Evidence of singularities for a family of contour dynamics equations

Abstract

In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter 0 < ␣ < 1. The limiting case ␣ 3 0 corresponds to 2D Euler equations, and ␣ ‫؍‬ 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.alpha-patches ͉ quasi-geostrophic equation ͉ blow-up ͉ Euler equations ͉ self-similar behavior O ne of the most important open problems in mathematical fluid dynamics is whether the solutions to the Euler and Navier-Stokes equations modeling the evolution of incompressible inviscid and viscous fluids, respectively, may develop singularities in finite time. Several possible scenarios for a singularity have been proposed in the past (see ref. 1 for an account of some of them), although none of them led to a rigorous proof of the formation of singularities. One of these scenarios corresponds to the so-called vortex patch problem that we briefly describe below.A vortex patch consist of a 2D region D(t) (simply connected and bounded) that evolves with a velocity given at each instant of time bywhere the stream function is such that ϭ Ϫ⌬ , and the vorticity has a constant value 0 over D(t). Vortex patches are, therefore, weak solutions of Euler equations. The appearance of finite time singularities at the contour of D(t) was the subject of strong debate based on numerical results showing that the curvature of the boundary might grow superexponentially in time (see refs. 2-4). Nevertheless, work of Chemin (5) and Bertozzi and Constantin (6) rigorously proved global existence of regular solutions for the dynamics of the vortex patch, and, at least in this case, singularities cannot appear.A very natural singularity scenario would correspond to a vortex patch of the so-called surface quasi-geostrophic equation (see ref. 7) for which the relation between the stream function and potential temperature (that play the role that vorticity plays in 2D Euler equations) is ϭ (Ϫ⌬) 1/2 . The interest of this equation lies in its strong analogies to the 3D Euler equation as it has been argued in ref. 8 and its physical relevance as a model for the formation of temperature fronts in some geophysical contexts (see refs. 9 and 10).Hence, a natural question to ask is whether models for which ϭ (Ϫ⌬) 1Ϫ(␣/2) representing an interpolation between 2D Euler and quasi-geostrophic patches develop singularities for 0 Ͻ ␣ Յ 1. In this work, we provide numerical evidence showing that this scenario is indeed the case and describe the self-similar structure of such singularities. This result represents a previously undescribed singularity scenario for incompressible flows. The Model Following Zabusky et al. (4), we can invert the relation betweenand to obtain the following formula for the velocity with 0where x ជ(␥, t) is the position of points over C(t), the boundary of D(t), parameterized with ␥, and 0 ϭ ⅐c ␣ . Here is the value of in the patch and the factor c ␣ ϭ ⌫(␣/2)͞2 1Ϫ␣ ⌫(2...

Numerical Simulation of a Self-Similar Cascade of Filament Instabilities in the Surface Quasigeostrophic System

“…This is in contrast to the case of the corner singularity of Ref. [16], which is intrinsic to the case of the temperature patch and which has no counterpart in the case of an initially smooth θ distribution. The present results suggest, therefore, that the evolution of the initially smooth θ distribution may also develop a singularity in finite time, an open problem of considerable theoretical interest.…”

“…The evolution as the singularity is approached is clarified by introducing a rescaled time variable [16] τ ¼ − logðt s − tÞ. The repeated stages of rapid growth (instability development) followed by a saturation stage (filament extension prior to the onset of the next instability) occur multiple times, with the filament scale shrinking by a factor of around 20 each time; see Fig.…”

“…[16], involving the formation of a corner. While mathematically singular, it is important to recognize that this type of singularity has no counterpart in the evolution of the continuous θ distribution of most relevance to fluid motion: a continuous distribution may develop a corner in a level set of θ without any accompanying blowup of the temperature gradient.…”

Local Regularity for the Modified SQG Patch Equation

“…A computational study of the SQG and modified SQG patches by Córdoba, Fontelos, Mancho, and Rodrigo [7] (where the patch problem for the modified SQG equation first appeared) suggested finite time singularity formation, with two patches touching each other and simultaneously developing corners at the point of touch. A more careful numerical study by Mancho [20] suggests self-similar elements involved in this singularity formation process, but its rigorous confirmation and understanding is still lacking.…”

No-splash theorems for fluid interfaces

“…Numerical simulations (8) suggest that an assymptotically self-similar singularity arises in finite time; however, no mathematical rigorous proof of this is known.…”