On Galerkin approximations of the Navier-Stokes equations in the limit of large Grashof numbers
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911768118296576 |
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| author | Foias, Ciprian Hoang, Luan Jolly, Michael S. |
| author_facet | Foias, Ciprian Hoang, Luan Jolly, Michael S. |
| contents | We examine how stationary solutions to Galerkin approximations of the Navier--Stokes equations behave in the limit as the Grashof number $G$ tends to $\infty$. An appropriate scaling is used to place the Grashof number as a new coefficient of the nonlinear term, while the body force is fixed. A new type of asymptotic expansion, as $G\to\infty$, for a family of solutions is introduced. Relations among the terms in the expansion are obtained by following a procedure that compares and totally orders positive sequences generated by the expansion. The same methodology applies to the case of perturbed body forces and similar results are obtained. We demonstrate with a class of forces and solutions that have convergent asymptotic expansions in $G$. All the results hold in both two and three dimensions, as well as for both no-slip and periodic boundary conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15649 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Galerkin approximations of the Navier-Stokes equations in the limit of large Grashof numbers Foias, Ciprian Hoang, Luan Jolly, Michael S. Analysis of PDEs Mathematical Physics 35Q30, 76D05, 76F02, 35C20, 41A60 We examine how stationary solutions to Galerkin approximations of the Navier--Stokes equations behave in the limit as the Grashof number $G$ tends to $\infty$. An appropriate scaling is used to place the Grashof number as a new coefficient of the nonlinear term, while the body force is fixed. A new type of asymptotic expansion, as $G\to\infty$, for a family of solutions is introduced. Relations among the terms in the expansion are obtained by following a procedure that compares and totally orders positive sequences generated by the expansion. The same methodology applies to the case of perturbed body forces and similar results are obtained. We demonstrate with a class of forces and solutions that have convergent asymptotic expansions in $G$. All the results hold in both two and three dimensions, as well as for both no-slip and periodic boundary conditions. |
| title | On Galerkin approximations of the Navier-Stokes equations in the limit of large Grashof numbers |
| topic | Analysis of PDEs Mathematical Physics 35Q30, 76D05, 76F02, 35C20, 41A60 |
| url | https://arxiv.org/abs/2308.15649 |