Some recent results on the critical surface quasi-geostrophic equation: a review

Abstract

We review some recent results on the dissipative surface quasi-geostrophic equation, focusing on the critical case. We provide some background results and prove global existence of regular solutions.

Global regularity of logarithmically supercritical MHD system with improved logarithmic powers

“…Such a symmetric property has played a crucial role in various other fluid equations as well, e.g. in the process of Galerkin approximation for the surface quasigeostrophic equation in [6]. Here, the last equality used the obvious fact that w j w k = w k w j ; as clear as this observation is, it also allows one to immediately realize that an analogous identity is impossible for (b • ∇)u or (u • ∇)b within the MHD system (2a)-(2c) because in general u k b j = b k u j , and the major obstacle in extending the approach of [2] to the MHD system was that this symmetric property was in fact used in many parts of the proof of [2] (see Remark 3.1, (47), (48), (60), (61), (62)).…”

Dissipation Length Scale Estimates for Turbulent Flows: A Wiener Algebra Approach

“…For more results on the subcritical QG, see for instance [5], where analyticity is established for arbitrary initial data in H 2 , or [11], where a local smoothing effect is exploited to establish analyticity, or [2], where analytic Gevrey regularity is established for several other equations as well. For results on the analyticity of solutions for critical QG equations, see [10] and [24]. For results on the regularity of passive scalar equations see [34] or [35].…”

Bifurcating steady-state solutions of the dissipative quasi-geostrophic equation in Lagrangian formulation

“…When 0 < α < 1 2 (the supercritical case) , the dissipation is too weak to ensure the global existence of large regular solutions. Therefore for the efforts on finite blow up examinations, we refer to Kiselev [19,20] and references therein. However, the global existence is obtainable if additional conditions are applied (see, for example, Constantin and Wu [12] and Dong and Pavlovic [15]).…”