Singular Stochastic Heat Equations with Fractional Memory

Shuolin Zhang; \quad Zhaoyang Yin

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arXiv · 2610.23189

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 $σ\in(5/6,1]$ and $ν\in(6σ/(12σ-5),1]$. 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 $L^\infty$ blow-up criterion for every $ν\in(0,1]$.

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BibTeX RIS

Shuolin Zhang, \quad Zhaoyang Yin. 2026-10-16. Singular Stochastic Heat Equations with Fractional Memory. https://arxiv.org/abs/2610.23189

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