Abstract In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane.We consider equations of the form Δ u + W ⋅ ∇ u = 0 {\Delta u+W\cdot\nabla u=0} in ℝ 2 {\mathbb{R}^{2}} ,where W = W 1 + i W 2 {W=W_{1}+iW_{2}} with each W j {W_{j}} being real-valued.Under the assumptions that W j ∈ L q j {W_{j}\in L^{q_{j}}} for some q 1 ∈ [ 2 , ∞ ] {q_{1}\in[2,\infty]} , q 2 ∈ ( 2 , ∞ ] {q_{2}\in(2,\infty]} and that W 2 {W_{2}} exhibits rapid decay at infinity,we prove new global unique continuation estimates.This improvement is accomplished by reducing our equations to vector-valued Beltrami systems.Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.
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Improved asymptotic expressions for the eigenvalues of Laplace’s tidal equations
ABSTRACT Laplace’s tidal equations govern the angular dependence of oscillations in stars when uniform rotation is treated within the so-called traditional approximation. Using a perturbation expansion approach, I derive improved expressions for the eigenvalue associated with these equations, valid in the asymptotic limit of large spin parameter q. These expressions have a relative accuracy of order q−3 for gravito-inertial modes, and q−1 for Rossby and Kelvin modes; the corresponding absolute accuracy is of order q−1 for all three mode types. I validate my analysis against numerical calculations, and demonstrate how it can be applied to derive formulae for the periods and eigenfunctions of Rossby modes.
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- NSF-PAR ID:
- 10215542
- Date Published:
- Journal Name:
- Monthly Notices of the Royal Astronomical Society
- Volume:
- 497
- Issue:
- 3
- ISSN:
- 0035-8711
- Page Range / eLocation ID:
- 2670 to 2679
- 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/10.1093/mnras/staa2159
