Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space

Abstract

Abstract. We study the critical dissipative quasi-geostrophic equations in R 2 with arbitrary H 1 initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable modification. A decay in time estimate for higher order homogeneous Sobolev norms of solutions is also discussed.

Long Time Behavior of the Solution of the Two-Dimensional Dissipative QGE in Lei–Lin Spaces

“…e existence of a global weak solution was established by several researchers. e reader is referred to [3][4][5][6][7] and their references. Furthermore, in the subcritical case, Constantin and Wu [8] proved that every sufficiently smooth initial data give rise to a unique global smooth solution.…”

Global well‐posedness and blow‐up criterion for the periodic quasi‐geostrophic equations in Lei‐Lin‐Gevrey spaces

“…The global well‐posedness for large data in critical spaces was obtained by many authors. The reader is referred to several studies() and the references therein. For the critical case α =1/2, Caffarelli and Vasseur established the global existence of smooth solutions using De Giorgi method, and Kiselev‐Nazarov‐Volberg adressed it in a periodic case.…”

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

“…It follows from (1) and (15) that for every small > 0 and every r > 0 we have uniform in N and t ∈ [ , T ] bounds (17) θ N t r ≤ C(r, ). By (15) and (17) and the well known compactness criteria (see e.g. [7], Chapter 8), we can find a subsequence θ N j converging in C([ , T ], H r ) to some function θ.…”

“…Since and r are arbitrary one can apply the standard subsequence of subsequence procedure to find a subsequence (still denoted by θ N j ) which converges to θ in C((0, T ], H r ), for any r > 0. The limiting function θ must satisfy the estimates (15) and it is straightforward to check that it solves the SQG equation on (0, T ]. Thus, it remains to show that θ can be made to converge to θ 0 strongly in H s as t → 0.…”

“…Finally, in two independent works [22] and [1], it was proved that global smooth solutions exist for large initial data without additional assumptions of any kind. The paper [22] works in periodic setting and shows existence of smooth solutions for smooth initial data (the recent work [15] extended approach of [22] the whole space setting). The method of [22] is based on an elementary new idea: a nonlocal maximum principle.…”