On Singular Vortex Patches, I : Well-Posedness Issues / Tarek M. Elgindi and In-Jee Jeong.

Format:

Book

Author/Creator:

Elgindi, Tarek M., author.

Jeong, In-Jee, 1990- author.

Series:

Memoirs of the American Mathematical Society ; Volume 283.

Memoirs of the American Mathematical Society Series ; Volume 283

Language:

English

Subjects (All):

Differential equations.

Dynamics.

Fluid mechanics.

Physical Description:

1 online resource (102 pages)

Edition:

First edition.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2023]

Summary:

"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 [greater than or equal to] 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 [pi symbol]/2 for all time. Even in the case of vortex patches with corners of angle [pi symbol]/2 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 R[superscript]2 with interesting dynamical behavior such as cusping and spiral formation in infinite time"-- Provided by publisher.

Contents:

Background material

Global well-posedness for symmetic patches in an intermediate space

Global well-posedness for symmetric C1, a-patches with corners

Ill-posedness results for vortex patches with corners

Effective system for the boundary evolution near the corner.

Notes:

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Elgindi, Tarek M. On Singular Vortex Patches, I: Well-Posedness Issues

ISBN:

9781470474010