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 quasigeostrophic equation in critical mixed norm Lebesgue spaces.The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation.The phenomenon is a priori nontrivial due to the nonlocal structure of the equation.Our approach is based on Kato’s method using Picard’s interation,which can be apdated to the multi-dimensional case and other nonlinear non-local equations.We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.