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Abstract An anelastic numerical model is used to study the influences of fine structure (FS) in the wind and stability profiles on gravity wave (GW) propagation in the Mesosphere and Lower Thermosphere (MLT). Large amplitude GWs interacting with FS, that is, thin regions of enhanced wind and stability, evolve very differently depending on the precise vorticity source and sink terms for small‐scale motions induced by the FS gradients. The resulting small‐scale dynamics are deterministic, promoting local instabilities, dissipation, and momentum deposition at locations and orientations determined by the initial FS. The resulting momentum depositions yield significant changes to the background wind structure, having scales and amplitudes comparable to the effects of large‐scale features in the ambient atmosphere. The deterministic nature of the large‐scale impacts further suggests that they can be estimated without fully resolving the underlying instability dynamics. Given the significant amplitudes and ubiquitous occurrence of FS throughout the atmosphere, the influences of these important and diverse flow evolutions merit inclusion in broader modeling efforts.
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Degenerate stability of the Caffarelli–Kohn–Nirenberg inequality along the Felli–Schneider curve
Abstract We show that the Caffarelli–Kohn–Nirenberg (CKN) inequality holds with a remainder term that is quartic in the distance to the set of optimizers for the full parameter range of the Felli–Schneider (FS) curve. The fourth power is best possible. This is due to the presence of non-trivial zero modes of the Hessian of the deficit functional along the FS-curve. Following an iterated Bianchi–Egnell strategy, the heart of our proof is verifying a ‘secondary non-degeneracy condition’. Our result completes the stability analysis for the CKN-inequality to leading order started by Wei and Wu. Moreover, it is the first instance of degenerate stability for non-constant optimizers and for a non-compact domain.
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- Award ID(s):
- 1954995
- PAR ID:
- 10487914
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Calculus of Variations and Partial Differential Equations
- Volume:
- 63
- Issue:
- 2
- ISSN:
- 0944-2669
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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