Skip to main content
SpringerLink
Log in

A Parabolic Problem with a Fractional Time Derivative

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi–Nash–Moser Hölder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Caffarelli L., Chan C.H., Vasseur A.: Regularity theory for parabolic nonlinear integral operators. J. Am. Math. Soc. 24(3), 849–869 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Caffarelli, L.A., Cabré, X.: Fully nonlinear elliptic equations. In: American Mathematical Society Colloquium Publications, Vol. 43. American Mathematical Society, Providence, 1995

  3. del Castillo-Negrete D., Carreras B.A., Lynch V.E.: Fractional diffusion in plasma turbulence. Phys. Plasmas 11, 3854–3864 (2004)

    Article  ADS  Google Scholar 

  4. del Castillo-Negrete D., Carreras B.A., Lynch V.E.: Nondiffusive transport in plasma turbulene: a fractional diffusion approach. Phys. Rev. Lett. 94, 065003 (2005)

    Article  ADS  Google Scholar 

  5. Felsinger M., Kassmann M.: Local regularity for parabolic nonlocal operators. Commun. Partial Differ. Equ. 38(9), 1539–1573 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Metzler R., Klafter J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon, 1993 (Theory and applications, Edited and with a foreword by S. M. Nikol’skiĭ, Translated from the 1987 Russian original, Revised by the authors)

  8. Zacher R.: Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcial. Ekvac. 52(1), 1–18 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zacher R.: A De Giorgi–Nash type theorem for time fractional diffusion equations. Math. Ann. 356(1), 99–146 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zaslavsky G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371(6), 461–580 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luis Caffarelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexis Vasseur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luis Caffarelli.

Additional information

Communicated by F. Lin

Mark Allen is supported by NSG Grant DMS-1303632. Luis Caffarelli is supported by NSF Grant DMS-1500871. Alexis Vasseur is supported by NSF Grant DMS-1209420.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allen, M., Caffarelli, L. & Vasseur, A. A Parabolic Problem with a Fractional Time Derivative. Arch Rational Mech Anal 221, 603–630 (2016). https://doi.org/10.1007/s00205-016-0969-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-016-0969-z

Keywords

Search

Navigation