Irrotational approximation to the quasi-1-d gas flow

Under the assumptions that both initial data and the cross-section have sufficiently small total variation and that the initial data are supersonic (or are subsonic respectively), we prove that in any bounded domain the L¹ norm of the difference between solutions of the hyperbolic system of balance laws and the potential flow system of balance laws with the same initial data can be bounded by the cube of the sum over total variations of the initial data and the cross-section.