From second grade fluids to the Navier-Stokes equations

TL;DR

This paper investigates the convergence of second grade fluid models to the Navier-Stokes equations as a parameter approaches zero, providing conditions for both global and local-in-time convergence based on initial data assumptions.

Contribution

It establishes the conditions under which second grade fluids converge to Navier-Stokes solutions, including smallness criteria for initial velocities.

Findings

Global-in-time convergence under small initial data product

Local-in-time convergence without smallness assumption

Convergence results depend on initial velocity norms

Abstract

We consider the limit $\alpha\to0$ for a second grade fluid on a bounded domain with Dirichlet boundary conditions. We show convergence towards a solution of the Navier-Stokes equations under two different types of hypothesis on the initial velocity $u_0$. If the product $\|u_0\|_{L^2}\|u_0\|_{H^1}$ is sufficiently small we prove global-in-time convergence. If there is no smallness assumption we obtain local-in-time convergence up to the time $C/\|u_0\|_{H^1}^4$.