in the form of the classical weak-strong estimates.

However, by allowing the discontinuity in $\bar{u}$ to move with an artificial speed which depends on $u$, we can recover weak-strong type estimates. Within the context of the relative entropy method, this theory of stability up to a shift was initiated in [@VASSEUR2008323] by Vasseur. Over the last decade, this theory of stability up to a shift has been matured and developed by Vasseur and his team. The first result was for pure shock wave initial data for the scalar conservation laws [@Leger2011_original]. Further results include work on the scalar viscous conservation laws in both one space dimension [@MR3592682] and multiple [@multi_d_scalar_viscous_9122017]. Recently, work on the scalar conservation laws has allowed for many discontinuities to exist in the otherwise smooth $\bar{u}$ – with each discontinuity shifted in such a way as to maintain $L^2$ stability between $\bar{u}$ and an arbitrary weak solution $u$ entropic for at least one entropy. With this, it is possible to make comparisons between two solutions which satisfy only one entropy condition, and thus show that one entropy condition is enough for uniqueness. See [@2017arXiv170905610K] (and the references therein) for more details. To study the $L^2$ stability of 