Decay rates for mild solutions of QGE with critical fractional dissipation in L 2 (R 2 )

Source abstract

Abstract We investigate the long-time behavior of global mild solutions to the two-dimensional critical quasi-geostrophic equation with critical fractional dissipation in Sobolev–Gevrey spaces. A recent result by Melo, Rocha and dos Santos Costa establishes the decay of Gevrey–Sobolev norms under the additional assumption that the initial data possess a decay character r *( θ 0 ) ∈ (−1, ∞ ). In the present paper, we show that this assumption is, in fact, unnecessary. Our approach is based on a new analysis of the L 2 -asymptotic behavior of global solutions. Instead of relying on the decay characterization of the associated linear semigroup, we prove directly that every global solution satisfies lim sup t → ∞ ‖ θ ( t ) ‖ L 2 = 0 ${\mathrm{lim sup}}_{t\to \infty }{\Vert}\theta \left(t\right){{\Vert}}_{{L}^{2}}=0$ , without assuming the existence of the decay character of the initial data. The proof combines Duhamel’s representation formula, refined estimates for the nonlinear remainder, interpolation arguments in homogeneous Sobolev spaces, and the energy inequality, together with a continuation argument based on the uniqueness of global solutions with small initial data.