We study an infinite system of nonlinear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3D Euler and Navier-Stokes equations in a rough approximation of wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.
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