We present a general class of adaptive controllers for spatially invariant systems based on certainty-equivalence approach. At each step, the plant is estimated via the distributed projection algorithm as presented in Sarwar et al., that guarantees to result in a gradually varying spatiotemporal estimated plant for large enough time. Given an instance in space and time, the estimated plant is thought of as a linear spatially invariant system, with the defining operators fixed at that time and space instance. The spatiotemporal local controllers, assumed to exist, are designed to stabilize the corresponding frozen linear spatially invariant systems. We show that as long as the rates of spatiotemporal variations of the estimated plant and the spatiotemporal controller eventually become sufficiently small, a globally stable adaptive scheme can be guaranteed.