On the stationary Navier-Stokes equations in distorted pipes under energy-stable outflow boundary conditions
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arXiv
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| Format: | Preprint |
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2025
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| author | Falocchi, Alessio Silvestre, Ana Leonor Sperone, Gianmarco |
| author_facet | Falocchi, Alessio Silvestre, Ana Leonor Sperone, Gianmarco |
| contents | The steady motion of a viscous incompressible fluid in distorted pipes, of finite length, is modeled through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by an arbitrary member of the Lions-Magenes class with positive influx, and the fluid motion is subject to a directional do-nothing boundary condition on the outlet, together with the standard no-slip assumption on the remaining walls of the domain. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data (that is, inlet velocity and external force) by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument that employs Bernoulli's law for solutions of the stationary Euler equations, as well as some properties of harmonic divergence-free vector fields. Under a suitable smallness assumption on the data, we also prove the unique solvability of the boundary-value problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07331 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the stationary Navier-Stokes equations in distorted pipes under energy-stable outflow boundary conditions Falocchi, Alessio Silvestre, Ana Leonor Sperone, Gianmarco Analysis of PDEs 35Q30, 35G60, 76D05, 35M12 The steady motion of a viscous incompressible fluid in distorted pipes, of finite length, is modeled through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by an arbitrary member of the Lions-Magenes class with positive influx, and the fluid motion is subject to a directional do-nothing boundary condition on the outlet, together with the standard no-slip assumption on the remaining walls of the domain. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data (that is, inlet velocity and external force) by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument that employs Bernoulli's law for solutions of the stationary Euler equations, as well as some properties of harmonic divergence-free vector fields. Under a suitable smallness assumption on the data, we also prove the unique solvability of the boundary-value problem. |
| title | On the stationary Navier-Stokes equations in distorted pipes under energy-stable outflow boundary conditions |
| topic | Analysis of PDEs 35Q30, 35G60, 76D05, 35M12 |
| url | https://arxiv.org/abs/2506.07331 |