Abstract
Córdoba et al. (Ann Math 162(3):1377–1389, 2005) introduced a nonlocal active scalar equation as a one-dimensional analogue of the surface-quasigeostrophic equation. It has been conjectured, based on numerical evidence, that the solution forms a cusp-like singularity in finite time. Up until now, no active scalar with nonlocal flux is known for which cusp formation has been rigorously shown. In this paper, we introduce and study a nonlocal active scalar, inspired by the Córdoba–Córdoba–Fontelos equation, and prove that either a cusp- or needle-like singularity forms in finite time.
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Communicated by V. Šverák
Vu Hoang expresses his gratitude to the German Research Foundation (DFG) for continued support through grants FOR 5156/1-1 and FOR 5156/1-2. Vu Hoang also acknowledges partial support by NSF Grant NSF-DMS 1412023.
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Hoang, V., Radosz, M. Cusp Formation for a Nonlocal Evolution Equation. Arch Rational Mech Anal 224, 1021–1036 (2017). https://doi.org/10.1007/s00205-017-1094-3
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DOI: https://doi.org/10.1007/s00205-017-1094-3