Non-uniqueness of the Leray-Hopf solutions in the hyperbolic setting
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Bibliographic record
Abstract
The Leray-Hopf solutions to the Navier-Stokes equation are known to be unique on R 2 . We show the uniqueness of the Leray-Hopf solutions breaks down on H 2 (-a 2 ), the two dimensional hyperbolic space with constant sectional curvature -a 2 . We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation. Finally, as a corollary we also show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions. Contents 1. Introduction 43 Acknowledgements 54 2. Preliminaries 54 3. Exponential decay of the gradient of bounded harmonic functions 62 4. The proof that |F | 2 L 1 (H 2 (-a 2 )) is finite 64 5. Finite Dissipation 69 6. Proofs of the main results 71 References 75
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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