Existence, Uniqueness and Regularity of the Fractional Harmonic Gradient Flow in General Target Manifolds

In this paper, we continue to study the fractional harmonic gradient flow on S taking values in a general closed manifold N ⊂ Rn, addressing global existence and uniqueness of solutions of energy class with sufficiently small energy, adding to the existing body of knowledge pertaining to the half-harmonic gradient flow and expanding upon our previous work in [34]. We extend the techniques by Struwe in [30] and Rivière in [22] to the non-local framework analogous to [34] to derive uniqueness, employ commutator estimates as in [8] for regularity and follow [30] for a general existence result.

[1]  Y. Meyer,et al.  Compensated compactness and Hardy spaces , 1993 .

[2]  Alessandro Pigati,et al.  Free boundary minimal surfaces: a nonlocal approach , 2017, ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE.

[3]  T. Rivière,et al.  Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps , 2010, 1012.2438.

[4]  Changyou Wang,et al.  Weak solutions of geometric flows associated to integro-differential harmonic maps , 2015, 1512.08353.

[5]  A. Freire.  Uniqueness for the harmonic map flow from surfaces to general targets , 1995 .

[6]  T. Rivière.  Conservation laws for conformally invariant variational problems , 2006, math/0603380.

[7]  V. Millot,et al.  On a Fractional Ginzburg–Landau Equation and 1/2-Harmonic Maps into Spheres , 2013, 1307.7015.

[8]  F. Lio.  Fractional harmonic maps into manifolds in odd dimension n > 1 , 2010, 1012.2741.

[9]  William R. Parke.  What is Fractional Integration? , 1999, Review of Economics and Statistics.

[10]  Claudia Bucur,et al.  Some observations on the Green function for the ball in the fractional Laplace framework , 2015, 1502.06468.

[11]  L. Martinazzi,et al.  Blow-up analysis of a nonlocal Liouville-type equation , 2015, 1503.08701.

[12]  Richard S. Hamilton,et al.  Harmonic Maps of Manifolds with Boundary , 1975 .

[13]  A. Schikorra,et al.  On regularity theory for n∕p-harmonic maps into manifolds , 2017, 1709.02329.

[14]  Matthias Hieber,et al.  Heat Kernels and maximal L^p - L^q Estimates for Parabolic Evolution Equations , 1996 .

[15]  Michael Struwe,et al.  On the evolution of harmonic maps in higher dimensions , 1988 .

[16]  Hieber Matthias,et al.  Heat kernels and maximal lp—lqestimates for parabolic evolution equations , 1997 .

[17]  Michael Struwe,et al.  On the evolution of harmonic mappings of Riemannian surfaces , 1985 .

[18]  Armin Schikorra,et al.  Fractional div-curl quantities and applications to nonlocal geometric equations , 2017, Journal of Functional Analysis.

[19]  E. Lenzmann,et al.  Sharp commutator estimates via harmonic extensions , 2016, Nonlinear Analysis.

[20]  A. Schikorra,et al.  n/p-harmonic maps: Regularity for the sphere case , 2012, 1202.1151.