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In this work, we introduce a notion of dissipative weak solution for a systemdescribing the evolution of a heat-conducting incompressible non-Newtonianfluid. This concept of solution is based on the balance of entropy instead ofthe balance of energy and has the advantage that it admits a weak-stronguniqueness principle, justifying the proposed formulation. We provide a proofof existence of solutions based on finite element approximations, thusobtaining the first convergence result of a numerical scheme for the fullevolutionary system including temperature dependent coefficients and viscousdissipation terms. Then we proceed to prove the weak-strong uniqueness propertyof the system by means of a relative energy inequality.

Weak-strong Uniqueness for Heat Conducting non-Newtonian Incompressible Fluids