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This paper focuses on stochastic partial differential equations (SPDEs) undertwo-time-scale formulation. Distinct from the work in the existing literature,the systems are driven by $\alpha$-stable processes with $\alpha \in(1,2)$. Inaddition, the SPDEs are either modulated by a continuous-time Markov chain witha finite state space or have an addition fast jump component. The inclusion ofthe Markov chain is for the needs of treating random environment, whereas theaddition of the fast jump process enables the consideration of discontinuity inthe sample paths of the fast processes. Assuming either a fast changing Markovswitching or an additional fast-varying jump process, this work aims to obtainthe averaging principles for such systems. There are several distinctdifficulties. First, the noise is not square integrable. Second, in our setup,for the underlying SPDE, there is only a unique mild solution and as a result,there is only mild It\^{o}'s formula that can be used. Moreover, another newaspect is the addition of the fast regime switching and the addition of thefast varying jump processes in the formulation, which enlarges theapplicability of the underlying systems. To overcome these difficulties, asemigroup approach is taken. Under suitable conditions, it is proved that the$p$th moment convergence takes place with $p\in(1,\alpha )$, which is strongerthan the usual weak convergence approaches.

Two-time-scale stochastic partial differential equations driven by $\alpha$-stable noises: Averaging principles