On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion

Xinwei Yu,Zhichun Zhai

doi:10.3934/cpaa.2012.11.1809

Abstract

In this article we study local and global well-posedness of the Lagrangian Averaged Euler equations. We show local well-posedness in Triebel-Lizorkin spaces and further prove a Beale-Kato-Majda type necessary and sufficient condition for global existence involving the stream function. We also establish new sufficient conditions for global existence in terms of mixed Lebesgue norms of the generalized Clebsch variables.

Highlights

In [12, 13], Holm, Marsden and Ratiu introduced the 3D Lagrangian averaged Euler equations as follows:∂tu +u + (∇uα)T · u = −∇p, divu = 0. (1)Here the j−th component of ∇v · u is (∇v · u)j = 3 k=1∂j vkuk, and the relation between the velocity u and the averaged velocity uα is given by uα = (1 − α2 )−1u. (2)It is easy to see that when α = 0 (1) reduces to the 3D incompressible Euler equations.Similar to the 3D Euler equations, (1) enjoys a “vorticity formulation” after taking curl of both sides and denoting ω = ∇ × u:∂tω +ω = ∇uα · ω, ω(0) = ω0. (3)
A novel approach to the global well-posedness problem is pioneered by Hou and Li in [15], with inspiration from the classical Clebsch representation of vorticity. They show that, if the initial vorticity ω can be written in terms of two level set functions as follows, ω(0, x) = ω0(φ0, ψ0)∇φ0 × ∇ψ0, this representation remains true for later times, ω(t, x) = ω0(φ, ψ)∇φ × ∇ψ, as long as the level set functions φ, ψ evolve according to φt +φ = 0, φ(0, x) = φ0(x), ψt +ψ = 0, ψ(0, x) = ψ0(x)
As pointed out in Hou-Li [15], the above Clebsch variables/level set formulation is a direct consequence of the Lagrangian structure of the flow and applies to the 3D Lagrangian averaged Euler equations

Summary

Introduction

In [12, 13], Holm, Marsden and Ratiu introduced the 3D Lagrangian averaged Euler equations as follows:. ∂j vkuk, and the relation between the velocity u and the averaged velocity uα is given by uα = (1 − α2 )−1u. Similar to the 3D Euler equations, (1) enjoys a “vorticity formulation” after taking curl of both sides and denoting ω = ∇ × u:. ∂tω + (uα · ∇)ω = ∇uα · ω, ω(0) = ω0. Note that (3) has the same form as the vorticity formulation for the 3D Euler equations, except that the transporting velocity u has been replaced by the “averaged” velocity uα = (1 − α2 )−1u. For convenience of the readers, we summarize the relations between u, ω, the averaged velocity uα and the stream function ψ:. Project supported in part by Natural Science and Engineering Research Council of Canada

XINWEI YU AND ZHICHUN ZHAI

We observe that

It now follows that when

The bound on