Global well-posedness for a generalized Keller-Segel system with degenerate dissipation and mixing

Abstract

We study the mixing effect for a generalized Keller-Segel system with degenerate dissipation and advection by a weakly mixing. Here the attractive operator has weak singularity, namely, the negative derivative appears in the nonlinear term by singular integral. Without advection, the solution of equation blows up in finite time. We show that the global well-posedness of solution with large advection. Since dissipation term degenerate into the damping, the enhanced dissipation effect of mixing no longer occurs, we prove that the mixing effect can weak the influence of nonlinear term. In this case, the mixing effect is similar with inviscid damping of shear flow. Combining to the mixing effect and damping effect of degenerate dissipation, the global $L^\infty$ estimate of solution is established.