Nonlinear Geometric Partial Differential Equations: Entire Solutions and Regularity

非线性几何偏微分方程：全解和正则性

• 批准号: 1600658
• 负责人: Panagiota Daskalopoulos
• 金额: $ 29.89万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600658&HistoricalAwards=false
• 关键词: Nonlinear; Geometric; Partial; Differential; Equations

Some of the most important problems in mathematics and physics are related to the understanding of singularities. Singularities can range anywhere from black holes in astrophysics to turbulence in fluid mechanics to the accumulation of cancer cells in biomedical research. Such physical phenomena are often described by differential equations that involve time and space. Studying the qualitative behavior of the solutions of these equations frequently deepens one's understanding of the related physical problems. To study a singularity of a solution one uses a so-called blow-up procedure that allows one to focus attention near the singularity and to exploit the scaling properties that the differential equation enjoys. Because of the change in the scaling of space and time, this process leads to a new solution that is defined for all space and time, in other words to a "global solution." The classification of global solutions, when possible, sheds new insight into the singularity and thus into the underlying physical phenomenon. This project addresses the questions of existence, uniqueness, and qualitative behavior of global solutions to nonlinear geometric elliptic and parabolic partial differential equations. Emphasis is given to the classification of ancient solutions, the construction of new ancient solutions from the gluing of solitons, and the study of fully nonlinear extrinsic geometric flows in the complete noncompact case. The interplay between analytical and geometric techniques will be a crucial factor in carrying out the research. The project links a wide range of active fields of mathematics, including nonlinear partial differential equations, differential geometry, and classical analysis. The principal investigator also intends to seek applications of the mathematical results to other disciplines such as quantum field theory and image processing. Results will be disseminated to the research community at various meetings and by publication of research articles. Special emphasis will be given to the training of Ph.D. students.

数学和物理学中的一些最重要的问题与对奇点的理解有关。奇点可以出现在任何地方，从天体物理学中的黑洞到流体力学中的湍流，再到生物医学研究中癌细胞的积累。这类物理现象通常用涉及时间和空间的微分方程来描述。研究这些方程解的定性行为常常加深人们对相关物理问题的理解。为了研究一个解的奇点，人们使用了一个所谓的爆破过程，它允许人们将注意力集中在奇点附近，并利用微分方程所具有的标度性质。由于空间和时间尺度的变化，这一过程导致了一个新的解决方案，这是为所有空间和时间定义的，换句话说，导致了一个“全球解决方案”。“在可能的情况下，整体解的分类为奇点提供了新的见解，从而为潜在的物理现象提供了新的见解。 本计画探讨非线性几何椭圆型与抛物型偏微分方程整体解的存在性、唯一性与定性行为。重点是古代的解决方案的分类，建设新的古代解决方案从胶合的孤子，和研究完全非线性的外部几何流在完全非紧的情况下。分析和几何技术之间的相互作用将是进行研究的一个关键因素。该项目连接了广泛的数学活跃领域，包括非线性偏微分方程，微分几何和经典分析。首席研究员还打算将数学结果应用于其他学科，如量子场论和图像处理。研究结果将在各种会议上和通过发表研究文章向研究界传播。将特别重视博士生的培养。学生