# On a non-solenoidal approximation to the incompressible Navier-Stokes equations

## TL;DR

This paper analyzes the long-term behavior of a non-solenoidal approximation to the incompressible Navier-Stokes equations, revealing that it diverges significantly from true solutions over time, especially for large t.

## Contribution

It provides a sharp asymptotic profile for the model's solutions, showing that the approximation's decay rate differs from the classical Navier-Stokes equations as time progresses.

## Key findings

- Solutions decay slower than Navier-Stokes solutions for large time
- The model is not a good long-term approximation of Navier-Stokes
- Asymptotic profile characterizes divergence over time

## Abstract

We establish an asymptotic profile that sharply describes the behavior as $t\to\infty$ for solutions to a non-solenoidal approximation of the incompressible Navier-Stokes equations introduced by Temam. The solutions of Temam's model are known to converge to the corresponding solutions of the classical Navier-Stokes, e.g., in $L^3\_{\rm loc} (R^+ \times R^3)$, provided $\epsilon\to0$, where $\epsilon>0$ is the physical parameter related to the artificial compressibility term. However, we show that such model is no longer a good approximation of Navier-Stokes for large times: indeed, its solutions can decay much slower as $t\to+\infty$ than the corresponding solutions of Navier-Stokes.