Dissipative Quasi-Geostrophic Equation: Local Well-posedness, Global Regularity and Similarity Solutions

Abstract

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 H 2-2α with arbitrary initial data θ 0 ∈ H 2-2α , where α ∈ (0,1) is the fractional power of -A in the dissipative term of 2D QGE. Then, we give a sufficient condition that the H 5 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 H 2-2α do not exist.