Abstract
A system of the Burgers equations of the two-velocity hydrodynamics is constructed. We consider the Cauchy problem in the case of a one-dimensional system. We have obtained a formula for solving the Cauchy problem and the estimate of the stability of this solution. 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. The existence and uniqueness of solutions to the Cauchy problem for the one-dimensional systems of the Burgers type are proved using the method of weak approximation.
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