A Posteriori Modeling Error Estimates for the Assumption of Perfect Incompressibility in the Navier--Stokes Equation

Abstract

We derive a posteriori estimates for the modeling error caused by the assumption of perfect incompressibility in the incompressible Navier--Stokes equation: Real fluids are never perfectly incompressible but always feature at least some low amount of compressibility. Thus, their behavior is described by the compressible Navier--Stokes equation, the pressure being a steep function of the density. We rigorously estimate the difference between an approximate solution to the incompressible Navier--Stokes equation and any weak solution to the compressible Navier--Stokes equation in the sense of Lions (without assuming any additional regularity of solutions). Heuristics and numerical results suggest that our error estimates are of optimal order in the case of “well-behaved” flows and divergence-free approximations of the velocity field. Thus, we expect our estimates to justify the idealization of fluids as perfectly incompressible also in practical situations.