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PreprintCC BY 4.0arXiv

Rough Stochastic Viscous Camassa-Holm equation

  • Xiaoxin Chen
  • Yiyao Lian
  • Zhaoyang Yin
  • Shuolin Zhang
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Abstract

In this paper, we establish well-posedness results for the viscous Camassa--Holm equation driven by the additive Gaussian noise (1−∂x2)−ηξ, where ξ denotes the space-time white noise and η>−1/12. This formulation includes spatially colored noise for η>0, space-time white noise for η=0, and rough noise for −1/12<η<0. More precisely, we establish a pathwise solution theory via approximation by equations driven by regularized noises. The main difficulty arises from the singular interactions between the nonlinear terms and the irregular stochastic convolution. By introducing suitable enhanced stochastic objects and applying paracontrolled techniques, we obtain local well-posedness for the full range η>−1/12. Furthermore, when η>1/4, we establish global well-posedness by deriving suitable energy estimates.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026rough,
  title = {{Rough Stochastic Viscous Camassa-Holm equation}},
  author = {Xiaoxin Chen and Yiyao Lian and Zhaoyang Yin and Shuolin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23198},
  eprint = {2610.23198},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23198},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Rough Stochastic Viscous Camassa-Holm equation
AU  - Xiaoxin Chen
AU  - Yiyao Lian
AU  - Zhaoyang Yin
AU  - Shuolin Zhang
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23198
UR  - https://arxiv.org/abs/2610.23198
AB  - In this paper, we establish well-posedness results for the viscous Camassa--Holm equation driven by the additive Gaussian noise (1−∂_x^2)^−ηξ, where ξ denotes the space-time white noise and η>−1/12. This formulation includes spatially colored noise for η>0, space-time white noise for η=0, and rough noise for −1/12<η<0. More precisely, we establish a pathwise solution theory via approximation by equations driven by regularized noises. The main difficulty arises from the singular interactions between the nonlinear terms and the irregular stochastic convolution. By introducing suitable enhanced stochastic objects and applying paracontrolled techniques, we obtain local well-posedness for the full range η>−1/12. Furthermore, when η>1/4, we establish global well-posedness by deriving suitable energy estimates.
ER  -