Half-harmonic gradient flow: aspects of a non-local geometric PDE. (English) Zbl 07817693
Summary: 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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