Asymptotics and singularities of evolution PDEs

我建议继续我最近在两个主题上的工作，这两个主题都与发展偏微分方程组(PDE)的渐近性和奇性有关。A.色散偏微分方程组在不稳定孤立波附近的动力学：当色散偏微分方程组的孤波是稳定的时，有密集的研究表明附近的解应该如何在空间局部地松弛到孤立波。对不稳定孤立波附近的动力学的详细描述要少得多，除非非线性是关键的。我已经成功地处理了几个特例，希望能攻击更一般的情况。我预计人们需要使用更复杂的规范形约化，基于哈密顿力学方法。B.不可压缩的3D Navier-Stokes方程：突出的正则性问题是超临界的，因为非线性不能由扩散项控制。首先排除爆破率与自相似率相当的奇异解是至关重要的，这使得它们变得至关重要。大多数已知的正则性结果在这一类中都假设很小。我最近的工作建立了这类中轴对称流动的正则性，没有小的假设。我希望将这一结果推向略微超临界的情况。我最近的另一个结果是关于大型正向离散自相似解的存在性。我计划研究它们的分叉，这与非唯一性问题有关。与此相关，我也对具有非零边界流量条件的定常流动的存在性问题感兴趣。

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.