A Hele-Shaw Limit Without Monotonicity

We study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition. This model can be seen as a simplified description of non-monotone motions in tumor growth and crowd motion, generalizing the congestion-only motions studied in recent literature (\cite{AKY}, \cite{PQV}, \cite{KP}, \cite{MPQ}). We characterize the limit density, which solves a free boundary problem of Hele-Shaw type in terms of the limit pressure. The novel feature of our result lies in the characterization of the limit pressure, which solves an obstacle problem at each time in the evolution

[1]  Morton E. Gurtin,et al.  On interacting populations that disperse to avoid crowding: The effect of a sedentary colony , 1984 .

[2]  E. Giusti.  Minimal surfaces and functions of bounded variation , 1977 .

[3]  Liquid drops sliding down an inclined plane , 2012, 1203.2942.

[4]  Ing. Rafael Laguardia.  Boundary layer formation in the transition from the Porous Media Equation to a Hele-Shaw flow , 2000 .

[5]  J. R. King,et al.  The mesa problem: diffusion patterns for ut=⊇. (um⊇u) as m→+∞ , 1986 .

[6]  Fernando Quirós,et al.  Asymptotic behaviour of the porous media equation in an exterior domain , 1999 .

[7]  Inwon C. Kim,et al.  Quasi-static evolution and congested crowd transport , 2013, 1304.3072.

[8]  B. Perthame,et al.  The Hele–Shaw Asymptotics for Mechanical Models of Tumor Growth , 2013, Archive for Rational Mechanics and Analysis.

[9]  Inwon C. Kim.  Uniqueness and Existence Results on the Hele-Shaw and the Stefan Problems , 2003 .

[10]  The Hele-Shaw problem as a \Mesa" limit of Stefan problems: Existence, uniqueness, and regularity of the free boundary , 2004, math/0410131.

[11]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[12]  D. Hilhorst,et al.  A sensity dependent diffussion equation in population dynamics: stabilization to equilibrium , 1986 .

[13]  B. Perthame,et al.  A HELE-SHAW problem for tumor growth , 2015, 1512.06995.

[14]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[15]  F. Quirós,et al.  Boundary layer formation in the transition from the porous media equation to a hele-shaw flow , 2003 .

[16]  Thomas P. Witelski.  Segregation and mixing in degenerate diffusion in population dynamics , 1997 .

[17]  Herbert Amann,et al.  Compact embeddings of vector valued Sobolev and Besov spaces , 2000 .

[18]  D. Aronson.  The porous medium equation , 1986 .

[19]  Inwon C. Kim,et al.  Porous medium equation to Hele-Shaw flow with general initial density , 2015, 1509.06287.

[20]  F. Santambrogio,et al.  A MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE , 2010, 1002.0686.

[21]  Filippo Santambrogio,et al.  Crowd motion and evolution PDEs under density constraints , 2018 .

[22]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[23]  Inwon C. Kim,et al.  Singular limit of the porous medium equation with a drift , 2017, Advances in Mathematics.

[24]  L. Caffarelli.  The obstacle problem revisited , 1998 .

[25]  Ljll,et al.  Free boundary limit of a tumor growth model with nutrient , 2020, 2003.10731.