Abstract: We introduce a framework for models of the limit order book based on a system of two stochastic partial differential equations (SPDEs) coupled by non-linear interaction at a free boundary. The two SPDEs model the evolution of the order book profile on the bid and ask side respectively, while the boundary interaction represents the (in general non-linear) effect of bid-ask-imbalance on price changes. Our model can be considered an extension of the stochastic Stefan problem introduced by Kim, Sowers and Zheng. Despite of the Non-Lipschitz drift introduced by the boundary interaction we show existence of a solution up to a stopping time of the general model; extending results of Kim, Sowers and Zheng. Finally we analyze a stationary version of the model and show that the average order book profile of a model proposed by Bouchaud, Mezard and Potters can be recovered as a special case. This talk is based on joint work with Marvin Müller.