Existence of global weak solutions to 2D reduced gravity two-and-a-half layer model

We study the global existence of weak solutions to a reduced gravity two-and-a-half layer model appearing in oceanic fluid dynamics in two-dimensional torus. Based on Faedo–Galerkin method and weak convergence method, we construct the global weak solutions which are renormalized in velocity variable, where the technique of renormalized solutions was introduced by Lacroix-Violet and Vasseur (2018). Besides, we prove that the renormalized solutions are weak solutions, which satisfy the basic energy inequality and Bresch–Desjardins entropy inequality, but not the Mellet–Vasseur type inequality. In the proof, we use the reduced gravity two-and-a-half layer model with drag forces and capillary term as approximate system. It should be pointed out that only when the capillary term vanishes, we prove the existence of renormalized solution to the approximation system, which is different from Lacroix-Violet and Vasseur (2018) with the quantum potential.