Singular limit problem for the Keller–Segel system and drift–diffusion system in scaling critical spaces

• Published in: Journal of evolution equations Vol. 20; no. 2; pp. 421 - 457
• Main Authors: Kurokiba, Masaki, Ogawa, Takayoshi
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2020; Springer Nature B.V
• Subjects: Analysis; Cauchy problems; Drift; Function space; Interpolation; Mathematics; Mathematics and Statistics; Relaxation time; Thermodynamics; Singular limit problem; Drift–diffusion system; 78A38; Global well-posedness; Primary 35K15; Maximal regularity; Secondary 35K55; Critical space; Scaling invariance; 35Q60; Keller–Segel equation
• ISSN: 1424-3199; 1424-3202
• DOI: 10.1007/s00028-019-00527-3

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
τ
→
∞
. 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
B
˙
q
,
σ
s
(
R
n
)
and
F
˙
q
,
σ
s
(
R
n
)
.