the entropy dissipation (see ), which we cannot Gronwall in comparison with the quadratic terms. The answer is to create a shift function which not only neutralizes entropy production at the discontinuity of the solution, but also creates additional negative entropy (see ) we can use to cancel out the linear term in the Gronwall argument (see ). Regarding the idea of additional negative entropy caused by a shift, see [@2017arXiv171207348K].

This work is related to the *generalized Riemann problem*, which concerns solutions with initial data which is piecewise-smooth instead of simply piecewise-constant across a single jump discontinuity. For existence and uniqueness results for the generalized Riemann problem, see [@MR2070131; @MR1680921]. However, these results have small data limitations.

Previous results in this direction include Chen, Frid, and Li [@MR1911734] where for the full Euler system, they show uniqueness and long-time stability for perturbations of Riemann initial data among a large class of entropy solutions (locally $BV$ and without smallness conditions) for the $3\times3$ Euler system in Lagrangian coordinates. They also show uniqueness for solutions piecewise-Lipschitz in $x$. For an extension to the relativistic Euler equations, see Chen and Li [@MR2068444]. However, these papers do not give $L^2$ stability results for 