Optimal convergence rates of the supercritical surface quasi-geostrophic equation. (December 2018) - Jia, Yan
Abstract: This study is concerned with the optimal convergence rates of the supercritical surface quasi-geostrophic equation. When the global weak solution θ of the supercritical surface quasi-geostrophic equation satisfies ∇ θ ∈ L r ( 0, ∞ ; B ̇ p, ∞ 0 ( R 2 ) ) for 2 p + α r = α, 2 α < p < ∞, then even for large initial data perturbation, every weak solution θ ̃ ( x, t ) of the perturbed quasi-geostrophic equation converges to θ ( x, t ) with the optimal algebraic convergence rate ‖ θ ̃ ( t ) − θ ( t ) ‖ L 2 ≤ C ( 1 + t ) − 1 α t > 0 . The findings are mainly based on some new estimates for the trilinear form in Besov spaces and the generalized Fourier splitting method.