Large-Time Behaviour of Solutions to the Surface Quasi-Geostrophic Equations

We construct a linear approximation of the solution to the Surface Quasi-Geostrophic Equation in $\mathbb{R}^2$, and obtain a convergence rate in $L^p$ between the solution and this approximation with respect to time. We also demonstrate that the nonlinear term of the solution is bounded sharply in $L^p$ by the same function of time.