2022/03/25 by He, Qingyou, Li, Hai-Liang, Perthame, Benoît · 1 citation
#35B45 #35K57 #35K65 #35Q92 #76N10 #76T99 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2203.13709
We complete previous results about the incompressible limit of both the n-dimensional (n≥3) compressible Patlak-Keller-Segel (PKS) model and its stationary state. As in previous works, in this limit, we derive the weak form of a geometric free boundary problem of Hele-Shaw type, also called congested flow. In particular, we are able to take into account the unsaturated zone, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation. Not only our analysis uses a completely different framework than previous approaches, but we also establish a novel uniform L3 estimate of the pressure gradient, regularity à la Aronson-Bénilan, and a uniform L1 estimate for the time derivative of the pressure. Furthermore, for the Hele-Shaw problem, we prove the uniqueness of solutions, meaning that the incompressible limit of the PKS model is unique. In addition, we establish the corresponding incompressible limit of the stationary state for the PKS model with a given mass, where, different from the case of PKS model, we obtain the uniform bound of pressure and the uniformly bounded support of density.