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

TL;DR

This paper investigates the well-posedness of the Fractional Rough Burgers equation driven by space-time noise in $H^s$ space, establishing local and global results depending on the dissipation parameter.

Contribution

It provides new well-posedness results for the fractional rough Burgers equation in different dissipation regimes, including para-controlled solutions for lower dissipation.

Findings

Local well-posedness for $\gamma ext{ in } (rac{4}{3},2]$

Global well-posedness for $\gamma ext{ in } (rac{5}{3},2]$

Para-controlled solutions for $\gamma ext{ in } (rac{5}{4},rac{4}{3}]$

Abstract

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