  We establish the global existence and the asymptotic behavior for the 2D
incompressible isotropic elastodynamics for sufficiently small, smooth initial
data in the Eulerian coordinates formulation.The main tools used to derive the
main results are, on the one hand, a modified energy method to derive the
energy estimate and on the other hand, a Fourier transform method with a
suitable choice of $Z-$ norm to derive the sharp $L^\infty-$ estimate. We
mention that the global existence of the same system but in the Lagrangian
coordinates formulation was recently obtained by Lei. Our goal is to improve
the understanding of the behavior of solutions. Also we present a different
approach to study $2D$ nonlinear wave equations from the point of view in
frequency space.
