarco Hieber, Matthias Palma, Filippo Zöchling, Tarek |
| contents | The problem of continuous data assimilation for semilinear parabolic equations based on partial observations corrupted by noise is investigated. The noise is allowed to be multiplicative, with additive noise arising as a special case. In a general Gelfand triple framework, an abstract theory for the nudging equation is developed that covers both weak and strong formulations. Mean square convergence of the assimilation error is proved under suitable assumptions, and, under additional integrability conditions on the noise, a uniform almost sure convergence result is established. Finally, the framework is applied to several PDE models, including the 2D Navier-Stokes, 2D magnetohydrodynamics, 2D quasi-geostrophic, and 1D Allen-Cahn equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10322 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Continuous Data Assimilation for Semilinear Parabolic Equations with Multiplicative Observation Noise Bröcker, Jochen Del Sarto, Gianmarco Hieber, Matthias Palma, Filippo Zöchling, Tarek Analysis of PDEs Mathematical Physics Probability 35K58, 60H15, 93C20, 35B40, 35B41, 37L30 The problem of continuous data assimilation for semilinear parabolic equations based on partial observations corrupted by noise is investigated. The noise is allowed to be multiplicative, with additive noise arising as a special case. In a general Gelfand triple framework, an abstract theory for the nudging equation is developed that covers both weak and strong formulations. Mean square convergence of the assimilation error is proved under suitable assumptions, and, under additional integrability conditions on the noise, a uniform almost sure convergence result is established. Finally, the framework is applied to several PDE models, including the 2D Navier-Stokes, 2D magnetohydrodynamics, 2D quasi-geostrophic, and 1D Allen-Cahn equations. |
| title | Continuous Data Assimilation for Semilinear Parabolic Equations with Multiplicative Observation Noise |
| topic | Analysis of PDEs Mathematical Physics Probability 35K58, 60H15, 93C20, 35B40, 35B41, 37L30 |
| url | https://arxiv.org/abs/2605.10322 |