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 m -fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as m ≥ 3. m\geq 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 π 2 \frac {\pi }{2} for all time. Even in the case of vortex patches with corners of angle π 2 \frac {\pi }{2} or with corners which are only locally m 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 2 \mathbb {R}^2 with interesting dynamical behavior such as cusping and spiral formation in infinite time.
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Joint inversion of compact operators
Abstract Joint inversion of multiple data types was studied as early as 1975 in [K. Vozoff and D. L. Jupp,Joint inversion of geophysical data,Geophys. J. Internat. 42 1975, 3, 977–991],where the authors used the singular value decomposition to determine the degree of ill-conditioning of joint inverse problems. The authors demonstrated in several examples that combining two physical models in a joint inversion, and by effectively stacking discrete linear models, improved the conditioning as compared to individual inversions. This work extends the notion of using the singular value decomposition to determine the conditioning of discrete joint inversion to using the singular value expansion to determine the well-posedness of joint operators. We provide a convergent technique for approximating the singular values of continuous joint operators. In the case of self-adjoint operators, we give an algebraic expression for the joint singular values in terms of the singular values of the individual operators. This expression allows us to show that while rare, there are situations where ill-posedness may be not improved through joint inversion and in fact can degrade the conditioning of an individual inversion. The expression also quantifies the benefits of including repeated measurements in an inversion. We give an example of joint inversion with two moderately ill-posed Green’s function solutions, and quantify the improvement over individual inversions. This work provides a framework in which to identify data types that are advantageous to combine in a joint inversion.
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- Award ID(s):
- 1720472
- NSF-PAR ID:
- 10188169
- Date Published:
- Journal Name:
- Journal of Inverse and Ill-posed Problems
- Volume:
- 28
- Issue:
- 1
- ISSN:
- 0928-0219
- Page Range / eLocation ID:
- 105 to 118
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/jiip-2019-0068
