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Incompressible limit of porous media equation with chemotaxis and growth
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abstract
We revisit the problem of proving the incompressible limit for the compressible porous media equation with Newtonian drift and growth. The question is motivated by models of living tissues development including chemotaxis. We extend the problem, already treated by the authors and several other contributions, in using a simplified approach, in treating dimensions two or higher, and in incorporating the pressure driven growth term. We also complete the analysis with stronger $L^4$ estimates on the pressure gradient. The major difficulty is to prove the strong convergence of the pressure gradient which is obtained here by a new observation on an algebraic relation involving the pressure gradient for weak limits.
Forward citations
Cited by 2 Pith papers
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On the incompressible limit of Keller-Segel system with volume-filling effects
For the volume-filling Keller-Segel system, the incompressible limit is a Hele-Shaw free-boundary problem for K>1 and a hyperbolic Keller-Segel system for K≤1.
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Hele-Shaw limit of chemotaxis-Navier-Stokes flows
For the chemotaxis-Navier-Stokes system with porous medium diffusion, as m→∞ the solutions converge to a Hele-Shaw free boundary problem whose pressure satisfies the complementarity relation P∞(ΔP∞ − ∇·(χ(c∞)∇c∞))=0.
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