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 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.

Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model

“…It is worth observing that, in view of ( 29), the strong solution additionally satisfies (30); moreover, thanks to the last condition in (27a), both the weak and the strong solution comply with (25).…”

“…3.2 does not depend on ε in the case of the additional assumption (24). The strong convergence (62) preserves this pointwise lower bound, which results in (25). In this case, the sequence {log θ ε } is bounded in L ∞ (0, T ; L q (Ω)) due to the lower bound and (69).…”

“…For thermodynamical systems this idea goes back to Dafermos [10]. In the context of fluid dynamics, the relative energy approach has also been used to show the stability of a stationary solution [17], the convergence to a singular limit [23], or to derive a posteriori estimates for simplified models [25].…”

Analysis of a model for the dynamics of microswimmer suspensions

“…and u n as in (7). Note that the mass matrix in this case is just the identity since the eigenfunctions w i are orthonormal with respect to the L 2 -inner product.…”

Analysis of a model for the dynamics of microswimmer suspensions

“…Such a relative energy inequality can be employed to a variety of uses, such as showing the stability of equilibria (e.g. in Feireisl [2]), deriving a posteriori estimates for modelling errors (see Fischer [3]) or as the basis of a generalised concept of solution (e.g. in Lasarzik [5]).…”