We consider a singular limit problem for the Cauchy problem of the Keller–Segel equation in a critical function space. We show that a solution to the Keller–Segel system in a scaling critical function space converges to a solution to the drift–diffusion system of parabolic–elliptic type (the simplified Keller–Segel model) in the critical space strongly as the relaxation time \(\tau \rightarrow \infty \). For the proof of singular limit problem, we employ generalized maximal regularity for the heat equation and use it systematically with the sequence of embeddings between the interpolation spaces \(\dot{B}^s_{q,\sigma }(\mathbb {R}^n)\) and \(\dot{F}^s_{q,\sigma }(\mathbb {R}^n)\).

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临界空间中Keller-Segel系统和漂移扩散系统的奇异极限问题

我们考虑临界函数空间中Keller-Segel方程的Cauchy问题的奇异极限问题。我们证明了，随着弛豫时间\的增加，在缩放关键函数空间中的Keller-Segel系统的解在收敛到收敛到关键空间中抛物线-椭圆型漂移-扩散系统（简化的Keller-Segel模型）的解。 （\ tau \ rightarrow \ infty \）。为了证明奇异极限问题，我们对热方程采用广义最大正则性，并将其与插值空间\（\ dot {B} ^ s_ {q，\ sigma}（\ mathbb {R } ^ n）\）和\（\ dot {F} ^ s_ {q，\ sigma}（\ mathbb {R} ^ n）\）。