Global regularity for a modified critical dissipative quasi-geostrophic equation

By Peter Constantin, Gautam Iyer, Jiahong Wu. 2008

127 citations recorded by the source corpus.

In this paper, we consider the modified quasi-geostrophic equation \begin{gather*} \del_t \theta + (u \cdot \grad) \theta + \kappa \Lambda^\alpha \theta = 0 u = \Lambda^{\alpha - 1} R^{\perp}\theta. \end{gather*} with $\kappa > 0$, $\alpha \in (0,1]$ and $\theta_0 \in \lp{2}(\R^2)$. We remark that the extra $\Lambda^{\alpha - 1}$ is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regularity of strong solutions to this system.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9b499e0b712f1b2714a000948568533628550b13

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

On the differentiability of the solution to an equation with drift and fractional diffusion (2010, background): “• Combined with [17], it implies the result in [6].”
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]).”
On the regularity of a class of generalized quasi-geostrophic equations (2010, methodology): “When α ∈]−1, 0[, it is just the modified quasi-geostrophic equation introduced in [8] by Constantin, Iyer and Wu.”
Finite time singularity for the modified SQG patch equation (2016, methodology): “We start with a definition of the Hölder and Sobolev norms of the boundaries of domains in R2 that will make the notions of C1,γ and H3 patches precise.”
Holder estimates for advection fractional-diffusion equations (2010, methodology): “This result is used in [6] to obtain the well posedness of a modified supercitical quasi-geostrophic equation.”