Nonlinear Landau Damping in Collisionless Plasma and Inviscid Fluid

The long-time nonlinear evolution of generic initial perturbations in stable Vlasov plasma and two-dimensional (2D) ideal fluid is studied. Even without dissipation, these systems relax to new steady states (Landau damping). The asymptotic damping laws are found to be algebraic, such as $t^{-1\mathrm{}}$ for 1D plasma potential, or $t^{-5\mathrm{}/2}$ for evolving stream function in a flow with nonvanishing shear. The rate of the relaxation is fast so that phase-space/fluid-element displacement in certain directions is uniformly small, implying that decaying Vlasov and 2D fluid turbulences are not ergodic.