Existence and Uniqueness of the Solution to the Dissipative 2D Quasi-Geostrophic Equations in the Sobolev Space

A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation

“…Recently, Chae and Lee [5] studied the super-critical case and proved the global well-posedness for small data in the Besov spacesḂ 2−2α 2,1 . Very recently, Cordoba-Cordoba [13], Ning [17,18] studied the existence and uniqueness in the Sobolev spaces H s , s ≥ 2 − 2α, α ∈ [0, 1 2 ]. Wu [24,25] studied the well-posedness in general Besov space B s p,q , s > 2(1 − α), p = 2 N .…”

“…In fact, if θ(t, x) and u(t, x) are solutions of (1.1), then θ λ (t, x) = λ 2α−1 θ(λ 2α t, λx) and u λ (t, x) = λ 2α−1 u(λ 2α t, λx) are also solutions of (1.1 [5] and [17]. Remark 1.3 Wu [24,25] proved the well-posedness of (1.1) …”

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

“…The critical dissipative quasi-geostrophic equations is an interesting model for investigating existence issues on genuine 3D Navier-Stokes equations. Recently, this equation has been studied intensively, see [4], [5], [6], [7], [13], [14], [8], [16], [17], [18], [19], [20], [21] and references therein.…”

“…In the critical case, Constantin, Córdoba and Wu [5] gave a construction of global regular solutions for initial data in H 1 under a smallness assumption of L ∞ norm of the data. For other results about local well-posedness and small-data global well-posedness in various function spaces, see also Chae and Lee [3], Ju [13], [15], Miura [16] and reference therein. For the issue of global regularity with large initial data, breakthrough only occurred recently.…”

On the uniqueness for sub-critical quasi-geostrophic equations

“…[2,5,6,7,8,9,10,12,14] and their references. Concerning the uniqueness of LerayHopf weak solutions, some results were obtained in [6,8,9] by assuming further LerayProdi-Serrin or integral-regularity type restrictions. Later on, in a remarkable paper, the authors of [2] show that Leray-Hopf weak solutions with arbitrary data θ 0 ∈ L 2 (R 2 ) and 1/2 ≤ γ ≤ 1 become smooth for any t > 0, and as a consequence those solutions are unique.…”