2022/03/05 by Raymond K. M. Chu, Chu, Raymond
Computer Science · Mathematics · Physics and Astronomy · #35Q92 #35R35 #76D27 #76S05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2203.02644
openalex publication_date 2022/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate a generalized Hele-Shaw equation with a source and drift terms where the density is constrained by an upper-bound density constraint that varies in space and time. By using a generalized porous medium equation approximation, we are able to construct a weak solution to the generalized Hele-Shaw equations under mild assumptions. Then we establish uniqueness of weak solutions to the generalized Hele-Shaw equations. Our next main result is a pointwise characterization of the density variable in the generalized Hele-Shaw equations when the system is in the congestion case. To obtain such a characterization for the congestion case, we derive a new uniform lower bounds on the time derivative pressure of the generalized porous medium equation via a refined Aronson-Benilan estimate that implies monotonicity on the density and pressure.