In this talk, we prove rigidity for the one dimensional, quintic nonlinear Schrodinger equation with mass equal to the mass of the soliton. We prove that non-scattering solutions are either the soliton or a pseudoconformal transformation of the soliton.

The long-time behavior of solutions of the three-dimensional Navier-Stokes equations in a periodic domain is studied. The time-dependent body force decays, as time $t$ tends to infinity, in a complicated but coherent manner. In fact, it is assumed to have a general and complicated asymptotic expansion which involves complex powers of $e^t$, $t$, $\ln t$, or other iterated logarithmic functions of $t$. We prove that all Leray-Hopf weak solutions admit an asymptotic expansion which is independent of the solutions and is uniquely determined by the asymptotic expansion of the body force. The proof makes use of the complexifications of the Gevrey-Sobolev spaces together with those of the Stokes operator and the bilinear form of the Navier-Stokes equations.

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