Errors in approximate solutions of Cauchy's problem for a first-order quasilinear equation

The proximity is investigated of the solution of Cauchy's problem for the equation u t ? +(?(u ? )) x = ?u xx ? (??(u ? ) > 0) to the solution of Cauchy's problem for the equation u t + (?(u)) x = 0, when the solution of the latter problem has a finite number of lines of discontinuity in the strip 0 = t = T. It is proved that, everywhere outside a fixed neighborhood of the lines of discontinuity, we have |u ? -u| = Ce, where the constant C is independent ofe. Similar inequalities are derived for the first derivatives of u ? -u.