We prove $L^{ \infty}_{ t}W^{ 1,p}_{ x}$ Sobolev estimates in the Keller-Segel system with linear diffusion in any dimensionby proving a functional inequality, inspired by the Brezis-Gallou\"et-Wainger inequality. These estimates are also valid at the discrete level in the Jordan-Kinderlehrer-Otto (JKO) scheme. By coupling this result with the diffusion properties of a functional according to Bakry-Emery theory, we deduce the $L^2_t H^{ 2}_{ x}$ convergence of the scheme, thereby extending the recent result of Santambrogio and Toshpulatov in the context of the Fokker-Planck equation to the Keller-Segel system.
Categories: math.AP