Refined blow-up criteria for the full compressible Navier–Stokes equations involving temperature

In this paper, inspired by the study of energy flux in local inequality 3D incompressible Navier–Stokes equations, we improve almost all blow-up criteria involving temperature to allow its scaling invariant space for full compressible equations. Enlightening regular via pressure $$\Pi =\frac{\text{ divdiv }}{-\Delta }(u_{i}u_{j})$$ equations on bounded domain, generalize Beirao da Veiga’s result (Chin Ann Math Ser B 16:407–412, 1995) from isentropic system case away vacuum.