I was reading one of Kato's papers on Navier-stokes equations. A mild solution can be denoted as $u= e^{t\Delta}u_0 + \int_{0}^{t} e^{(t-s)\Delta} \mathbb P\nabla \cdot(u \otimes u)ds$, where $\mathbb P$ is the Leray projector. Is it possible to denote a solution like this for 2D QG equation? Thanks.
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Consider the quasi-geostrophic equation $$\theta_t + (-\Delta)^\gamma\theta +(u\cdot \nabla) \theta=0 ,\ \ t>0,$$ with $\nabla\cdot u=0$ and initial datum $\theta_0(x),\ x\in\mathbb{R}^2$. For instance, in
Carrillo, José A.; Ferreira, Lucas C. F. The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations. Nonlinearity 21 (2008), no. 5, 1001--1018,
eq. (3.1), they use the following integral formulation of the quasi-geostrophic equation $$\theta(t) = G_\gamma(t)\theta_0 - \int_0^t \nabla G_\gamma(t-s)(\theta R[\theta])(s) ds ,$$ where $\hat{G_\gamma}(t,\xi) = e^{-|\xi|^{2\gamma}t}$ and $u=R[\theta]$, the Riesz transform of the potential temperature.
