Abstract We consider the singular limit problem of the Cauchy problem to the Keller-Segel equation in the two dimensional critical space. It is shown that the solution to the Keller-Segel system in the scaling critical function space converges to the solution to the drift-diffusion system of parabolic-elliptic equations (the simplified Keller-Segel equation) in the critical space strongly as the relaxation time parameter τ → ∞ . For the proof, we show generalized maximal regularity for the heat equations and use it systematically with the sequence of embeddings between the interpolation spaces B ˙ q , σ s ( R 2 ) and F ˙ q , σ s ( R 2 ) for the proof of singular limit problem.

中文翻译：

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二维Keller-Segel系统在临界空间标度上的奇异极限问题

摘要 我们在二维临界空间考虑Keller-Segel方程的柯西问题奇异极限问题。结果表明，尺度临界函数空间中Keller-Segel系统的解与临界空间中抛物线-椭圆方程（简化的Keller-Segel方程）漂移-扩散系统的解强烈收敛于弛豫时间参数 τ → ∞ 。对于证明，我们展示了热方程的广义最大正则性，并将其与插值空间 B ˙ q , σ s ( R 2 ) 和 F ˙ q , σ s ( R 2 ) 之间的嵌入序列系统地用于证明奇异极限问题。