Fourier–Galerkin approximation of the solutions of the 2D Euler equations with bounded vorticity

In this paper we prove the uniform-in-time [Formula: see text] convergence for the Fourier–Galerkin approximation to Yudovich solutions of the 2D Euler equations. Precisely, we show that both the approximating velocity and the approximating vorticity converge strongly in [Formula: see text]. Moreover, for the convergence of the velocity we provide an explicit rate of convergence. The proofs are based on a relative entropy approach and the Osgood lemma. Related results under different assumptions on the vorticity are also proved.