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<p>We study a porous medium equation that models tissue growth in a heterogeneous environment. We show that, in the incompressible limit, solutions converge to those of a weak form of a Hele-Shaw type free boundary problem. To obtain enough compactness to take the limit, we establish an <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>4</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$L^{4}$</EquationSource> </InlineEquation> bound on the gradient of the pressure and an estimate of Aronson-Bénilan type.</p>

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The Incompressible Limit of an Inhomogeneous Model of Tissue Growth

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月14日

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  • Anthony Sulak,
  • Olga Turanova

摘要

We study a porous medium equation that models tissue growth in a heterogeneous environment. We show that, in the incompressible limit, solutions converge to those of a weak form of a Hele-Shaw type free boundary problem. To obtain enough compactness to take the limit, we establish an L 4 $L^{4}$ bound on the gradient of the pressure and an estimate of Aronson-Bénilan type.