A Carleman contraction method for inverse initial data recovery in the Navier-Stokes equations with unknown body force
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908988941008896 |
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| author | Nguyen, Phuong M. Nguyen, Loc H. |
| author_facet | Nguyen, Phuong M. Nguyen, Loc H. |
| contents | We solve an inverse initial data problem for the incompressible Navier-Stokes system. The objective is to recover the initial velocity and pressure from lateral boundary observations, without assuming that the time-independent body force is known. To eliminate this unknown force, we differentiate the momentum equation with respect to time and then apply a Legendre polynomial-exponential time-dimensional reduction. This procedure yields a coupled system of elliptic equations for the expansion coefficients. We then construct a contractive map for this reduced system on a suitable admissible set equipped with a Carleman-weighted norm. Its fixed point yields an approximate solution of the time-dimensional reduction model, and the contraction property gives rise to a globally convergent Picard iteration. Finally, we present a numerical algorithm based on this framework and numerical experiments showing accurate reconstructions of the initial velocity and pressure from synthetic boundary data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09934 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Carleman contraction method for inverse initial data recovery in the Navier-Stokes equations with unknown body force Nguyen, Phuong M. Nguyen, Loc H. Analysis of PDEs We solve an inverse initial data problem for the incompressible Navier-Stokes system. The objective is to recover the initial velocity and pressure from lateral boundary observations, without assuming that the time-independent body force is known. To eliminate this unknown force, we differentiate the momentum equation with respect to time and then apply a Legendre polynomial-exponential time-dimensional reduction. This procedure yields a coupled system of elliptic equations for the expansion coefficients. We then construct a contractive map for this reduced system on a suitable admissible set equipped with a Carleman-weighted norm. Its fixed point yields an approximate solution of the time-dimensional reduction model, and the contraction property gives rise to a globally convergent Picard iteration. Finally, we present a numerical algorithm based on this framework and numerical experiments showing accurate reconstructions of the initial velocity and pressure from synthetic boundary data. |
| title | A Carleman contraction method for inverse initial data recovery in the Navier-Stokes equations with unknown body force |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.09934 |