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L2-contraction and asymptotic stability of Cahn-Hilliard fronts

  • Bongsuk Kwon
  • Wanyong Shim
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Abstract

We study the stability of transition fronts for the one-dimensional Cahn-Hilliard equation. More precisely, we prove that for any Hölder initial datum sufficiently close to a front in the L2 norm, the corresponding solution of the Cahn-Hilliard equation exists globally and converges, up to a dynamical shift, to the front as time tends to infinity. The result holds for a class of double-well potentials in a C4-neighborhood of the standard quartic potential. For the proof, we develop an L2-stability framework adapted to Cahn-Hilliard fronts. The key ingredient is a nontrivial second-order Poincaré-type inequality that reveals a coercive structure in the indefinite quadratic energy form associated with the linearized operator about the front, once the dynamical shift is taken into account. This yields an L2-contraction estimate and, via a far-field semigroup argument, asymptotic orbital stability of the front in the unweighted L2 topology. The stability analysis requires only the L2-smallness of the initial perturbation; no higher-order smallness, spatial localization, or moment assumptions are imposed.

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Cite

BibTeXRIS
BibTeX
@misc{kwon2026l2contraction,
  title = {{$L^2$-contraction and asymptotic stability of Cahn-Hilliard fronts}},
  author = {Bongsuk Kwon and Wanyong Shim},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.15658},
  eprint = {2606.15658},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.15658},
}
RIS
TY  - UNPB
TI  - L^2-contraction and asymptotic stability of Cahn-Hilliard fronts
AU  - Bongsuk Kwon
AU  - Wanyong Shim
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2606.15658
UR  - https://arxiv.org/abs/2606.15658
AB  - We study the stability of transition fronts for the one-dimensional Cahn-Hilliard equation. More precisely, we prove that for any Hölder initial datum sufficiently close to a front in the L^2 norm, the corresponding solution of the Cahn-Hilliard equation exists globally and converges, up to a dynamical shift, to the front as time tends to infinity. The result holds for a class of double-well potentials in a C^4-neighborhood of the standard quartic potential. For the proof, we develop an L^2-stability framework adapted to Cahn-Hilliard fronts. The key ingredient is a nontrivial second-order Poincaré-type inequality that reveals a coercive structure in the indefinite quadratic energy form associated with the linearized operator about the front, once the dynamical shift is taken into account. This yields an L^2-contraction estimate and, via a far-field semigroup argument, asymptotic orbital stability of the front in the unweighted L^2 topology. The stability analysis requires only the L^2-smallness of the initial perturbation; no higher-order smallness, spatial localization, or moment assumptions are imposed.
ER  -