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

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