This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar
By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as
- Award ID(s):
- 2054686
- NSF-PAR ID:
- 10327289
- Date Published:
- Journal Name:
- Discrete & Continuous Dynamical Systems
- Volume:
- 42
- Issue:
- 5
- ISSN:
- 1078-0947
- Page Range / eLocation ID:
- 2199
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity\begin{document}$ \theta $\end{document} is of lower singularity, i.e.,\begin{document}$ u $\end{document} , where\begin{document}$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $\end{document} is a logarithmic smoothing operator and\begin{document}$ p $\end{document} . We complete this study by considering the more singular regime\begin{document}$ \beta \in [0, 1] $\end{document} . The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.\begin{document}$ \beta\in(1, 2) $\end{document} -
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, focusing on the real rank one case, i.e., equipped with a diagram involution. We construct explicitly a relative braid group action of type on the affine quantum group . Real and imaginary root vectors for are constructed, and a Drinfeld type presentation of is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine quantum groups in the sequels. -
Abstract Let us fix a prime
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