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Weak solutions of geometric flows associated to integro-differential harmonic maps

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Abstract

The purpose of this note is to prove the existence of global weak solutions to the flow associated to integro-differential harmonic maps into spheres and Riemannian homogeneous manifolds.

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References

  1. Chen, Y.M.: The weak solutions to the evolution problems of harmonic maps. Math. Z. 201(1), 69–74 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, Y.M., Hong, M.C., Hungerbühler, N.: Heat flow of \(p\)-harmonic maps with values into spheres. Math. Z. 215(1), 25–35 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, Y.M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201(1), 83–103 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Da Lio, F.: Fractional harmonic maps into manifolds in odd dimension \(n>1\). Calc. Var. PDE 48(3–4), 421–445 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Da Lio, F., Rivière, T.: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math. 227(3), 1300–1348 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Da Lio, F., Rivière, T.: Three-term commutator estimates and the regularity of \(\frac{1}{2}\)-harmonic maps into spheres. Anal. PDE 4(1), 149–190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Freire, A.: Global weak solutions of the wave map system to compact homogeneous spaces. Manuscr. Math. 91(4), 525–533 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hélein, F.: Regularity of weakly harmonic maps from a surface into a manifold with symmetries. Manuscr. Math. 70(2), 203–218 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hardt, R., Lin, F.-H.: Partially constrained boundary conditions with energy minimizing mappings. Commun. Pure Appl. Math. 42(3), 309–334 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lin, F., Wang, C.: The analysis of harmonic maps and their heat flows. World Scientific Publishing Co. Pte. Ltd, Hackensack (2008)

    Book  MATH  Google Scholar 

  11. Moore, J.D., Schlafly, R.: On equivariant isometric embeddings. Math. Z. 173, 119–133 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Millot, V., Sire, Y.: On a fractional Ginzburg–Landau equation and 1/2-harmonic maps into spheres. Arch. Ration. Mech. Anal. 215(1), 125–210 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pu, X., Guo, B.: The fractional Landau–Lifshitz–Gilbert equation and the heat flow of harmonic maps. Calc. Var. Partial Differ. Equ. 42(1–2), 1–19 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Schikorra, A.: Regularity of n/2-harmonic maps into spheres. J. Differ. Equ. 252, 1862–1911 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Schikorra, A.: Integro-differential harmonic maps into spheres. Commun. PDE 40(1), 506–539 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shatah, J.: Weak solutions and development of singularities of the \(su(2)\)-\(\sigma \) model. Commun. Pure Appl. Math. 41, 459–469 (1998)

    Article  MathSciNet  Google Scholar 

  17. Simon, L.: Theorems on regularity and singularity of energy minimizing maps. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, Based on Lecture Notes by Norbert Hungerbühler (1996)

  18. Struwe, M.: On the evolution of harmonic maps in higher dimensions. J. Differ. Geom. 28(3), 485–502 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Freiburg, Freiburg im Breisgau, Germany

    Armin Schikorra

  2. Johns Hopkins University, Krieger Hall, Baltimore, MD, USA

    Yannick Sire

  3. Purdue University, West Lafayette, IN, USA

    Changyou Wang

Authors
  1. Armin Schikorra
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  2. Yannick Sire
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  3. Changyou Wang
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Correspondence to Yannick Sire.

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Schikorra, A., Sire, Y. & Wang, C. Weak solutions of geometric flows associated to integro-differential harmonic maps. manuscripta math. 153, 389–402 (2017). https://doi.org/10.1007/s00229-016-0899-y

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  • DOI: https://doi.org/10.1007/s00229-016-0899-y

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