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

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 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 $$L^p-L^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:mrow> </mml:math> framework. In particular assuming the boundedness of the quantities $$(\varrho ^{-1}, \varvec{u})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ϱ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and of a suitable norm of $$\nabla _x \varrho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∇</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>ϱ</mml:mi> </mml:mrow> </mml:math> the solution remains regular and the blow-up does not occur. We develop the condition on $$\nabla _x \varrho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∇</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>ϱ</mml:mi> </mml:mrow> </mml:math> 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.