Cauchy Problem for System of the Burgers Equations Arising in the Two-velocity Hydrodynamics

A system of the Burgers equations of the two-velocity hydrodynamics is derived. We consider the Cauchy problem in the case of a one-dimensional system and the estimate of the stability of the solution is obtained. We have obtained a formula for the Cauchy problem for the one-dimensional system of equations that arises in the two-velocity hydrodynamics. It is shown that with disappearance of the kinetic friction coefficient, which is responsible for the energy dissipation, this formula turns to the famous Cauchy problem for the one-dimensional Burgers equation.