Refined hyper-contractivity and uniqueness for the Keller–Segel equations

This note considers the hyper-contractivity and the uniqueness of the weak solutions to the two dimensional Keller–Segel equations. We prove a refined hyper-contractive property and consequently obtain the uniqueness of global weak solutions provided that the initial data satisfy ∫R2ρ0dx<8π, ∫R2|x|ρ0dx<∞ and ∫R2ρ0logρ0dx<∞. We also extend the result to higher dimensions.