On incompressible heat-conducting viscoelastic rate-type fluids with
stress-diffusion and purely spherical elastic response

We prove the existence of large-data global-in-time weak solutions to an
evolutionary PDE system describing flows of incompressible
\emph{heat-conducting} viscoelastic rate-type fluids with stress-diffusion,
subject to a stick-slip boundary condition for the velocity and a homogeneous
Neumann boundary condition for the extra stress tensor. In the introductory
section we develop the thermodynamic foundations of the proposed model, and we
document the role of thermodynamics in obtaining critical structural relations
between the quantities of interest. These structural relations are then
exploited in the mathematical analysis of the governing equations. In
particular, the definition of weak solution is motivated by the thermodynamic
basis of the model.
The extra stress tensor describing the elastic response of the fluid is in
our case purely spherical, which is a simplification from the physical point of
view. The model nevertheless exhibits features that require novel mathematical
ideas in order to deal with the technically complex structure of the associated
internal energy and the more complicated forms of the corresponding entropy and
energy fluxes. The paper provides the first rigorous proof of the existence of
large-data global-in-time weak solutions to the governing equations for
\emph{coupled thermo-mechanical processes} in viscoelastic rate-type fluids.