A Smooth, Recurrent, Non-Periodic Viscosity Solution of the Hamilton-Jacobi Equation
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908365223886848 |
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| author | Charfi, Skander |
| author_facet | Charfi, Skander |
| contents | Viscosity solutions of the Hamilton-Jacobi equation were introduced by Lions and Crandall. For Tonelli Hamiltonians, these solutions are generated by the Lax-Oleinik operator. It is known that this operator converges in the autonomous framework, but this convergence fails in the general cases. In this paper, we introduce a method to construct smooth, recurrent, non-periodic viscosity solutions on fixed compact manifolds $M$ of dimension 2 or higher. Additionally, we provide a detailed description of the non-wandering set of the Lax-Oleinik operator and identify its action on various omega-limit sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08700 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Smooth, Recurrent, Non-Periodic Viscosity Solution of the Hamilton-Jacobi Equation Charfi, Skander Dynamical Systems Analysis of PDEs Optimization and Control Viscosity solutions of the Hamilton-Jacobi equation were introduced by Lions and Crandall. For Tonelli Hamiltonians, these solutions are generated by the Lax-Oleinik operator. It is known that this operator converges in the autonomous framework, but this convergence fails in the general cases. In this paper, we introduce a method to construct smooth, recurrent, non-periodic viscosity solutions on fixed compact manifolds $M$ of dimension 2 or higher. Additionally, we provide a detailed description of the non-wandering set of the Lax-Oleinik operator and identify its action on various omega-limit sets. |
| title | A Smooth, Recurrent, Non-Periodic Viscosity Solution of the Hamilton-Jacobi Equation |
| topic | Dynamical Systems Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2505.08700 |