Blow-up criterion for the compressible Navier--Stokes system with inflow-outflow boundary conditions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914182991970304 |
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| author | Abbatiello, Anna Meliani, Mostafa |
| author_facet | Abbatiello, Anna Meliani, Mostafa |
| contents | We consider the compressible Navier-Stokes system in three dimensions with general inflow-outflow boundary conditions, meaning that we prescribe a boundary velocity which has non-zero normal component and accordingly the density is prescribed on the inflow part of the boundary. We establish a blow-up criterion in a class of strong solutions in the $L^p-L^q$ framework. In particular assuming the boundedness of the quantities $(\varrho^{-1}, \bu)$ and of a suitable norm of $\nabla_x \varrho$ the solution remains regular and the blow-up does not occur. We develop the condition on $\nabla_x \varrho$ because we need a new approach in order to accommodate the inhomogeneous boundary conditions, as the standard estimates on the material time derivative works when the normal component of the boundary velocity is zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06150 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blow-up criterion for the compressible Navier--Stokes system with inflow-outflow boundary conditions Abbatiello, Anna Meliani, Mostafa Analysis of PDEs We consider the compressible Navier-Stokes system in three dimensions with general inflow-outflow boundary conditions, meaning that we prescribe a boundary velocity which has non-zero normal component and accordingly the density is prescribed on the inflow part of the boundary. We establish a blow-up criterion in a class of strong solutions in the $L^p-L^q$ framework. In particular assuming the boundedness of the quantities $(\varrho^{-1}, \bu)$ and of a suitable norm of $\nabla_x \varrho$ the solution remains regular and the blow-up does not occur. We develop the condition on $\nabla_x \varrho$ because we need a new approach in order to accommodate the inhomogeneous boundary conditions, as the standard estimates on the material time derivative works when the normal component of the boundary velocity is zero. |
| title | Blow-up criterion for the compressible Navier--Stokes system with inflow-outflow boundary conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.06150 |