The paper by Caffarelli and Vasseur focuses on drift-diffusion equations with fractional diffusion, specifically motivated by the critical dissipative quasi-geostrophic equation. They prove that solutions to these equations with $L^2$ initial data and minimal assumptions on the drift are locally Hölder continuous. As an application, they show that solutions to the quasi-geostrophic equation with initial $L^2$ data and critical diffusion $(-\Delta)^{1/2}$ are locally smooth in any space dimension. The main theorems establish a priori estimates for the solutions. The first theorem (From $L^2$ to $L^\infty$) states that if a solution $\theta$ satisfies certain level set energy inequalities, then it is bounded in $L^\infty$. The second theorem (From $L^\infty$ to $C^\alpha$) shows that if a solution $\theta$ is bounded in $L^\infty$ and the velocity $v$ satisfies certain conditions, then $\theta$ is Hölder continuous in space. These theorems are used to derive the regularity of solutions to the quasi-geostrophic equation. The third theorem states that solutions to the quasi-geostrophic equation with initial data in $L^2$ and critical diffusion are bounded in $C^\alpha$ for any time $t_0 > 0$. Higher regularity is established using standard potential theory, and the fundamental solution of the operator $\partial_t + \Lambda$ is shown to be the Poisson kernel. The paper also discusses the existence theory for approximate solutions and provides corollaries that extend the results to different scenarios, including different norms and domains. The techniques used are similar to those in the De Giorgi-Nash-Moser method for treating boundary parabolic problems.The paper by Caffarelli and Vasseur focuses on drift-diffusion equations with fractional diffusion, specifically motivated by the critical dissipative quasi-geostrophic equation. They prove that solutions to these equations with $L^2$ initial data and minimal assumptions on the drift are locally Hölder continuous. As an application, they show that solutions to the quasi-geostrophic equation with initial $L^2$ data and critical diffusion $(-\Delta)^{1/2}$ are locally smooth in any space dimension. The main theorems establish a priori estimates for the solutions. The first theorem (From $L^2$ to $L^\infty$) states that if a solution $\theta$ satisfies certain level set energy inequalities, then it is bounded in $L^\infty$. The second theorem (From $L^\infty$ to $C^\alpha$) shows that if a solution $\theta$ is bounded in $L^\infty$ and the velocity $v$ satisfies certain conditions, then $\theta$ is Hölder continuous in space. These theorems are used to derive the regularity of solutions to the quasi-geostrophic equation. The third theorem states that solutions to the quasi-geostrophic equation with initial data in $L^2$ and critical diffusion are bounded in $C^\alpha$ for any time $t_0 > 0$. Higher regularity is established using standard potential theory, and the fundamental solution of the operator $\partial_t + \Lambda$ is shown to be the Poisson kernel. The paper also discusses the existence theory for approximate solutions and provides corollaries that extend the results to different scenarios, including different norms and domains. The techniques used are similar to those in the De Giorgi-Nash-Moser method for treating boundary parabolic problems.