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arXiv · 2512.06150

Blow-up criterion for the compressible Navier--Stokes system with inflow-outflow boundary conditions

Anna Abbatiello·Mostafa Meliani
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Abstract

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.

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Anna Abbatiello, Mostafa Meliani. 2025-12-05. Blow-up criterion for the compressible Navier--Stokes system with inflow-outflow boundary conditions. https://arxiv.org/abs/2512.06150

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