Publications [#379798] of Tarek M Elgindi

Papers Published

1. Elgindi, TM; Jeong, IJ, On Singular Vortex Patches, I: Well-Posedness Issues, Memoirs of the American Mathematical Society, vol. 283 no. 1400 (March, 2023), pp. 1-102 [doi]
   (last updated on 2024/11/20)

   Abstract:
   The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally m-fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as m ≥ 3. In this case, all of the angles involved solve a closed ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show that any other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle π2 for all time. Even in the case of vortex patches with corners of angle π2 or with corners which are only locally m-fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on R2 with interesting dynamical behavior such as cusping and spiral formation in infinite time.