Global well-posedness for the 2 D quasi-geostrophic equation in a critical Besov space

We show that the the 2 D quasi-geostrophic equation has global and unique strong solution, when the (large) data belongs in the critical, scale invariant space \dot{B}^{2-2\al}_{2, \infty}\cap L^{2/(2\al-1)}

Optimal solvability for the Dirichlet and Neumann problem in dimension two

We show existence and uniqueness for the solutions of the regularity and the Neumann problems for harmonic functions on Lipschitz domains with data in the Hardy spaces H^p, p>2/3, where This in turn implies that solutions to the Dirichlet problem with data in the Holder class C^{1/2}(\partial D) are themselves in C^{1/2}(D). Both of these results are sharp. In fact, we prove a more general statement regarding the H^p solvability for divergence form elliptic equations with bounded measurable coefficients. We also prove similar solvability result for the regularity and Dirichlet problem for the biharmonic equation on Lipschitz domains