For the the initial-boundary value problem (IBVP) for the isentropic gas dynamics written in Lagrangian coordinates in Riemann invariants we show how the necessary conditions for existence of global smooth solution can be obtained using technics due to P.\,Lax. Under these conditions we formulate the existence result in the class of piecewise-smooth functions. A priori estimates with respect to the input data for the difference scheme approximating this problem are obtained. Stability estimates are proved using only limitations for the initial and boundary conditions corresponding the differential problem. Estimates of stability in the general case have been obtained only for the finite instant of time $t