Solvability conditions of the Cauchy problem for a system of first-order quasi-linear equations, where ${f_1}(t,x), {f_2}(t,x), {S_1}, {S_2}$ are given functions

We consider a Cauchy problem for a system of two quasilinear first order partial differential equations with continuous and bounded free terms. Theorems on the local and nonlocal existence and uniqueness of solutions to the Cauchy problem are formulated and proved. The sufficient conditions for the existence and uniqueness of a local solution of the Cauchy problem in the initial coordinates at which the solution has the same smoothness with respect to $ x $ as the initial functions of the Cauchy problem are determined. The sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem in the initial coordinates (for a given finite interval $t\in[0,T]$) are determined. Local existence and uniqueness theorem of the solution of the Cauchy problem for a system of quasilinear first order partial differential equations with continuous and bounded free terms is proved with the method of an additional argument. The investigation of a nonlocal solvability of the Cauchy problem is based on the method of an additional argument. The proof of the nonlocal solvability of the Cauchy problem for a system of quasilinear first order partial differential equations with continuous and bounded free terms relies on global estimates.