Early-time resonances in the three-dimensional wall-bounded axisymmetric Euler and related equations

Abstract

We investigate the complex-time analytic structure of solutions of the three-dimensional (3D)-axisymmetric, wall-bounded, incompressible Euler equations, by starting with the initial data proposed in Luo and Hou [“Potentially singular solutions of the 3D axisymmetric Euler equations,” Proc. Natl. Acad. Sci. U. S. A. 111(36), 12968–12973 (2014)], to study a possible finite-time singularity. We use our pseudospectral Fourier–Chebyshev method [Kolluru et al., “Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,” Phys. Rev. E 105(6), 065107 (2022)], with quadruple-precision arithmetic, to compute the time-Taylor-series coefficients of the flow fields, up to a high order. We show that the resulting approximations display early-time resonances; the initial spatial location of these structures is different from that for the tygers, which we have obtained in Kolluru et al. [“Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,” Phys. Rev. E 105(6), 065107 (2022)]. We then perform asymptotic analysis of the Taylor-series coefficients, by using generalized ratio methods, to extract the location and nature of the convergence-limiting singularities and demonstrate that these singularities are distributed around the origin, in the complex-t2 plane, along two curves that resemble the shape of an eye. We obtain similar results for the one-dimensional (1D) wall-approximation (of the full 3D-axisymmetric Euler equation) called the 1D Hou–Luo model, for which we use Fourier-pseudospectral methods to compute the time-Taylor-series coefficients of the flow fields. Our work examines the link between tygers, in Galerkin-truncated pseudospectral studies, and early-time resonances, in truncated time-Taylor expansions of solutions of partial differential equations, such as those we consider.