Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908386092646400 |
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| author | Stevenson, Noah Tice, Ian |
| author_facet | Stevenson, Noah Tice, Ian |
| contents | We give the first mathematical construction of two-dimensional traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations for a single finite depth layer of constant density fluid. Our construction is based on a rigorous justification of the formal shallow water limit, which postulates that in a certain scaling regime the full free boundary traveling Navier-Stokes system of PDEs reduces to a governing system of ODEs. We find heteroclinic orbits solving these ODEs and, through a delicate fixed point argument employing the Stokes problem in thin domains and a nonautonomous orbital perturbation theory, use these ODE solutions as the germs from which we build bore PDE solutions for sufficiently shallow layers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_24562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations Stevenson, Noah Tice, Ian Analysis of PDEs Primary 35Q35, 35C07, 76D33, Secondary 35B40, 35J66, 76A20, 76L05, 34C37, 35A24 We give the first mathematical construction of two-dimensional traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations for a single finite depth layer of constant density fluid. Our construction is based on a rigorous justification of the formal shallow water limit, which postulates that in a certain scaling regime the full free boundary traveling Navier-Stokes system of PDEs reduces to a governing system of ODEs. We find heteroclinic orbits solving these ODEs and, through a delicate fixed point argument employing the Stokes problem in thin domains and a nonautonomous orbital perturbation theory, use these ODE solutions as the germs from which we build bore PDE solutions for sufficiently shallow layers. |
| title | Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations |
| topic | Analysis of PDEs Primary 35Q35, 35C07, 76D33, Secondary 35B40, 35J66, 76A20, 76L05, 34C37, 35A24 |
| url | https://arxiv.org/abs/2505.24562 |