Blow-up criterion for the 3-D inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion

In this paper, we investigate the blow-up criterion of an evolutionary partial differential equation system controlling the flows of incompressible viscoelastic rate-type fluids with stress-diffusion, where the extra stress tensor describing the elastic response of the fluid is purely spherical. By utilizing this criterion, the global well-posedness of the system can be readily obtained. Despite being a physical simplification, this model exhibits features that necessitate novel mathematical approaches to tackle the technically complex structure of the associated internal energy, as well as the more complicated forms of the corresponding entropy and energy fluxes. The paper provides the first rigorous proof of the existence of a global solution to the model under small initial data.