On the approximation of vorticity fronts by the Burgers–Hilbert equation

Article type: Research Article

Authors: Hunter, John K.^{a; *} | Moreno-Vasquez, Ryan C.^a | Shu, Jingyang^b | Zhang, Qingtian^c

Affiliations: [a] Department of Mathematics, University of California at Davis, CA, USA. E-mails: [email protected], [email protected] | [b] Department of Mathematics, Temple University, PA, USA. E-mail: [email protected] | [c] Department of Mathematics, West Virginia University, WV, USA. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

Keywords: Incompressible Euler equations, vorticity fronts, asymptotic equations, modified energy

DOI: 10.3233/ASY-211724

Journal: Asymptotic Analysis, vol. 129, no. 2, pp. 141-177, 2022