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Porous medium flow with both a fractional potential pressure and fractional time derivative

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Abstract

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional and is of Caputo-type, which takes into account “memory”. The precise model is

$$D_t^\alpha u - div\left( {u - {{\left( { - \Delta } \right)}^{ - \sigma }}u} \right) = f,0 < \sigma < \frac{1}{2}.$$

This paper poses the problem over {t ∈ R+, x ∈ Rn} with nonnegative initial data u(0, x) ≥ 0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x) have exponential decay at infinity is proved. The main result is Hölder continuity for such weak solutions.

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Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA

    Mark Allen, Luis Caffarelli & Alexis Vasseur

Authors
  1. Mark Allen
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  2. Luis Caffarelli
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  3. Alexis Vasseur
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Corresponding author

Correspondence to Mark Allen.

Additional information

Dedicated to Haim Brezis on the occasion of his 70th birthday

This work was supported by NSG grant DMS-1303632, NSF grant DMS-1500871 and NSF grant DMS-1209420.

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Allen, M., Caffarelli, L. & Vasseur, A. Porous medium flow with both a fractional potential pressure and fractional time derivative. Chin. Ann. Math. Ser. B 38, 45–82 (2017). https://doi.org/10.1007/s11401-016-1063-4

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  • DOI: https://doi.org/10.1007/s11401-016-1063-4

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