Affiliation:
1. Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL, USA
2. 150 West University Blvd, Melbourne, FL 32901, USA
Abstract
<abstract><p>The goal of this paper is to discuss some of the results in the author's previous papers and expand upon the work there by proving two new results: a global weak existence result as well as a first bubbling analysis for the half-harmonic gradient flow in finite time. In addition, an alternative local existence proof to the one provided in <sup>[<xref ref-type="bibr" rid="b47">47</xref>]</sup> is presented based on a fixed-point argument. This preliminary bubbling analysis leads to two potential outcomes for the possibility of finite-time bubbling until a conjecture by Sire, Wei and Zheng, see <sup>[<xref ref-type="bibr" rid="b40">40</xref>]</sup>, is settled: Either there always exists a global smooth solution to the half-harmonic gradient flow without concentration of energy in finite-time, which still allows for the formation of half-harmonic bubbles as $ t \to +\infty $, or finite-time bubbling may occur in a similar way as for the harmonic gradient flow due to energy concentration in finitely many points. In the first part of the introduction to this paper, we provide a survey of the theory of harmonic and fractional harmonic maps and the associated gradient flows. For clarity's sake, we restrict our attention to the case of spherical target manifolds $ S^{n-1} $, but our discussion extends to the general case after taking care of technicalities associated with arbitrary closed target manifolds $ N $ (cf. <sup>[<xref ref-type="bibr" rid="b48">48</xref>]</sup>).</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)
Subject
Applied Mathematics,Mathematical Physics,Analysis
Reference48 articles.
1. A. Audrito, On the existence and Hölder regularity of solutions to some nonlinear Cauchy-Neumann problems, arXiv: 2107.03308.
2. L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq., 32 (2007), 1245–1260. https://doi.org/10.1080/03605300600987306
3. K.-C. Chang, W. Y. Ding, R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom., 36 (1992), 507–515. https://doi.org/10.4310/jdg/1214448751
4. R. Coifman, P. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, Journal de mathématiques pures et appliquées, 72 (1993), 247–286.
5. F. Da Lio, Compactness and bubbles analysis for half-harmonic maps into spheres, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 201–224. https://doi.org/10.1016/j.anihpc.2013.11.003
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献