Abstract
We show the existence of infinitely many admissible weak solutions for the incompressible porous media equations for all Muskat-type initial data with \({C^{3,\alpha}}\)-regularity of the interface in the unstable regime and for all non-horizontal data with \({C^{3,\alpha}}\)-regularity in the stable regime. Our approach involves constructing admissible subsolutions with piecewise constant densities. This allows us to give a rather short proof where it suffices to calculate the velocity and acceleration at time zero - thus emphasizing the instantaneous nature of non-uniqueness due to discontinuities in the initial data.
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The authors gratefully acknowledge the support of the ERC Grant Agreement No. 724298.
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Förster, C., Székelyhidi, L. Piecewise Constant Subsolutions for the Muskat Problem. Commun. Math. Phys. 363, 1051–1080 (2018). https://doi.org/10.1007/s00220-018-3245-2
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DOI: https://doi.org/10.1007/s00220-018-3245-2