Bifurcations of viscous boundary layers in the half space
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908580964204544 |
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| author | Bian, Dongfen Grenier, Emmanuel Iooss, Gérard |
| author_facet | Bian, Dongfen Grenier, Emmanuel Iooss, Gérard |
| contents | It is well-established that shear flows are linearly unstable provided the viscosity is small enough, when the horizontal Fourier wave number lies in some interval, between the so-called lower and upper marginally stable curves. In this article, we prove that, under a natural spectral assumption, shear flows undergo a Hopf bifurcation near their upper marginally stable curve. In particular, close to this curve, there exists space periodic traveling waves solutions of the full incompressible Navier-Stokes equations. For the linearized operator, the occurrence of an essential spectrum containing the entire negative real axis causes certain difficulties which are overcome. Moreover, if this Hopf bifurcation is super-critical, these time and space periodic solutions are linearly and nonlinearly asymptotically stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bifurcations of viscous boundary layers in the half space Bian, Dongfen Grenier, Emmanuel Iooss, Gérard Analysis of PDEs 35Q30, 35B32 It is well-established that shear flows are linearly unstable provided the viscosity is small enough, when the horizontal Fourier wave number lies in some interval, between the so-called lower and upper marginally stable curves. In this article, we prove that, under a natural spectral assumption, shear flows undergo a Hopf bifurcation near their upper marginally stable curve. In particular, close to this curve, there exists space periodic traveling waves solutions of the full incompressible Navier-Stokes equations. For the linearized operator, the occurrence of an essential spectrum containing the entire negative real axis causes certain difficulties which are overcome. Moreover, if this Hopf bifurcation is super-critical, these time and space periodic solutions are linearly and nonlinearly asymptotically stable. |
| title | Bifurcations of viscous boundary layers in the half space |
| topic | Analysis of PDEs 35Q30, 35B32 |
| url | https://arxiv.org/abs/2510.06715 |