We consider a Cauchy problem for a system of two quasilinear first order partial differential equations with special right-hand sides. We obtain the conditions of a nonlocal solvability of this Cauchy problem. The study of the nonlocal solvability of the Cauchy problem for a system of two quasilinear differential equations with special right-hand sides is based on the method of an additional argument. The proof of the nonlocal solvability of the Cauchy problem for a system of two quasilinear first order partial differential equations with special right-hand sides relies on global estimates.

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We consider a Cauchy problem for a system of two first order quasilinear differential equations with right-hand sides 1 = 2 (,) + 2 () (,), 2 = 2 (,). We study the solvability of the Cauchy problem on the base of an additional argument method. We obtain the sufficient conditions for the existence and uniqueness of a local solution to the Cauchy problem in terms of the original coordinates coordinates for a system of two first order quasilinear differential equations with right-hand sides 1 = 2 (,) + 2 () (,), 2 = 2 (,), under which the solution has the same smoothness in as the initial functions in the Cauchy problem does. A theorem on the local existence and uniqueness of a solution to the Cauchy problem is formulated and proved. The theorem on the local existence and uniqueness of a solution to the Cauchy problem for a system of two first order quasilinear differential equations with right-hand sides 1 = 2 (,) + 2 () (,), 2 = 2 (,) is proved by the additional argument method. We obtain the sufficient conditions of the existence and uniqueness of a nonlocal solution to the Cauchy problem in terms of the initial coordinates for a system of two first order quasilinear differential equations with right-hand sides 1 = 2 (,) + 2 () (,), 2 = 2 (,). A theorem on the nonlocal existence and uniqueness of the solution of the Cauchy problem is formulated and proved. The proof of the nonlocal solvability of the Cauchy problem for a system of two quasilinear first order partial differential equations with right-hand sides 1 = 2 (,) + 2 () (,), 2 = 2 (,) is based on global estimates.

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The Cauchy problem for a system of first-order quasilinear equations with special right-hand sides is considered. The study of solvability of this system in the original coordinates is based on the method of additional argument. It is proved that the local solution of such system exists and that its smoothness is not lower than the smoothness of the initial conditions. For system of two equations non-local solutions are considered that are continued by finite number of steps from the local solution. Sufficient conditions for the existence of such non-local solution are derived. The proof of the non-local resolvability of the system relies on original global estimates.