Well-posedness of half-harmonic map heat flows for rough initial data

2025/04/09 by Koch, Kilian, Christof Melcher, Melcher, Christof · 1 citation
Mathematics · #35A01 #35K55 #35R11 #58J35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2504.06933

openalex publication_date 2025/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We adopt the Koch-Tataru theory for the Navier-Stokes equations, based on Carleson measure estimates, to develop a scaling-critical low-regularity framework for half-harmonic map heat flows. This nonlocal variant of the harmonic map heat flow has been studied recently in connection with free boundary minimal surfaces. We introduce a new class of initial data for the flow, broader than the conventional energy or Sobolev spaces considered in previous work, for which we establish existence, uniqueness, and continuous dependence. The class particularly includes homogeneous initial data that give rise to self-similar expanders.

Cited by

• Analyzing the Geometry of Immersions of Co-Dimension One via Shape Operator Dynamics

Related

• Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary
• Harmonic map heat flow with rough boundary data
• Global dissipative half-harmonic flows into spheres: small data in\n critical Sobolev spaces
• Regularity and long-time behavior of nonlocal heat flows
• Diffusive stability and self-similar decay for the harmonic map heat flow