  Let a 1-d system of hyperbolic conservation laws, with two unknowns, be
endowed with a convex entropy. We consider the family of small $BV$ functions
which are global solutions of this equation. For any small $BV$ initial data,
such global solutions are known to exist. Moreover, they are known to be unique
among $BV$ solutions verifying either the so-called Tame Oscillation Condition,
or the Bounded Variation Condition on space-like curves. In this paper, we show
that these solutions are stable in a larger class of weak (and possibly not
even $BV$) solutions of the system. This result extends the classical
weak-strong uniqueness results which allow comparison to a smooth solution.
Indeed our result extends these results to a weak-$BV$ uniqueness result, where
only one of the solutions is supposed to be small $BV$, and the other solution
can come from a large class. As a consequence of our result, the Tame
Oscillation Condition, and the Bounded Variation Condition on space-like curves
are not necessary for the uniqueness of solutions in the $BV$ theory, in the
case of systems with 2 unknowns. The method is $L^2$ based. It builds up from
the theory of a-contraction with shifts, where suitable weight functions $a$
are generated via the front tracking method.
