Interface problems in fluids and nonlinear waves

本项目研究流体力学和非线性波动方程中的界面问题，重点研究这些问题解的奇异极限，包括：1)有表面张力和无表面张力的无粘性流体流动中的自由边界问题；磁流体力学中的自由边界问题；3)具有奇异势的非线性波动方程解的奇异极限尖锐界面Minkowski空间中极小曲面的存在性和正则性。这些问题的一个共同点是它们都有一个有几何解释的变分公式。通过关注这些非线性双曲偏微分方程的解析方面和几何方面之间的关系，PI将开发新的方法和技术来系统地获得能量估计，自由边界规则边界和适定性。研究将基于变分原理产生的几何方法，或者基于研究非线性偏微分方程解的存在性、唯一性、规律性和长期行为时产生的问题和分析方法的几何性质。这些问题是研究双曲型偏微分方程自由边界正则性和快速振荡解的典型问题。它们的决议应该是对我们了解这一领域的重要贡献。它们是一组紧密相连的问题，需要新的方法和技术，这些方法和技术在方案设计的其他领域可能很有用。流体界面问题出现在许多物理、医学和工程模型中。在水波和天体物理学（包括恒星形状）的研究中出现了涉及流体-真空界面的问题。涉及流体-流体界面的问题出现在多相流体流动中，而涉及流体-可变形结构界面的问题出现在生物医学建模中，如细胞变形。磁流体动力学中的界面问题是核聚变产能理论和实践研究的核心问题。在这个项目中研究的自由边界问题可能会揭示用于限制等离子体的磁场中存在的不稳定性。我们还将研究热等离子体和冷等离子体之间的界面，即传输屏障。对这些问题进行系统的数学分析，应该有助于我们理解各种不稳定性的起源。病态问题的识别也应该对建模考虑和应用非常有益。

This project is concerned with the study of interface problems arising in fluid mechanics and nonlinear wave equations with emphasis given to the study of singular limits of solutions to these problems which include 1) free boundary problems arising in inviscid fluid flows with and without surface tension; 2) free boundary problems arising in magneto-hydro-dynamics; and 3) existence and regularity of minimal surfaces in Minkowski spaces viewed as sharp interfaces of singular limits of solutions to nonlinear wave equation with singular potentials. A common thread of these problems is that they have a variational formulation which has a geometric interpretation. By focusing on the relationship between the analytical aspects and geometric aspects of these nonlinear hyperbolic PDEs, the PI will develop new methods and techniques to systematically obtain energy estimates, free boundary regularity bounds, and well-posedness. The research will be based on the geometric methods that arise either from variational principles or from the geometric nature of the problems and analytical methods that arise in studying existence, uniqueness, regularity, and long-time behavior of solutions to nonlinear PDEs. These problems are prototypical for the study of free boundary regularity and rapidly-oscillating solutions in hyperbolic partial differential equations. Their resolution should be an important contribution to our understanding of this area. They are a cohesive set of problems that will require new methods and techniques that may well be useful in other area of PDEs. Fluid interface problems arise in many physical, medical, and engineering models. Problems involving fluid-vacuum interfaces arise in the study of water waves and astrophysics including the shape of stars. Problems involving fluid-fluid interfaces arise in multi-phase fluid flows while problems involving fluid-deformable structure interfaces arise in biomedical modeling such as cell deformation. Interface problems in magneto-hydro-dynamics are central to the theoretical and practical study of producing energy by fusion. The free boundary problems to be studied in this project may shed light on the instabilities present in the magnetic fields used for confining plasmas. We will also study the interface between hot and cool plasmas, i.e., the transport barrier. A systematic mathematical analysis of these problems should help us understand how various types of instabilities originate. The identification of ill-posed problems should also be very beneficial for modeling considerations and applications.