in this talk, i will consider the incompressible navier-stokesequations in the mild solution setting. using this setting, i will show how to obtain analyticity of mild solutions using the gevrey regularity technique. this regularity enables to get decay rates of weak solutions of the navier-stokes equations. this idea can be applied to other dissipative equations with analytic nonlinearities. i will finally consider the regularity of the flow map of mild solutions using the log-lipschitz regularity of the velocity field.