Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equation

We analyze the well-posedness of the initial value problem for the dissipative quasi-geostrophic equations in the subcritical case. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. While the only small self-similar solution in the strong L^p{cal L}^{p} space is the null solution, infinitely many self-similar solutions do exist in weak- L^p{cal L}^{p} spaces and in a recently introduced [7] space of tempered distributions. The asymptotic stability of solutions is obtained in both spaces, and as a consequence, a criterion of self-similarity persistence at large times is obtained.