pure shock wave initial data in the systems case, the technique of a-contraction was introduced [@MR3519973; @MR3479527; @MR3537479; @serre_vasseur; @Leger2011]. For a general overview of theory of shifts and the relative entropy method, see [@MR3475284 Section 3-5]. By considering stability up to a shift, the method of relative entropy can also be used to study the asymptotic limit when the limit is discontinuous (see [@MR3333670] for the scalar case, [@MR3421617] for systems). There is a long history of using the relative entropy method to study the asymptotic limit. However, without the theory of shifts, it appears that only limits which are Lipschitz continuous can be studied (see [@MR1842343; @MR1980855; @MR2505730; @MR1115587; @MR1121850; @MR1213991; @MR2178222; @MR2025302] and [@VASSEUR2008323] for a survey).

The present article is a further step in the program of stability up to a shift.

In this paper, we continue the ideas introduced in [@scalar_move_entire_solution]. In [@scalar_move_entire_solution], it is shown that the generalized characteristics of $u$ can be used as shift functions to kill growth in $L^2$ between a piecewise smooth solution $\bar{u}$ and weak solution to entropic for the entropy $\eta$. Further, using the generalized characteristic as a shift function provides various benefits over using the previous shift 