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PreprintCC BY 4.0arXiv

Singular Stochastic Heat Equations with Fractional Memory

  • Shuolin Zhang
  • \quad Zhaoyang Yin
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Abstract

We extend the paracontrolled distribution approach of Gubinelli, Imkeller, and Perkowski to singular stochastic heat equations with Caputo fractional time derivatives. By establishing Schauder estimates for the time-fractional heat flow and controlling history terms, we address the difficulties arising from temporal memory and the lack of the semigroup property. For the one-dimensional rough Burgers-type equation driven by space-time white noise with spatial dissipation (−Δ)σ, we prove global well-posedness for σ∈(5/6,1] and ν∈(6σ/(12σ−5),1]. For the heat equation with nonlinear multiplicative spatial white noise on the two-dimensional torus, we establish local well-posedness of renormalized solutions and an L∞ blow-up criterion for every ν∈(0,1].

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026singular,
  title = {{Singular Stochastic Heat Equations with Fractional Memory}},
  author = {Shuolin Zhang and \quad Zhaoyang Yin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23189},
  eprint = {2610.23189},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23189},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Singular Stochastic Heat Equations with Fractional Memory
AU  - Shuolin Zhang
AU  - \quad Zhaoyang Yin
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23189
UR  - https://arxiv.org/abs/2610.23189
AB  - We extend the paracontrolled distribution approach of Gubinelli, Imkeller, and Perkowski to singular stochastic heat equations with Caputo fractional time derivatives. By establishing Schauder estimates for the time-fractional heat flow and controlling history terms, we address the difficulties arising from temporal memory and the lack of the semigroup property. For the one-dimensional rough Burgers-type equation driven by space-time white noise with spatial dissipation (−Δ)^σ, we prove global well-posedness for σ∈(5/6,1] and ν∈(6σ/(12σ−5),1]. For the heat equation with nonlinear multiplicative spatial white noise on the two-dimensional torus, we establish local well-posedness of renormalized solutions and an L^∞ blow-up criterion for every ν∈(0,1].
ER  -