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We prove that there exists a~large-data and global-in-time weak solution toa~system of partial differential equations describing an unsteady flow of anincompressible heat-conducting rate-type viscoelastic stress-diffusive fluidfilling up a~mechanically and thermally isolated container of any dimension. Toovercome the~principle difficulties connected with ill-posedness ofthe~diffusive Oldroyd-B model in three dimensions, we assume that the~fluidadmits a~strengthened dissipation mechanism, at least for excessive elasticdeformations. All the~relevant material coefficients are allowed to dependcontinuously on the~temperature, whose evolution is captured bya~thermodynamically consistent equation. In fact, the~studied model is derivedfrom scratch using only the~balance equations for linear momentum and energy,the~formulation of the~second law of thermodynamics and the~constitutiveequation for the~internal energy. The~latter is assumed to be a~linear functionof temperature, which simplifies the~model. The~concept of our weak solutionincorporates both the~temperature and entropy inequalities, and also the~localbalance of total energy provided that the~pressure function exists.

Coupling the Navier-Stokes-Fourier equations with the Johnson-Segalman stress-diffusive viscoelastic model: Global-in-time and large-data analysis