  Little is known about global existence of large-variation solutions to Cauchy
problems for systems of conservation laws in one space dimension. Besides
results for $L^\infty$ data via compensated compactness, the existence of
global BV solutions for arbitrary BV data remains an outstanding open problem.
In particular, it is not known if isentropic gas dynamics admits an a priori
variation bound which applies to all BV data.
  In a few cases such results are available: scalar equations, Temple class
systems, $2\times 2$-systems satisfying Bakhvalov's condition, and, in
particular, isothermal gas dynamics. In each of these cases the equations admit
a TVD (Total Variation Diminishing) field: a scalar function defined on state
space whose spatial variation along entropic solutions does not increase in
time.
  In this paper we consider strictly hyperbolic $2\times 2$-systems and derive
a representation result for scalar fields that are TVD across all pairwise wave
interactions, when the latter are resolved as in the Glimm scheme. We then use
this to show that isentropic gas dynamics with a $\gamma$-law pressure function
does not admit any nontrivial TVD field of this type.
