Global well-posedness of critical surface quasigeostrophic equation on the sphere

Abstract

In this paper we prove global well-posedness of the critical surface quasigeostrophic equation on the two dimensional sphere building on some earlier work of the authors. The proof relies on an improving of the previously known pointwise inequality for fractional laplacians as in the work of Constantin and Vicol for the euclidean setting.

On the degenerate boussinesq equations on surfaces

“…In passing, we remark that despite the lack of literature on the analysis of Boussinesq equations on manifolds, various PDEs of hydrodynamic models have nevertheless been studied on manifolds, including the Navier-Stokes equations [40], the rotating Euler equations [55], and the SQG (surface quasi-geostrophic) equations [6,7], even for critical cases without the smallness assumption on the initial data.…”

On the blow up of a non-local transport equation in compact manifolds

“…Later, the global regularity in the critical case was shown in [16] for arbitrary initial data in Sobolev spaces by adapting the method of continuity in [24]. In [27], finite time singularities of smooth solutions was shown in the range 0 < γ < 1 2 . For the intermediate case, 1 2 ≤ γ < 1, whether solutions may blow up in finite time is an intriguing open problem.…”

Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators

“…For example, in [3,4], A. Córdoba and D. Córdoba studied regularity and -decay for solutions. In [5] the well-posedness of quasigeostrophic equation was treated on the sphere, on general riemannian manifolds in [6] or the 2D stochastic quasigeostrophic equation on the torus T 2 in [7]. This equation is also denominated as advection-fractional diffusion; see for example [8], or it may be classified as a fractional Fokker-Planck equation [9].…”

“…This equation is also denominated as advection-fractional diffusion; see for example [8], or it may be classified as a fractional Fokker-Planck equation [9]. However we follow the usual terminology of quasigeostrophic equation which has appeared in our main references [1][2][3][4][5][6][7].…”