Dynamic Free-Boundary Problems

动态自由边界问题

基本信息

  • 批准号:
    1566578
  • 负责人:
  • 金额:
    $ 34.57万
  • 依托单位:
  • 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 2020-06-30
  • 项目状态:
    已结题

项目摘要

A "dynamic free-boundary problem" is the problem of finding the solution of a partial differential equation in a domain whose evolution is a priori unknown. One example is the problem of modeling melting ice, where the interface of ice and water is determined dynamically by the distribution of temperature (i.e., the solution of the heat equation) in the water region. This project addresses some fundamental questions concerning free-boundary problems, such as the existence and long-time behavior of solutions. This is an important topic, since without a proper mathematical theory it is difficult to develop accurate and trustworthy numerical methods for dealing with concrete physical problems. A good example is the problem of liquid drops sliding on a tilted surface. In this case the shape of the drop can change drastically when the velocity is increased, and a singularity (corner) develops at the rear of the drop when the velocity exceeds a certain critical value. At higher speeds the tail of the drop may break into another component (pearling). Accurate modeling of the motion of drops is therefore a highly complex question in fluid mechanics, with many important applications in engineering. Another example is in the crowd motion of individuals or cars in congested areas exiting through part of the boundary of the confining domain (e.g., a room or a highway); in these examples, the exit pattern is heavily influenced by the shape of the domain as well as the location of the exit. The project aims towards a better understanding of the properties of these problems and seeks to provide a framework for developing accurate computer-based numerical simulations of the processes. The problems studied in this project arise in a variety of physical phenomena, including the phase change between liquid and solid, the motion of capillary drops, congested crowd motion, and tumor growth. Particular focus will be on the asymptotic behavior of solutions starting with general initial data, either in the context of homogenization and long-time behavior or in the "stiff-pressure" limit of nonlinear diffusion. In addition to standard methods in partial differential equations, such as integral estimates, it will often be necessary to introduce geometric methods to understand the pointwise behavior of the moving interface. The first part of the project concerns volume-preserving geometric motions and their large-time behavior. The challenge lies in possible topological changes of the interface caused by the merging and splitting of fronts. For this reason most results in this area hold only for convex surfaces. The principal investigator will introduce a modified version of the moving-planes method to investigate a more general class of interfaces and to study their convergence to equilibrium. The second subproject concerns the evolution of capillary drops on a flat or tilted surface in the quasi-static approximation regime. While many models have been proposed in the study of dynamic capillary drops, the analysis of such models is still in its early stages, due the wide range of behavior with respect to parameters such as the roughness of the surface or the volume of the drop. The aim here is to address the well-posedness and long-time behavior of solutions in various regimes, to classify possible singularities, and to investigate the process's transitional behavior upon the change of parameters. The principal investigator also proposes to study the emergence of congested zones in collective motions, for example the evolution of jammed regions in crowd motions with a density constraint, or tumor growth in the motion of cancer cells with anti-crowding pressure. The plan is to characterize the motion law of the congested zones as well as to investigate the stability and long-time behavior of the solutions. Finally, the principal investigator proposes to study interface homogenization problems in random media, where the inhomogeneity is present either in the advection vector field or in the latent heat. The goal is to understand how the inhomogeneities in the system interact with the geometry of the interfaces to affect the macroscopic behavior of the solutions. New approaches will be introduced to the investigation, using tools such as energy estimates and concentration inequalities.
“动态自由边界问题”是在其演化是先验未知的区域中求偏微分方程解的问题。一个例子是模拟融化的冰,其中冰和水的界面是由水域内的温度分布(即热方程的解)动态确定的。这个项目解决了一些与自由边界问题有关的基本问题,例如解的存在性和长期行为。这是一个重要的课题,因为如果没有适当的数学理论,就很难开发出准确可靠的数值方法来处理具体的物理问题。一个很好的例子是液滴在倾斜的表面上滑动的问题。在这种情况下,当速度增加时,液滴的形状可能会发生巨大变化,当速度超过某个临界值时,液滴的后部会出现奇点(拐角)。在较高的速度下,液滴的尾部可能会破碎成另一种成分(珍珠)。因此,液滴运动的精确建模在流体力学中是一个非常复杂的问题,在工程上有许多重要的应用。另一个例子是拥挤地区的个人或汽车通过限制域(例如,房间或高速公路)的部分边界退出的人群运动;在这些例子中,出口模式严重受域的形状以及出口的位置的影响。该项目旨在更好地了解这些问题的性质,并力求提供一个框架,以便对这些过程进行准确的计算机数值模拟。本项目研究的问题出现在各种物理现象中,包括液固相变、毛细液滴运动、拥挤的人群运动和肿瘤生长。将特别关注从一般初始数据开始的解的渐近行为,无论是在均化和长时间行为的背景下,还是在非线性扩散的“刚性压力”极限下。除了偏微分方程中的标准方法,如积分估计,经常需要引入几何方法来了解运动界面的逐点行为。该项目的第一部分涉及体积保持几何运动及其大时间行为。挑战在于锋面的合并和分裂可能导致界面的拓扑变化。因此,该区域中的大多数结果仅适用于凸面。首席研究人员将介绍一种改进的移动平面法,以研究更一般类型的界面,并研究它们的收敛到平衡。第二个分项目涉及在准静态近似条件下平面或倾斜表面上毛细液滴的演化。虽然在动态毛细液滴的研究中已经提出了许多模型,但由于与表面粗糙度或液滴体积等参数有关的行为范围很广,这类模型的分析仍处于早期阶段。这里的目的是解决解在不同区域的适定性和长期行为,对可能的奇点进行分类,并研究过程在参数变化时的过渡行为。首席研究员还建议研究集体运动中拥挤区的出现,例如在密度受限的人群运动中拥堵区域的演变,或在反拥挤压力下癌细胞运动中的肿瘤生长。该计划是为了刻画拥堵区域的运动规律,以及研究解的稳定性和长期行为。最后,主要研究人员建议研究随机介质中的界面均匀化问题,其中不均匀存在于平流矢量场或潜热中。目标是了解系统中的非均质性如何与界面的几何形状相互作用,以影响溶液的宏观行为。调查将引入新的方法,使用能源估计和浓度不平等等工具。

项目成果

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Inwon Kim其他文献

Regularity and Nondegeneracy for Tumor Growth with Nutrients
Mean Field Limit for Congestion Dynamics in One Dimension
一维拥塞动力学的平均场极限
  • DOI:
  • 发表时间:
    2024
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Inwon Kim;A. Mellet;Jeremy Sheung
  • 通讯作者:
    Jeremy Sheung
Head and Tail Speeds of Mean Curvature Flow with Forcing

Inwon Kim的其他文献

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{{ truncateString('Inwon Kim', 18)}}的其他基金

Dynamic Free Boundary Problems
动态自由边界问题
  • 批准号:
    2153254
  • 财政年份:
    2022
  • 资助金额:
    $ 34.57万
  • 项目类别:
    Standard Grant
Dynamic Free Boundary Problems
动态自由边界问题
  • 批准号:
    1900804
  • 财政年份:
    2019
  • 资助金额:
    $ 34.57万
  • 项目类别:
    Standard Grant
Nonlinear Partial Differential equations and boundary conditions.
非线性偏微分方程和边界条件。
  • 批准号:
    1300445
  • 财政年份:
    2013
  • 资助金额:
    $ 34.57万
  • 项目类别:
    Continuing Grant
Free Boundary Problems and nonlinear PDEs
自由边界问题和非线性偏微分方程
  • 批准号:
    0970072
  • 财政年份:
    2010
  • 资助金额:
    $ 34.57万
  • 项目类别:
    Standard Grant
Free Boundary Problems and Viscosity solutions
免费边界问题和粘度解决方案
  • 批准号:
    0700732
  • 财政年份:
    2007
  • 资助金额:
    $ 34.57万
  • 项目类别:
    Standard Grant
Free boundary problems and Viscosity solutions.
自由边界问题和粘度解决方案。
  • 批准号:
    0627896
  • 财政年份:
    2006
  • 资助金额:
    $ 34.57万
  • 项目类别:
    Standard Grant
Free boundary problems and Viscosity solutions.
自由边界问题和粘度解决方案。
  • 批准号:
    0401436
  • 财政年份:
    2004
  • 资助金额:
    $ 34.57万
  • 项目类别:
    Standard Grant

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