Asymptotics and singularities of evolution PDEs

我打算继续我最近在两个课题上的工作，这两个课题都与发展偏微分方程的渐近性和奇异性有关。不稳定孤立波附近色散偏微分方程的动力学：当色散偏微分方程的孤立波是稳定的，有深入的研究表明附近的解决方案应该如何放松孤立波，局部空间。不稳定孤立波附近的动力学的详细描述是鲜为人知的，除了当非线性是关键。我已经成功地处理了几个特殊情况，并希望攻击更一般的情况。我预计需要使用更复杂的规范形式约简，基于Hamilton力学方法。不可压缩三维Navier-Stokes方程：突出的正则性问题是超临界的，在这个意义上，非线性不能由扩散项控制。首先必须排除其爆破速率与自相似速率相当的奇异解，这使得它们成为临界解。大多数已知的规律性结果假设在这类小。我最近的工作建立了这类轴对称流动的正则性，而没有小性假设。我希望把这个结果推到稍微超临界的情况。我的另一个最近的结果是存在大向前离散自相似解决方案。我计划研究他们的分歧，这是连接到非唯一性问题。与此相关，我也对具有非零边界通量条件的定常流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.