Positive heat-flow observations on spherical shells: cutoffs, optimal preparation and concentration
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Abstract
On a fixed spherical shell, we determine the last time at which nonnegative initial temperatures can produce one common signed whole-boundary heat flux. Admissible times form (0,τ), with a finite unattained cutoff. The boundary heat-kernel law fixes the shrinking concentration scale. After unit ground-state normalization, every weighted finite-q preparation problem, 1<q<∞, has a unique optimizer described by a scalar moment equation, and the cutoff limit has sharp costs and universal near-minimizer profiles. The conclusions include quantitative ground-weighted probability stability, second-moment control and physical angular rigidity. Separate endpoint arguments give atomic probability concentration at the L^1 endpoint and a filled-layer profile at the cap endpoint. Exactly feasible examples show the limits of higher-moment and strong cap-topology conclusions. An interior convex-hull criterion treats simultaneous finite schedules, and the smooth compatible class has the same norm infima. The shell-dependent boundary law connects feasibility, optimal preparation and concentration throughout the paper.
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Research details
- Subjects
- Mathematics · Analysis and Spectral Theory
- Keywords
- heat equation, overdetermined boundary flux, spherical shells, positive initial data, convex moment optimization, concentration, Wasserstein stability
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
On a fixed spherical shell, tailored nonnegative preparations realize one common signed whole-boundary flux exactly at times (0,τ), with finite unattained cutoff. Unit ground normalization gives unique weighted finite-q optimizers, sharp costs, finite-schedule criteria and the stated finite-power/endpoint concentration and angular rigidity.
Disclosures
AI-assisted tools were used in research exploration, internal analytic checks, literature comparisons and manuscript preparation. Internal assistant reviews are not external human refereeing or formal proof-assistant verification.
Reproduction
The 29-page analytic manuscript, eleven editable TeX files, README.md and SHA256SUMS.txt are distributed together in the Heat_Flow_Shells directory. Run tectonic main.tex from that directory using the usual LaTeX resources. The source includes the complete bibliography; no external research folders, numerical programs or datasets are required. SHA256SUMS.txt uses paths relative to its own directory.