Singular limit of the porous medium equation with a drift

We study the “stiff pressure limit” of a nonlinear drift-diffusion equation, where the density is constrained to stay below the maximal value one. The challenge lies in the presence of a drift and the consequent lack of monotonicity in time. In the limit a Hele-Shaw-type free boundary problem emerge...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2019-06, Vol.349, p.682-732
Main Authors: Kim, Inwon, Požár, Norbert, Woodhouse, Brent
Format: Article
Language:English
Subjects:
Hele-Shaw problem
Porous medium equation
Singular limit
Tumor growth
Viscosity solutions
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Summary:We study the “stiff pressure limit” of a nonlinear drift-diffusion equation, where the density is constrained to stay below the maximal value one. The challenge lies in the presence of a drift and the consequent lack of monotonicity in time. In the limit a Hele-Shaw-type free boundary problem emerges, which describes the evolution of the congested zone where density equals one. We discuss pointwise convergence of the densities as well as the BV regularity of the limiting free boundary.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2019.04.017