On the compactness of the reduced-gravity two-and-a-half layer equations

We consider the reduced-gravity two-and-a-half model in oceanic fluid dynamics, and prove the stability of weak solutions in periodic domain Ω=T2. The proof is based on the uniform a priori estimates and the method of weak convergence, and the limit is carried out with the help of a new higher regularity estimate of the velocity, which is derived by constructing a special test function.