GLOBAL WELL-POSEDNESS FOR A GENERALIZED KELLER-SEGEL SYSTEM WITH DEGENERATE DISSIPATION AND MIXING

We study the mixing effect for a generalized Keller-Segel system with degenerate dissipation and advection. Here the attractive operator has weak singularities, namely, the singular integral in the nonlinear term contains negative derivative. Without advection, the solution of equation blows up in finite time. We show that the solution is globally well-posed with large, weakly mixed flows. Since the dissipation term degenerates and turns into damping, the enhanced dissipation effect of mixing no longer occurs. We prove that the mixing effect can weaken the influence of nonlinear term. In this case, the mixing effect is similar to inviscid damping of shear flow. Combining the mixing effect and damping effect of degenerate dissipation, we establish the global L-infinity estimate of solution.