The structure of dissipative viscous system of conservation laws

This assumption allows us to make a priori estimates of weak solutions satisfying the entropy inequality ∂tu+ divQ(u) ≤ 0, such that u(·, t)− ū is square integrable. The purpose of this paper is to study viscous extensions of (1) that are compatible with the entropy, in the sense that there is an entropy inequality for classical solutions of the extended system. In our weakest setting, such an inequality stands at the level of spatial integrals only. Somehow, we justify requirements made by Kawashima in his thesis [9], using only natural assumptions.