Asymptotics and singularities of evolution PDEs

进化偏微分方程的渐近性和奇点

• 批准号: 261356-2013
• 负责人: Tsai, TaiPeng
• 金额: $ 1.68万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635320
• 关键词: Asymptotics; singularities; evolution; PDEs

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.

我打算继续我最近在两个主题上的工作，这两个主题都与演化偏微分方程（PDEs）的渐近性和奇异性有关。色散偏微分方程在不稳定孤立波附近的动力学：当色散偏微分方程的一个孤立波是稳定的时，有大量的研究表明，在空间局部，附近的解应该如何弛豫到孤立波。不稳定孤立波附近动力学的详细描述很少为人所知，除非非线性是临界的。我已经成功地处理了几个特殊的情况，并希望攻击更普遍的情况。我认为需要使用更复杂的范式化简，基于哈密顿力学方法。不可压缩三维Navier-Stokes方程：突出的正则性问题是超临界的，即非线性不能由扩散项控制。重要的是首先要排除爆破率与自相似率相当的奇异解，这使得它们至关重要。在这类中，大多数已知的正则性结果都假定较小。我最近的工作在没有小假设的情况下建立了这类轴对称流动的规律性。我希望把这个结果推到稍微超临界的情况。我最近的另一个结果是关于大的前向离散自相似解的存在性。我打算研究它们的分叉，这与非唯一性问题有关。与此相关，我对非零边界流的存在性问题也很感兴趣。