Homology of algebras of families of pseudodifferential operators

We compute the Hochschild, cyclic, and periodic cyclic homology groups of algebras of families of Laurent complete symbols on manifolds with corners. We show in particular that the spectral sequence associated with Hochschild homology degenerates at $E^2$ and converges to Hochschild homology. As a byproduct, we deduce an identification of the space of residue traces on fibrations by manifolds with corners. In the process, we prove several general results about algebras of complete symbols on manifolds with corners.Comment: 29 pages, LaTe

Global existence of the two-dimensional QGE with sub-critical dissipation

In this paper, we study the sub-critical dissipative quasi-geostrophic equations $({\bf S}_\alpha)$. We prove that there exists a unique local-in-time solution for any large initial data $\theta_0$ in the space ${\bf{\mathcal X}}^{1-2\alpha}(\mathbb R^2)$ defined by (\ref{ec}). Moreover we show that $({\bf S}_\alpha)$ has a global solution in time if the norms of the initial data in ${\bf{\mathcal X}}^{1-2\alpha}(\mathbb R^2)$ are bounded by $1/4$. Also, we prove a blow-up criterion of the non global solution of $({\bf S}_\alpha)$.Comment: 16 page