Hele-Shaw flow as a singular limit of a Keller-Segel system with nonlinear diffusion (Q6607673)

The classical parabolic-elliptic Keller-Segel system is considered, which describes the diffusion of a population of bacteria with density \(\rho\). The ``continuity'' equation contains the term \(\rho^m\) -- where \(m\) is giving the ``nonlinearity degree'' of the model. The chemical's diffusivity is denoted by the small parameter \(\epsilon^2\). The concentration solves an elliptic equation. In the previous papers [\textit{I. Kim} et al., Nonlinearity 36, No. 2, 1082--1119 (2023; Zbl 1506.35282); Trans. Am. Math. Soc. 377, No. 1, 395--429 (2024; Zbl 1530.35131)], the authors considered \(m=1, \rho \leq 1 \) and (with some additional hypothesis) obtained the \(\Gamma\)-convergence of the energy functional toward the perimeter functional, when \(\epsilon \rightarrow 0\). In the present paper, this property is proved for \(m>2\) and without the restriction \(\rho \leq 1\). The main result is given in Theorem 2.4: when \(\epsilon \rightarrow 0\), the solution of the considered system converges to a (weak) solution of the Hele-Shaw free boundary problem with surface tension. The main mathematical tools are: the Legendre transform of the convex functions, the Modica-Mortola theorem, the Lebesgue's dominated convergence theorem, the Lions-Aubin compactness result. Some important facts about \(BV\) functions are briefly described in the last section.