Asymptotics and singularities of evolution PDEs

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I propose to continue my recent work in two subjects, both of which are concerned with the asymptotics and singularities of evolution partial differential equations (PDEs).A. Dynamics of dispersive PDEs near unstable solitary waves: When a solitary wave of a dispersive PDE is stable, there are intensive studies showing how nearby solutions should relax to the solitary wave, locally in space. The detailed description of the dynamics near an unstable solitary wave is much less known, except when the nonlinearity is critical. I have succeeded in treating several special cases, and hope to attack the more general situations. I expect that one needs to use more sophisticated normal form reduction, based on Hamiltonian mechanic approach.B. Incompressible 3D Navier-Stokes equations: The outstanding regularity problem is supercritical in the sense that the nonlinearity cannot be controlled by the diffusion term. It is essential to first exclude singular solutions whose blow-up rates are comparable to the self-similar rate, which makes them critical. Most known regularity results assume smallness in this class. My recent work establishes regularity for axisymmetric flows in this class without smallness assumption. I hope to push this result to slightly supercritical case. My other recent result is on the existence of large forward discrete self-similar solutions. I plan to study their bifurcation, which is connected to the nonuniqueness question. Related, I am also interested in the existence problem of stationary flows with nonzero boundary flux condition.