Stability and maximal regularity of the diffusion equation on unbounded surfaces

Abstract

This paper studies both stability and maximal regularity of the diffusion equation on an unbounded surface without any assumptions on the curvature. We show that the surface diffusion system admits a unique global-in-time strong solution satisfying L²-asymptotic stability when the slope of the surface is gentle. We also derive maximal L²-regularity of the solution when the exterior force satisfies both surface divergence form and tangential condition. Moreover, we investigate the Hölder continuity with respect to time of the solution. The key idea of showing our stability and maximal regularity results is to apply nice properties of both Laplace and weighted Laplace-Beltrami operators in order to derive the properties of the Laplace-Beltrami operator on the unbounded surface.

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BibTeX RIS

BibTeX

@misc{koba2026stability,
  title = {{Stability and maximal regularity of the diffusion equation on unbounded surfaces}},
  author = {Hajime Koba},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26622},
  eprint = {2609.26622},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26622},
}

RIS

TY  - UNPB
TI  - Stability and maximal regularity of the diffusion equation on unbounded surfaces
AU  - Hajime Koba
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26622
UR  - https://arxiv.org/abs/2609.26622
AB  - This paper studies both stability and maximal regularity of the diffusion equation on an unbounded surface without any assumptions on the curvature. We show that the surface diffusion system admits a unique global-in-time strong solution satisfying L^2-asymptotic stability when the slope of the surface is gentle. We also derive maximal L^2-regularity of the solution when the exterior force satisfies both surface divergence form and tangential condition. Moreover, we investigate the Hölder continuity with respect to time of the solution. The key idea of showing our stability and maximal regularity results is to apply nice properties of both Laplace and weighted Laplace-Beltrami operators in order to derive the properties of the Laplace-Beltrami operator on the unbounded surface.
ER  -

Abstract

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