Global solvability and convergence of the Euler-Poincaré regularization of the two-dimensional Euler equations

Abstract

We study the Euler-Poincaré equations that are the regularized Euler equations derived from the Euler-Poincaré framework. It is noteworthy to remark that the Euler-Poincaré equations are a generalization of two well-known regularizations, the vortex blob method and the Euler-α equations. We show the global existence of a unique weak solution for the two-dimensional (2D) Euler-Poincaré equations with the initial vorticity in the space of Radon measure. This is a remarkable feature of these equations since the existence of weak solutions with the Radon measure initial vorticity has not been established in general for the 2D Euler equations. We also show that weak solutions of the 2D Euler-Poincaré equations converge to those of the 2D Euler equations in the limit of the regularization parameter when the initial vorticity belongs to the space of integrable and bounded functions.

Universality of the Anomalous Enstrophy Dissipation at the Collapse of Three Point Vortices on Euler--Poincaré Models

“…We first investigate the properties of K ε . As shown in [8], under the assumptions of Theorem 2.1, K ε belongs to C 0 (R 2 ) and it is quasi-Lipschitz continuous with K ε (0) = 0. It is also important to remark that K ε is defined by…”

“…where u ε = K * ω ε owing to the Biot-Savart formula. It has been shown in [8] that the initial value problem of (2.4) has a unique global weak solution in the space of Radon measures M(R 2 ) on R 2 , in which the following equation for the Lagrangian flow map η ε associated with the regular velocity is considered.…”

“…The assumptions of Theorem 2.1 are satisfied with two well-known regularization of the Euler equations; the Euler-α equations for h(x) = K 0 (|x|)/(2π) and the vortex blob method for h(x) = 1/(π(|x| 2 + 1) −2 ). See [8,13].…”

“…And the EP system is formally equivalent to the Euler equations when ε is exactly zero. The existence of the unique global solutions to the Euler-Poincaré equations for initial vorticity distributions in M(R 2 ) has been established in [8] as to the Euler-α equations. Furthermore, since the Euler-Poincaré equations share common mathematical structures with the Euler-α equations, the evolution of the weak solution can be investigated in terms of the dynamics of ε-point vortices, which is introduced in this paper.…”

Singular solutions of nonlinear hydrodynamic equations arising in turbulence theory

“…Gotoda [38] has shown the existence of a unique global weak solution of (2.4) for initial vorticity distributions belonging to the space of Radon measures M(R 2 ). Suppose that the regularization function h ∈ C 1 (R 2 \ {0}) vanishes as |x| → ∞, and it satisfies the following conditions.…”