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

2025/12/05 by Abbatiello, Anna, Meliani, Mostafa
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.06150

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 ∇_x \varrho the solution remains regular and the blow-up does not occur. We develop the condition on ∇_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.

Citations

• L^p-L^q existence for the open compressible MHD system
• Conditional regularity for the Navier-Stokes-Fourier system with Dirichlet boundary conditions
• A Blow-up Criterion for Two Dimensional Compressible Viscous Heat-Conductive Flows
• Hitchhiker's guide to the fractional Sobolev spaces
• A blow-up criterion for compressible viscous heat-conductive flows
• A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations
• Parabolic and elliptic equations with VMO coefficients

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