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

TL;DR

This paper proves the global existence and boundedness of solutions for a generalized Keller-Segel system with degenerate dissipation and mixing, showing mixing can prevent blow-up despite weak dissipation.

Contribution

It demonstrates that mixing effects can ensure global well-posedness in a Keller-Segel system with degenerate dissipation, extending understanding of nonlinear PDE behavior under weak dissipation.

Findings

Global solutions exist despite degenerate dissipation.

Mixing weakens nonlinear effects, preventing blow-up.

Enhanced dissipation effects are not necessary for global regularity.

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.