Half-harmonic gradient flow: aspects of a non-local geometric PDE

Jerome D. Wettstein; Jerome D. Wettstein

doi:10.3934/mine.2023058

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-38. doi: 10.3934/mine.2023058

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Half-harmonic gradient flow: aspects of a non-local geometric PDE

Jerome D. Wettstein ^{1,2
,
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1.
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL, USA
2.
150 West University Blvd, Melbourne, FL 32901, USA

Received: 16 December 2021 Revised: 13 October 2022 Accepted: 17 October 2022 Published: 07 November 2022

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 ^[47] 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 ^[40], 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. ^[48]).
- fractional Laplacian,
- half-harmonic map,
- gradient flow,
- finite-time bubbling,
- non-local PDE
Citation: Jerome D. Wettstein. Half-harmonic gradient flow: aspects of a non-local geometric PDE[J]. Mathematics in Engineering, 2023, 5(3): 1-38. doi: 10.3934/mine.2023058

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Abstract

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 ^[47] 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 ^[40], 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. ^[48]).

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