Behavior of Solutions of 2D Quasi-Geostrophic Equations

Abstract

We study solutions to the 2D quasi-geostrophic (QGS) equationand prove global existence and uniqueness of smooth solutions if α ∈ ( 1 2 , 1]; weak solutions also exist globally but are proven to be unique only in the class of strong solutions. Detailed aspects of large time approximation by the linear QGS equation are obtained.

On the Euler equations of incompressible fluids

“…The magnitude |∇ ⊥ θ| obeys the same equation as (12), and the stretching factor α has a principal value representation like (13) [58]). In the critical case, it was known for some time that if the initial data are small in L ∞ they remain small and the solution is regular ( [44], [62]).…”

A pointwise estimate for fractionary derivatives with applications to partial differential equations

“…In particular, it is applied to the quasigeostrophic equation with critical viscosity t ϩ R͑ ͒⅐ٌ Ќ ϭ Ϫ ⌳ , where ٌ Ќ ϭ (ϪѨ ͞Ѩx 2 , Ѩ ͞Ѩx 1 ), R( ) ϭ (R 1 ( ), R 2 ( )), and R j denotes the j th -Riesz transform in R 2 (see refs. [1][2][3][4][5][6].…”

On Existence and Space-Time Behavior of Dissipative 2D Quasi-Geostrophic Flows

“…Most of those works study mainly asymptotic behavior of Y in various function spaces by means of the modern techniques in real analysis. Among them, the subcritical case, i.e., the case where 1=2 < a a 1, has been most investigated; see, e.g., [5], [13], [16] and references cited therein. In particular, Carrillo and Ferreira [4] for some 2=g 0 < p a y, where and throughout this paper we set g 0 ¼ 2a À 1.…”