On Well-Posedness of 2D Dissipative Quasi-Geostrophic Equation in Critical Mixed Norm Lebesgue Spaces

We establish local and global well-posedness of the 2D dissipative quasi-geostrophic equation in critical mixed norm Lebesgue spaces. The result demonstratesthe persistence of the anisotropic behavior of the initial data under the evolution ofthe 2D dissipative quasi-geostrophic equation. The phenomenon is a priori nontrivialdue to the nonlocal structure of the equation. Our approach is based on Kato's methodusing Picard's iteration, which can be adapted to the multi-dimensional case andother nonlinear non-local equations. We develop time decay estimates for solutions offractional heat equation in mixed norm Lebesgue spaces that could be useful for otherproblems.