Carleman estimates for the Korteweg-de Vries equation with piecewise constant main coefficient
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908360143536128 |
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| author | Loyola, Cristóbal |
| author_facet | Loyola, Cristóbal |
| contents | In this article, we investigate observability-related properties of the Korteweg-de Vries equation with a discontinuous main coefficient, coupled by suitable interface conditions. The main result is a novel two-parameter Carleman estimate for the linear equation with internal observation, assuming a monotonicity condition on the main coefficient. As a primary application, we establish the local exact controllability to the trajectories by employing a duality argument for the linear case and a local inversion theorem for the nonlinear equation. Secondly, we establish the Lipschitz-stability of the inverse problem of retrieving an unknown potential using the Bukhge{\uı}m-Klibanov method, when some further assumptions on the interface are made. We conclude with some remarks on the boundary observability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07264 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Carleman estimates for the Korteweg-de Vries equation with piecewise constant main coefficient Loyola, Cristóbal Analysis of PDEs Optimization and Control 35Q53, 35R05, 35R30, 93B05, 93B07 93C20 In this article, we investigate observability-related properties of the Korteweg-de Vries equation with a discontinuous main coefficient, coupled by suitable interface conditions. The main result is a novel two-parameter Carleman estimate for the linear equation with internal observation, assuming a monotonicity condition on the main coefficient. As a primary application, we establish the local exact controllability to the trajectories by employing a duality argument for the linear case and a local inversion theorem for the nonlinear equation. Secondly, we establish the Lipschitz-stability of the inverse problem of retrieving an unknown potential using the Bukhge{\uı}m-Klibanov method, when some further assumptions on the interface are made. We conclude with some remarks on the boundary observability. |
| title | Carleman estimates for the Korteweg-de Vries equation with piecewise constant main coefficient |
| topic | Analysis of PDEs Optimization and Control 35Q53, 35R05, 35R30, 93B05, 93B07 93C20 |
| url | https://arxiv.org/abs/2505.07264 |