Analysis of Nonlinear Partial Differential Equations

非线性偏微分方程分析

• 批准号: EP/E035027/1
• 负责人: John Ball
• 金额: $ 344.82万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2007
• 资助国家: 英国
• 起止时间: 2007 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FE035027%2F1
• 关键词: Analysis; Nonlinear; Partial; Differential; Equations

Partial differential equations (PDEs) are equations that relate the partial derivatives, usually with respect to space and time coordinates, of unknown quantities. They are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of phenomena in the physical, natural and social sciences, often arising from fundamental conservation laws such as for mass, momentum and energy. Significant application areas include geophysics, the bio-sciences, engineering, materials science, physics and chemistry, economics and finance. Length-scales of natural phenomena modelled by PDEs range from sub-atomic to astronomical, and time-scales may range from nanoseconds to millennia. The behaviour of every material object can be modelled either by PDEs, usually at various different length- or time-scales, or by other equations for which similar techniques of analysis and computation apply. A striking example of such an object is Planet Earth itself.Linear PDEs are ones for which linear combinations of solutions are also solutions. For example, the linear wave equation models electromagnetic waves, which can be decomposed into sums of elementary waves of different frequencies, each of these elementary waves also being solutions. However, most of the PDEs that accurately model nature are nonlinear and, in general, there is no way of writing their solutions explicitly. Indeed, whether the equations have solutions, what their properties are, and how they may be computed numerically are difficult questions that can be approached only by methods of mathematical analysis. These involve, among other things, precisely specifying what is meant by a solution and the classes of functions in which solutions are sought, and establishing ways in which approximate solutions can be constructed which can be rigorously shown to converge to actual solutions. The analysis of nonlinear PDEs is thus a crucial ingredient in the understanding of the world about us.As recognized by the recent International Review of Mathematics, the analysis of nonlinear PDEs is an area of mathematics in which the UK, despite some notable experts, lags significantly behind our scientific competitors, both in quantity and overall quality. This has a serious detrimental effect on mathematics as a whole, on the scientific and other disciplines which depend on an understanding of PDEs, and on the knowledge-based economy, which in particular makes increasing use of simulations of PDEs instead of more costly or impractical alternatives such as laboratory testing.The proposal responds to the national need in this crucial research area through the formation of a forward-looking world-class research centre in Oxford, in order to provide a sharper focus for fundamental research in the field in the UK and raise the potential of its successful and durable impact within and outside mathematics. The centre will involve the whole UK research community having interests in nonlinear PDEs, for example through the formation of a national steering committee that will organize nationwide activities such as conferences and workshops.Oxford is an ideal location for such a research centre on account of an existing nucleus of high quality researchers in the field, and very strong research groups both in related areas of mathematics and across the range of disciplines that depend on the understanding of nonlinear PDEs. In addition, two-way knowledge transfer with industry will be achieved using the expertise and facilities of the internationally renowned mathematical modelling group based in OCIAM which, through successful Study Groups with Industry, has a track-record of forging strong links to numerous branches of science, industry, engineering and commerce. The university is committed to the formation of the centre and will provide a significant financial contribution, in particular upgrading one of the EPSRC-funded lectureships to a Chair

偏微分方程（英语：Partial differential equations，PDE）是一种将未知量的偏导数（通常是关于空间和时间坐标）联系起来的方程。它们在几乎所有的数学应用中都无处不在，它们为物理、自然和社会科学中的现象提供了自然的数学描述，通常来自于质量、动量和能量等基本守恒定律。重要的应用领域包括物理学、生物科学、工程、材料科学、物理和化学、经济和金融。偏微分方程所模拟的自然现象的长度尺度从亚原子到天文学不等，时间尺度可以从纳秒到千年不等。每一个物质对象的行为都可以通过偏微分方程来建模，通常是在各种不同的长度或时间尺度上，或者通过类似的分析和计算技术应用的其他方程来建模。一个突出的例子就是地球本身。线性偏微分方程是解的线性组合也是解的方程。例如，线性波动方程对电磁波进行建模，电磁波可以被分解为不同频率的基本波的总和，这些基本波中的每一个也是解。然而，大多数精确模拟自然的偏微分方程都是非线性的，一般来说，没有办法显式地写出它们的解。事实上，这些方程是否有解，它们的性质是什么，以及如何用数值计算，这些都是只有通过数学分析方法才能解决的难题。这些涉及，除其他事项外，精确地指定什么是解决方案和寻求解决方案的功能类别，并建立方法，可以构造近似解决方案，可以严格证明收敛到实际解决方案。因此，非线性偏微分方程的分析是理解我们周围世界的一个重要组成部分。正如最近的《国际数学评论》所承认的那样，非线性偏微分方程的分析是英国数学领域的一个领域，尽管有一些著名的专家，但无论是在数量上还是在整体质量上，都远远落后于我们的科学竞争对手。这对整个数学，对依赖于对偏微分方程的理解的科学和其他学科，以及对以知识为基础的经济，特别是越来越多地使用偏微分方程的模拟，而不是更昂贵或不切实际的替代方案，如实验室测试。该提案通过形成一个前瞻性的世界，回应了国家在这一关键研究领域的需求-在牛津大学的一流的研究中心，以提供一个更清晰的重点在该领域的基础研究在英国和提高其成功和持久的影响力的潜力内外数学。该中心将使对非线性偏微分方程感兴趣的整个英国研究界参与进来，例如通过成立一个全国指导委员会，组织全国性的活动，如会议和讲习班。牛津大学是建立这样一个研究中心的理想地点，因为该领域现有的高素质研究人员的核心，以及在数学相关领域和依赖于非线性偏微分方程理解的学科范围内非常强大的研究小组。此外，将利用设在OCIAM的国际知名数学建模小组的专业知识和设施，实现与工业界的双向知识转移，该小组通过成功的工业研究小组，与科学，工业，工程和商业的许多分支建立了密切的联系。该大学致力于建立该中心，并将提供大量财政捐助，特别是将EPSRC资助的一个讲师职位升级为教席

项目成果

Electrical Impedance Tomography by Elastic Deformation

• DOI: 10.1137/070686408
• 发表时间: 2008-06
• 期刊: SIAM J. Appl. Math.
• 作者: H. Ammari;E. Bonnetier;Yves Capdeboscq;M. Tanter;M. Fink

Progress on the strong Eshelby's conjecture and extremal structures for the elastic moment tensor

• DOI: 10.1016/j.matpur.2010.01.003
• 发表时间: 2009-09
• 期刊: Journal de Mathématiques Pures et Appliquées
• 作者: H. Ammari;Yves Capdeboscq;Hyeonbae Kang;Hyundae Lee;G. Milton;Habib Zribi

Mathematical models and reconstruction methods in magneto-acoustic imaging

• DOI: 10.1017/s0956792509007888
• 发表时间: 2009-06-01
• 期刊: EUROPEAN JOURNAL OF APPLIED MATHEMATICS
• 影响因子: 1.9
• 作者: Ammari, Habib;Capdeboscq, Yves;Kozhemyak, Anastasia
• 通讯作者: Kozhemyak, Anastasia

Two asymptotic models for arrays of underground waste containers

地下废物容器阵列的两个渐近模型

• DOI: 10.1080/00036810902922590
• 发表时间: 2009
• 期刊: Applicable Analysis
• 影响因子: 1.1
• 作者: Allaire G

Microwave Imaging by Elastic Deformation

• DOI: 10.1137/110828241
• 发表时间: 2011-12
• 期刊: SIAM J. Appl. Math.
• 作者: H. Ammari;Yves Capdeboscq;F. Gournay;A. Rozanova-Pierrat;Faouzi Triki