A note on Monge–Ampère Keller–Segel equation

This note studies the Monge–Ampère Keller–Segel equation in a periodic domain TdTd(d≥2)(d≥2), a fully nonlinear modification of the Keller–Segel equation where the Monge–Ampère equation det(I+∇2v)=u+1det(I+∇2v)=u+1 substitutes for the usual Poisson equation Δv=uΔv=u. The existence of global weak solutions is obtained for this modified equation. Moreover, we prove the regularity in L∞(0,T;L∞∩W1,1+γ(Td))L∞(0,T;L∞∩W1,1+γ(Td)) for some γ>0γ>0.