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On the local solvability of the two-dimensional Hele-Shaw problem with fractional derivative with respect to time

Abstract

We study the two-dimensional quasistationary Stefan probem (the Hele-Shaw problem) in which the motion of the free boundary is described by a “fractional” Darcy law. We prove the existence and uniqueness of a classical solution to the free boundary problem for a small time interval.

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Correspondence to N. V. Vasil’eva.

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Original Russian Text © N.V. Vasil’eva, N.V. Krasnoshchek, 2014, published in Matematicheskie Trudy, 2014, Vol. 17, No. 2, pp. 102–131.

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Vasil’eva, N.V., Krasnoshchek, N.V. On the local solvability of the two-dimensional Hele-Shaw problem with fractional derivative with respect to time. Sib. Adv. Math. 25, 276–296 (2015). https://doi.org/10.3103/S1055134415040057

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Keywords

  • quasistationary Stefan problem
  • anomalous diffusion
  • Caputo derivative
  • regularizer
  • coercive estimate