Modeling the dynamics of real-world physical systems is critical for spatiotemporal prediction tasks, but challenging when data is limited. The scarcity of real-world data and the difficulty in reproducing the data distribution hinder directly applying meta-learning techniques. Although the knowledge of governing partial differential equations (PDE) of data can be helpful for the fast adaptation to few observations, it is mostly infeasible to exactly find the equation for observations in real-world physical systems. In this work, we propose a framework, physics-aware meta-learning with auxiliary tasks, whose spatial modules incorporate PDE-independent knowledge and temporal modules utilize the generalized features from the spatial modules to be adapted to the limited data, respectively. The framework is inspired by a local conservation law expressed mathematically as a continuity equation and does not require the exact form of governing equation to model the spatiotemporal observations. The proposed method mitigates the need for a large number of real-world tasks for meta-learning by leveraging spatial information in simulated data to meta-initialize the spatial modules. We apply the proposed framework to both synthetic and real-world spatiotemporal prediction tasks and demonstrate its superior performance with limited observations.