On singular vortex patches, I well-posedness issues / Tarek M. Elgindi, In-Jee Jeong.
| Author/creator | Elgindi, Tarek M. |
| Other author | Jeong, In-Jee, 1990- |
| Format | Electronic |
| Publication Info | Providence, Rhode Island : American Mathematical Society, 2023. |
| Description | v, 89 pages : illustrations ; 26 cm. |
| Supplemental Content | Full text available from Ebook Central - Academic Complete |
| Subjects |
| Series | Memoirs of the American Mathematical Society, 0065-9266 ; Volume 283, Number 1400 |
| 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. |
| 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 [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. |
| Bibliography note | Includes bibliographical references (pages 85-89). |
| Access restriction | Available only to authorized users. |
| Technical details | Mode of access: World Wide Web |
| Genre/form | Electronic books. |
| LCCN | 2023012960 |
| ISBN | 9781470456825 (paperback) |
| ISBN | (pdf) |
Availability
| Library | Location | Call Number | Status | Item Actions |
|---|---|---|---|---|
| Electronic Resources | Access Content Online | ✔ Available |