On the Navier–Stokes equations: Existence theorems and energy equalities

Currently available results on the solvability of the Navier–Stokes equations for incompressible non-Newtonian fluids are presented. The order of nonlinearity in the equations may be variable; the only requirement is that it must be a measurable function. Unsteady and steady equations are considered. A lot of attention is paid to the recovery of energy balance, whose violation is theoretically admissible, in particular, in the three-dimensional classical unsteady Navier–Stokes equation. When constructing a weak solution by a limit procedure, a measure arises as a limit of viscous energy densities. Generally speaking, the limit measure contains a nonnegative singular (with respect to the Lebesgue measure) component. It is this singular component that maintains energy balance. Sufficient conditions for the absence of a singular component are studied: in this case, the standard energy equality holds. In many respects, only the regular component of the limit measure is important: in the natural form it is equal to the product of the viscous stress tensor and the gradient of a solution; if this natural form is retained, then the problem is solvable. Conditions are found for the validity of the indicated fundamental representation of the absolutely continuous component of the limit measure.