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

In this paper we prove the uniform-in-time L2 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 C(L2). 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.