The paper investigates a parabolic approximation of a linear unsteady Navier–Stokes problem for an incompressible viscous fluid. The proposed approach replaces the incompressibility constraint with an evolutionary pressure equation that includes a small regularization parameter. Such a transformation makes it possible to apply methods from the theory of parabolic differential equations and simplify the analysis of the mathematical model.

The study employs techniques of functional analysis, energy estimates, and the Galerkin method to examine the properties of the approximating system. Conditions ensuring the existence, uniqueness, and regularity of generalized and strong solutions are established. Particular attention is paid to the convergence of the approximating solution to the solution of the original Navier–Stokes system as the approximation parameter tends to zero. Quantitative estimates characterizing the convergence rate in various functional spaces are obtained.

To demonstrate the practical relevance of the proposed approach, a model problem related to transient flow in a pipeline system is considered. The results indicate that the parabolic approximation provides a satisfactory balance between computational efficiency and solution accuracy. The estimates obtained justify the method's applicability to the mathematical modeling of unsteady hydrodynamic processes and may serve as a basis for developing efficient numerical algorithms in computational fluid dynamics.