MEASURABLE SEMIGROUP SELECTION OF THE HEAT FLOW FOR HARMONIC MAPS

The paper establishes that there exist infinitely many measurable semigroups selecting global weak solutions to the harmonic map heat flow and carefully frames the semigroup identity to hold for every s ≥ 0 and almost every t ≥ 0, relying on compactness, upper semicontinuity, a splicing property valid at energy-inequality times, and a general semiflow-selection theorem of Cardona–Kapitanski. This is stated precisely in the definition of weak solution (energy inequality for a.e. t) and in Theorem 2.1 and its proof, which splice only at such times and then invoke the abstract selection theorem to obtain the semigroup property for a.e. t, including t = 0 (see the definition and Theorem 2.1 together with the proof discussion). The candidate solution reproduces the selection idea with a constructive lexicographic maximization over Laplace-type functionals, but it incorrectly assumes the time-shift/concatenation closure and semigroup identity for every t ≥ 0 (not just a.e. t) and contains a sign/factor error in the “memoryless” identity. It also misattributes the existence of infinitely many solutions to Coron rather than to the later work by Béthuel–Coron–Ghidaglia–Soyeur. These issues make the model’s proof as written incorrect, while the paper’s result and proof are sound.