[Paper Review] Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

Claude Bardos, Jasmine S. Linshiz+1 more|ArXiv.org|Feb 19, 2009

Navier-Stokes equation solutions65 references5 citations

TL;DR

This paper introduces a regularized Birkhoff-Rott-α model derived from the 2D Euler-α equations to study vortex sheet dynamics. It proves global existence and regularity of solutions for initially Lipschitz, Hölder, or higher-order Hölder continuous vortex sheets, and establishes convergence of Euler-α solutions to weak solutions of the original Euler equations for initial vorticity as a signed Radon measure.

ABSTRACT

We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-alpha equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-alpha equation, remains Lipschitz for all times, (ii) an initially Holder C^{1,beta}, 0 <= beta < 1, chord arc curve remains in C^{1,beta} for all times, and finally, (iii) an initially Holder C^{n,beta}, n <= 1, 0 < beta < 1, closed chord arc curve remains so for all times. In all these cases the weak Euler-alpha and the BR-alpha descriptions of the vortex sheet motion are equivalent.

Motivation & Objective

• To address the ill-posedness of the classical Birkhoff-Rott equation for vortex sheets due to Kelvin-Helmholtz instability.
• To establish global regularity and long-time existence for vortex sheet evolution under a regularization that preserves the physical structure of the dynamics.
• To prove convergence of solutions of the Euler-α equations to weak solutions of the original 2D Euler equations when initial vorticity is a signed Radon measure.
• To show equivalence between the weak Euler-α and BR-α descriptions under various initial regularity conditions on the vortex sheet.

Proposed method

• Introduces a Birkhoff-Rott-α (BR-α) equation derived from the 2D Euler-α equations, which provides a non-local, regularized velocity field via a smoothed kernel.
• Uses the Euler-α model as a viscous-like regularization that preserves the incompressibility and conservation laws of the original Euler system.
• Applies the theory of weak solutions and Radon measures to handle initial vorticity concentrated on a curve with fixed sign.
• Employs Hölder and Lipschitz regularity estimates on the vortex sheet curve to control the growth of curvature and derivatives.
• Applies the Lebesgue dominated convergence theorem and asymptotic analysis of the regularized kernel to control singular integrals.
• Establishes equivalence between the BR-α and weak Euler-α formulations under initial regularity assumptions via energy and derivative estimates.

Experimental results

Research questions

• RQ1 Can the Birkhoff-Rott equation for vortex sheets be regularized to ensure global existence and smooth evolution under physically relevant initial conditions?
• RQ2 Does the solution of the Euler-α equations converge to a weak solution of the original 2D Euler equations when initial vorticity is a signed Radon measure?
• RQ3 Under what regularity assumptions on the initial vortex sheet (e.g., Lipschitz, C^{1,β}, C^{n,β}) does the BR-α solution remain uniformly regular for all time?
• RQ4 Are the weak Euler-α and BR-α formulations equivalent for initial data with sufficient regularity and fixed-sign vorticity?
• RQ5 What is the threshold regularity that prevents singularity formation in vortex sheet evolution under the BR-α model?

Key findings

• For initial vorticity as a finite Radon measure of fixed sign, solutions of the Euler-α equations converge to a weak solution of the 2D Euler equations as α → 0.
• If the initial vortex sheet is Lipschitz and chord-arc, the BR-α solution remains Lipschitz for all time, with uniform bounds on the Lipschitz constant.
• If the initial vortex sheet is in C^{1,β} for 0 ≤ β < 1 and chord-arc, the BR-α solution remains in C^{1,β} for all time, with Hölder norm controlled uniformly.
• If the initial vortex sheet is in C^{n,β} for n ≥ 1 and 0 < β < 1, and closed and chord-arc, the BR-α solution remains in C^{n,β} for all time.
• Under the above regularity conditions, the weak Euler-α and BR-α descriptions of vortex sheet motion are equivalent.
• The regularization prevents curvature blow-up and avoids the finite-time singularity formation seen in the classical Birkhoff-Rott equation.