Generalized surface quasi‐geostrophic equations with singular velocities

By Dongho Chae, Peter Constantin, D. C'ordoba, Francisco Gancedo, Jiahong Wu. 2011

216 citations recorded by the source corpus.

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity field u related to the scalar θ by $u=\nabla^\perp\Lambda^{\beta-2}\theta$, where $1<\beta\le 2$ and $\Lambda=(-\Delta)^{1/2}$ is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch‐type solutions. The second family is a dissipative active scalar equation with $u=\nabla^\perp (\log(I-\Delta))^\mu\theta\ {\rm for}\ \mu>0$, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani. © 2012 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e98377281a260f93e615d7288119f1d6214df7e4

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Citations in the corpus

Finite time singularity for the modified SQG patch equation (2016, background): “) Local existence of such contour solutions in the more singular case α ∈ (12 , 1] was obtained by Chae, Constantin, Córdoba, Gancedo, and Wu [6].”
On Kato–Ponce and fractional Leibniz (2016, background): “We then write (this is the elegant trick used in [6]) ∫ (∇θ · Dγ−1∇⊥Dsθ)Dsθdx = −(1) 2 ∫ Dsθ(Dγ−1∇⊥ · (Dθ∇θ) − ∇θ · Dγ−1∇⊥Dsθ)dx.”
On the Loss of Continuity for Super-Critical Drift-Diffusion Equations (2012, background): “In fact, blow up in finite time does not seem to be known to hold not just for SQG, but for most (if not all) of the supercritical active scalar equations with divergence-free drift currently in the literature (see, for example [7, 8, 10, 22]).”
Global Weak Solutions for SQG in Bounded Domains (2016, methodology): “/ D 0: Next, we adapt the well-known commutator representation of the nonlinearity in SQG ([12], see also [2,3]) to take into account the lack of translation invariance of ƒ: LEMMA 3.”