I  present a new model for heat transfer in compressible fluid flows.
The model is derived from Hamilton's principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler--Lagrange equation.
The governing system is shown to be first-order hyperbolic and asymptotically consistent with the Euler equations for compressible heat conducting fluids.
The  governing equations admit the eigenfields that are neither linear degenerate nor genuinely non-linear in the sense of Lax.
In particular, we found expansion shocks and compression fans as possible solutions of the governing equations.
The expansion shocks satisfy the generalized Clausius -- Duhem inequality and Oleinik -- Liu admissibility condition.
Evidence of these properties is provided on a set of numerical test cases.

This is a joint work with F. Dhaouadi.