Results of Fractional Rough Burgers equation in $H^s$ space and its application

TL;DR

The paper analyzes well-posedness for the Fractional Rough Burgers equation on the torus driven by space-time white noise in Sobolev spaces. It develops two regimes: for higher dissipation $\gamma\in(\frac{4}{3},2]$, local well-posedness is shown (globality for $\gamma>\tfrac{3}{2}$) via a fixed-point framework; for lower dissipation $\gamma\in(\frac{5}{4},\frac{4}{3}]$, a para-controlled approach is developed to construct solutions with higher-regularity remainder. The Degasperis–Procesi shallow-water equation and its nonlocal rough variant are treated as applications of the FRB theory, with energy estimates supporting global results in appropriate ranges. The work provides a structured set of tools—Littlewood–Paley, stochastic regularity, and contraction mappings—alongside a para-controlled strategy to handle rough forcing, contributing to the broader understanding of dissipative SPDEs in Sobolev spaces. Overall, the paper advances rigorous treatment of rough stochastic PDEs with fractional dissipation and extends paracontrolled techniques to $H^s$ settings.

Abstract

In this paper, we study the well-posedness of Fractional Rough Burgers equation driven by space-time noise in $H^s(\m T)$ space. For the higher dissipation $γ\in(\frac{4}{3},2]$, we establish local well-posedness. Global well-posedness is further obtained when $γ$ is restricted to the interval $(\frac{3}{2}, 2]$. For the lower dissipation $γ\in(\frac{5}{4},\frac{4}{3}]$, we use the regularity analysis derivation the para-controlled solution.

Theorems & Definitions (60)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.2
• Remark 1.3
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Lemma 2.1: bahouriFourierAnalysisNonlinear2011