$L^2$ on unit circle

Question

I have a (presumably) very basic question concerning notation. I understand that any periodic function can be considered as a function on the unit circle $S^1$. I often read that a certain PDE holds on the circle (e.g. here https://arxiv.org/pdf/1504.00955.pdf ) Given this context, what is then the precise definition of $L^2(S^1)$? Are these just the measurable periodic functions on $\mathbb{R}$ such that the $L^2$-norm on some intervall $[-L,L]$ ($L$ being the length of the period of the fct.) is finite? I also don't really get how integrals of the form $ \int_{S^1} u(x) dx$ are defined in the context of the mentioned paper above. I am not really used to this $S^1$-notation. Can someone help?