Suppression of blow up by mixing in generalized Keller-Segel system with fractional dissipation and strong singular kernel

In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel equation with a fractional dissipation and advection by a weakly mixing (see Definition \ref{def:2.4}). Here the attractive kernel has strong singularity, namely, the derivative appears in the nonlinear term by singular integral. Without advection, the solution of equation blows up in finite time. Under a suitable mixing condition on the advection, we show the global existence of classical solution with large initial data in the case of the derivative of dissipative term is higher than that of nonlinear term. Since the attractive kernel is strong singularity, the weakly mixing has destabilizing effect in addition to the enhanced dissipation effect, which makes the problem more complicated and difficult. In this paper, we establish the $L^\infty$-criterion and obtain the global $L^\infty$ estimate of the solution through some new ideas and techniques. Combined with \cite{Shi.2019}, we discuss all cases of generalized Keller-Segel system with mixing effect, which was proposed by Kiselev, Xu (see \cite{Kiselev.2016}) and Hopf, Rodrigo (see \cite{Hopf.2018}). Based on more precise estimate of solution and the resolvent estimate of semigroup operator, we introduce a new method to study the enhanced dissipation effect of mixing in generalized parabolic-elliptic Keller-Segel equation with a fractional dissipation. And the RAGE theorem is no longer needed in our analysis.