On the question of vorticity evolution in liquid and gas

We consider the problem of a linear heterogeneous partial differential first order equation, which arises in the general spatial case when constructing the Friedmann velocity field for vorticity using the method proposed by the author in 2015. This method uses Friedman’s theorem, which requires the continuity of the second derivatives of the solution to the problem. It is shown that for some smoothness of the initial conditions, the continuity of the second derivatives of the coefficients and the right-hand side (heterogeneity) of the equation implies the existence of a solution and the continuity of its second derivatives in some three-dimensional domain containing a flat domain on which the initial conditions are given. We establish requirements for the smoothness of hydrodynamic functions that enter together with their derivatives into the above expressions of the coefficients the right side of the equation. As a result, a rigorous justification is given for an approach proposed in 2015 to consider the Friedman velocity.