In this paper, we consider a class of rough nonlinear evolution equations driven by infinite-dimensional $\gamma$-Hölder rough paths with $\mathrm{\gamma }\mathrm{\in }(1/3,1/2]$. First, we give a proper integral with respect to infinite-dimensional $\gamma$-Hölder rough paths by using rough paths theory. Second, we obtain the global in time solution and random dynamical system of rough evolution equation. Finally, we derive the existence of local unstable manifolds for rough evolution equations by a properly discretized Lyapunov–Perron method.

中文翻译：

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粗糙演化方程的不稳定流形

在本文中，我们考虑了一类由无限维驱动的粗略非线性演化方程$\gamma$-Hölder 粗糙路径与$\mathrm{\gamma }\mathrm{\in }(\mathrm{1个}/\mathrm{3个},\mathrm{1个}/\mathrm{2个}]$. 首先，我们给出关于无限维的适当积分$\gamma$-Hölder 通过使用粗略路径理论来粗略路径。其次，我们得到了粗糙演化方程的全局时间解和随机动力系统。最后，我们通过适当离散化的 Lyapunov–Perron 方法推导了粗糙演化方程的局部不稳定流形的存在性。