Decay estimates of solution to the two‐dimensional fractional quasi‐geostrophic equation.

This paper studies the fractional quasi‐geostrophic equation with modified dissipation term. We first prove the existence and uniqueness of smooth solution of the fractional quasi‐geostrophic equation by using the regularization method. Then, utilizing the Fourier‐splitting method, we obtain the long‐time behavior of the solution. More precisely, we show that under suitable assumptions on the initial data, for any multi‐index γ,m≥4$$ \gamma, m\ge 4 $$ and α∈12,1$$ \alpha \in \left(\frac{1}{2},1\right) $$, the solution ψ$$ \psi $$ satisfies C1(1+t)−1+|γ|2+2α≤∇γψ(·,t)L2≤C2(1+t)−1+|γ|2+2α,|γ|=0,1,...,m−1,t≥1,$$ {C}_1{\left(1+t\right)}^{-\frac{1+\mid \gamma \mid }{2+2\alpha }}\le {\left\Vert {\nabla}^{\gamma}\psi \left(\cdotp, t\right)\right\Vert}_{L^2}\le {C}_2{\left(1+t\right)}^{-\frac{1+\mid \gamma \mid }{2+2\alpha }},\mid \gamma \mid =0,1,\dots, m-1,t\ge 1, $$and C3(1+t)−1+|γ|2+2α≤||∇γψ(·,t)||L2≤C4(1+t)−|γ|2+2α,|γ|=m,t≥1.$$ {C}_3{\left(1+t\right)}^{-\frac{1+\mid \gamma \mid }{2+2\alpha }}\le {\left\Vert {\nabla}^{\gamma}\psi \Big(\cdotp, t\Big)\right\Vert}_{L^2}\le {C}_4{\left(1+t\right)}^{-\frac{\mid \gamma \mid }{2+2\alpha }},\mid \gamma \mid =m,t\ge 1. $$ [ABSTRACT FROM AUTHOR]