From second grade fluids to the Navier–Stokes equations

Abstract

We consider the limit α → 0 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 4 H 1 .

Weak solutions for theα-Euler equations and convergence to Euler

“…As for the α-Euler equations, we use the notation v = u − α∆u and q = curl v. Let us mention at this point that convergence towards a solution of the Euler equation when α, ν → 0 was proved in the case of the Navier boundary conditions without any condition on the relative sizes of ν and α in dimension two, see [4], and with the condition ν α bounded in dimension three, see [5]. In the case of the Dirichlet boundary conditions, convergence towards a solution of the Navier-Stokes equations when α → 0 and ν > 0 is fixed was proved in [3], see also [18].…”

The limit behavior of the second‐grade fluid equations

“…with no-penetration condition u E •n=0. The second author and his collaborators in Lopes Filho et al 8 proved that the convergence from Euler-α equations to Euler equations (5) with no-slip boundary conditions as α→0. For no-slip boundary conditions, the convergence of second-grade fluid equations to Euler equations is more complicated, as α→0,ν→0, simultaneously.…”

“…Since it is still a challenging problem for the vanishing viscosity limit of Navier-Stokes equations near a solid boundary, although we have obtained the convergence from second-grade fluid equations to Navier-Stokes equations in Busuioc 4 and Busuioc. 5 In Lopes Filho et al, 9 the second author and his collaborators studied the limit behaviors of the solution u (α,ν) to (1), (2) with (3) as both α and ν go to zero, which they can be illustrated according to the following Figure 1. Under suitable initial velocity assumptions, we observe that if the viscosity ν ¼ Oðα 2 Þ, as α→0, as shown in region 4, the solution u (α,ν) converges to the solution u E of Euler equations in L 2 space uniformly in time.…”

“…Let u (ν,α) be the solution of the problem (1), (2), and (3) with initial data satisfies (9), corresponding to ν>0,α>0. Let u E be the solution of Euler system (5) with the same initial data u 0 defined by (9) and the nopenetration boundary condition u E •n ¼ 0:…”

“…The term in can be absorbed by the pressure. The limit behavior as α →0 of second‐grade fluid equations to Navier‐Stokes equations was studied in Busuioc, Busuioc, and Cao and Titi . Second, as ν =0, it is well known that the inviscid second‐grade fluid model are called Euler‐ α equations.…”