abstract: This is recently completed joint work with Herbert Koch and Robert McCann. Higher order convergence rates for fast diffusion to Barenblatt in the relative infinity norm can be obtained by a blend of Schauder estimates, and dynamical systems style semigroup estimates. As the functional analytic desires for a dynamical systems approach are not automatically borne out by the nonlinearity of the equation, the blended approach of PDE methods and functional analytic techniques will provide the desired estimates. Spectral information of the linearization is simultaneously used in a scale of differently weighted H\"older spaces in order to accommodate the growth of the various eigenfunctions (relative to Barenblatt) into the function space in question, and also to offset the loss of selfadjointness such as to maintain the sharp spectral radius.
At this time, to avoid further technicalities, we measure the nonlinear dynamics by distance to the tangent space, rather than to a slow manifold, which, by the arithmetic of the eigenvalues, gives still several higher order terms; however I believe that improvement to yet higher orders in which linear combinations of eigenvalues enter into the asymptotics (up to the onset of the essential spectrum) should be within reach of the same methods.