Maximal regularity and a singular limit problem for the Patlak–Keller–Segel system in the scaling critical space involving BMO

Abstract

We consider a singular limit problem of the Cauchy problem for the Patlak–Keller–Segel equation in a scaling critical function space. It is shown that a solution to the Patlak–Keller–Segel system in a scaling critical function space involving the class of bounded mean oscillations converges to a solution to the drift-diffusion system of parabolic-elliptic type (simplified Keller–Segel model) strongly as the relaxation time parameter $$\tau \rightarrow \infty $$ τ → ∞ . For the proof, we show generalized maximal regularity for the heat equation in the homogeneous Besov spaces and the class of bounded mean oscillations and we utilize them systematically as well as the continuous embeddings between the interpolation spaces $$\dot{B}^s_{q,\sigma }({\mathbb {R}}^n)$$ B ˙ q , σ s ( R n ) and $$\dot{F}^s_{q,\sigma }({\mathbb {R}}^n)$$ F ˙ q , σ s ( R n ) for the proof of the singular limit. In particular, end-point maximal regularity in BMO and space time modified class introduced by Koch–Tataru is utilized in our proof.

Maximal regularity of the heat evolution equation on spatial local spaces and application to a singular limit problem of the Keller–Segel system

“…Lamarié-Rieusset [23] extended these results to the homogeneous Morrey space. Kurokiba-Ogawa [20][21][22] showed the singular limit problem (1.2) in the scaling critical Bochner-Lebesgue spaces for the large initial data and showed the appearance of the initial layer in the two and higher dimensional cases. Those results [20][21][22] cover the finite mass case, where the positive solution preserves the total mass, while the other results are not covering the finite mass case.…”

“…Namely such a singular limit problem also remains valid in the local uniform class that reflect a spatial structure of a solution to both problems. Meanwhile the singular limit was established in the scaling invariant classes in [20][21][22] by applying the maximal regularity estimate for the Cauchy problem of the heat equation with λ ≥ 0. Maximal regularity is a useful tool to see that the time local well-posedness of the problems (1.1) and (1.3) and it provides useful local estimates which are independent of the parameter τ > 0.…”

“…Hence it allowed us to show that the limit exists and it solves the limiting problem (1.3) in the critical Lebesgue space. Following basic method used in [20][21][22], we employ maximal regularity for the heat equation to show the singular limit problem (1.2).…”