  We consider non-dissipative (elastic) rate-type material models that are
derived within the Gibbs-potential-based thermodynamic framework. Since the
absence of any dissipative mechanism in the model prevents us from establishing
even a local-in-time existence result in two spatial dimensions for a spatially
periodic problem, we propose two regularisations. For such regularized problems
we obtain well-posedness of the planar, spatially periodic problem. In contrast
with existing results, we prove ours for a regularizing term present solely in
the evolution equation for the stress.
