Dissipative 2D quasi-geostrophic equation: Local well-posedness, global regularity and similarity solutions

The dissipative two dimensional Quasi-Geostrophic Equation (2D QGE) is studied. First, we prove existence and uniqueness of the solution, local in time, in the critical Sobolev space H2-2 alpha with arbitrary initial data theta(0) is an element of H2-2 alpha, where alpha is an element of (0, 1) is the fractional power of -Delta in the dissipative term of 2D QGE. Then, we give a sufficient condition that the H-s norm of the solution stays finite for any s > 0. This generalizes previous results by the author [18,20]. Finally, we prove that the Leray type similarity solutions which blow up in finite time in the critical Sobolev space H2-2 alpha do not exist.