Speaker
Description
(Filippo Santamborgio - Université Claude Bernard - Lyon 1)
This talk, based on a joint work (still to be finished) with Charles Elbar (Lyon) and Alejandro Fernandez Jimenez (Oxford), lies at the intersection of two well-known phenomena in parabolic equations. The first concerns nonlinear estimates on the second derivatives of solutions to diffusion equations: by looking at the PDE satisfied by the Laplacian of the logarithm of the solution of the heat equation (or of suitable powers of the solution for non-linear diffusion such as porous media equations) one can obtain lower bounds in the form of $\Delta(\log\rho)\geq -C/t$. The second, instead, concerns the critical mass in the parabolic-elliptic Keller-Segel chemotactic system where linear diffusion is coupled with advection by a potential generated by the convolution of the solution with the Poisson Kernel: it is well-known for this nonlinear equation that explosion in finite time or global existence depends on the mass (which is preserved in time) and the best estimates are obtained for small mass. In the talk I will show how, under small mass assumptions, it is also possible to obtain estimates, with instantaneous regularization, on the Laplacian of the pressure for the critical case, i.e. for the case of linear diffusion in 2D where the pressure is logarithmic or in power-case in higher dimension. The proofs I will show will mainly focus on the 2D case, easier to explain.
