Abstract:In this paper, we investigate the regularization effect of fractional stochastic forcing on Burgers-type equations with fractional dissipation, with an application to the Degasperis--Procesi (DP) equation. In particular, we consider the perturbation induced by the singular noise $|D|^{1/2}\xi$ and establish local well-posedness in the negative Sobolev space $H^{-1/4+\delta}$ for some small $\delta>0$. Due to the singular nature of the nonlinear interactions, classical solution theories cannot be directly applied. Inspired by the framework developed in \cite{hairer2013solving,gubinelli2017kpz}, we introduce a generalized solution theory based on an enhanced structure and derive the effective equations satisfied by these generalized solutions. Our main contribution is the establishment of a general framework for describing singular PDEs driven by rough data. Moreover, we prove the convergence of the associated non-Gaussian rough structures in the fractional dissipation setting. As an application, we apply this framework to the stochastic Degasperis--Procesi equation and obtain its local well-posedness in the low-regularity regime.
From: Shuolin Zhang [view email]
[v1]
Sat, 7 Feb 2026 10:57:02 UTC (28 KB)
[v2]
Tue, 8 Sep 2026 05:59:25 UTC (68 KB)
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