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

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 a negative 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 Lᵖ-Lq L p - L q framework. In particular assuming the boundedness of the quantities ( ⁻¹, u) ( ϱ - 1 , u ) and of a suitable norm of ∇ ₓ ∇ x ϱ the solution remains regular and the blow-up does not occur. We develop the condition on ∇ ₓ ∇ x ϱ because we need a new approach in order to accommodate the inhomogeneous boundary conditions, as the standard estimates on the material time derivative work when the normal component of the boundary velocity is zero.