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Small Scale Formations in the Incompressible Porous Media Equation

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Abstract

We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for a class of stratified steady states of IPM.

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Notes

  1. Observe that by Sard’s theorem [16], since \(\rho _0 \in C^2({\mathbb {T}}^2)\), the set of h such that \(\{\rho _0(x)=h\}\) contains a critical point has Lebesgue measure zero.

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Acknowledgements

The authors acknowledge partial support of the NSF-DMS grants 1715418, 1846745 and 2006372. AK has been partially supported by Simons Foundation. YY was partially supported by the Sloan Research Fellowship, the NUS startup grant A-0008382-00-00 and MOE Tier 1 grant A-0008491-00-00. This paper has been initiated at the AIM Square, and the authors thank AIM for support and collaborative opportunity.

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Authors and Affiliations

  1. Department of Mathematics, Duke University, 120 Science Dr., Durham, NC, 27708, USA

    Alexander Kiselev

  2. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Yao Yao

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  1. Alexander Kiselev
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  2. Yao Yao
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Correspondence to Alexander Kiselev.

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Communicated by J. Bedrossian.

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Kiselev, A., Yao, Y. Small Scale Formations in the Incompressible Porous Media Equation. Arch Rational Mech Anal 247, 1 (2023). https://doi.org/10.1007/s00205-022-01830-z

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