Uniform time of existence for the alpha Euler equations

Abstract

Abstract. We consider the α-Euler equations on a bounded three-dimensional domain with frictionless Navier boundary conditions. Our main result is the existence of a strong solution on a positive time interval, uniform in α, for α sufficiently small. Combined with the convergence result in [4], this implies convergence of solutions of the α-Euler equations to solutions of the incompressible Euler equations when α → 0. In addition, we obtain a new result on local existence of strong solutions for the incompressible Euler equations on bounded three-dimensional domains. The proofs are based on new a priori estimates in conormal spaces.

Global well-posedness and long-time dynamics for a higher order quasi-geostrophic type equation

“…Hence, we can insert this bound into (50) and integrate the resulting expression in time: after noticing that R(t) ˆt 0 e V(τ ) ν(τ ) 6 (1 + ν γ ) 2 τ 2 dτ + C ˆt 0 e V(τ ) f (τ ) H −1 dτ .…”

Weak solutions for theα-Euler equations and convergence to Euler

“…We refer to the recent book [8] for an extensive discussion of various aspects of the second grade fluids. 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].…”

From second grade fluids to the Navier–Stokes equations

“…We also proved the convergence in 3D but under the additional hypothesis that the solutions exist on a time interval independent of α and ν. We proved in [4] that the hypothesis of existence of a uniform time existence is verified if ν = 0 and α → 0. In the case of the Dirichlet boundary conditions, there is only the paper [15] which shows convergence in 2D.…”

“…For such a solution to exist, the initial data must verify a compatibility condition. Indeed, one can apply the Leray projector to (4) to obtain that ∂ t u − νP∆u + P(u · ∇u) = 0. Since u vanishes at the boundary, so does ∂ t u.…”