  The Lagrange-Flux schemes are Eulerian finite volume schemes that make use of
an approximate Riemann solver in Lagrangian description with particular upwind
convective fluxes. They have been recently designed as variant formulations of
Lagrange-remap schemes that provide better HPC performance on modern multicore
processors, see~[De Vuyst et al., OGST 71(6), 2016]. Actually Lagrange-Flux
schemes show several advantages compared to Lagrange-remap schemes, especially
for multidimensional problems: they do not require the computation of deformed
Lagrangian cells or mesh intersections as in the remapping process. The paper
focuses on the entropy property of Lagrange-Flux schemes in their semi-discrete
in space form, for one-dimensional problems and for the compressible Euler
equations as example. We provide pseudo-viscosity pressure terms that ensure
entropy production of order $O(|\Delta u|^3)$, where $|\Delta u|$ represents a
velocity jump at a cell interface. Pseudo-viscosity terms are also designed to
vanish into expansion regions as it is the case for rarefaction waves.
