Analysis of Nonlinear Partial Differential Equations

偏微分方程（PDEs）是关于未知量的偏导数的方程，通常是关于空间和时间坐标的。它们在数学的几乎所有应用中都无处不在，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，通常源于质量、动量和能量等基本守恒定律。重要的应用领域包括地球物理学、生物科学、工程学、材料科学、物理化学、经济学和金融学。偏偏方程模拟的自然现象的长度尺度从亚原子到天文，时间尺度从纳秒到千年。每种物质的行为都可以用偏微分方程来模拟，通常是在不同的长度或时间尺度上，或者用其他类似的分析和计算技术应用的方程来模拟。这类天体的一个显著例子就是地球本身。线性偏微分方程是解的线性组合也是解的偏微分方程。例如，线性波动方程模拟电磁波，它可以分解成不同频率的基本波的和，这些基本波中的每一个也都是解。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，没有办法明确地写出它们的解。事实上，这些方程是否有解，它们的性质是什么，以及如何用数值方法计算它们，这些都是很难的问题，只能用数学分析的方法来解决。除其他事项外，这些包括精确地指定解的含义和寻求解的函数类别，并建立可以构造近似解的方法，这些近似解可以严格地证明收敛于实际解。因此，对非线性偏微分方程的分析是理解我们周围世界的关键因素。正如最近的《国际数学评论》所认识到的那样，非线性偏微分方程的分析是英国数学的一个领域，尽管有一些著名的专家，但在数量和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，以及对知识经济产生了严重的有害影响，特别是对偏微分方程模拟的使用越来越多，而不是更昂贵或不切实际的替代方法，如实验室测试。该提案通过在牛津建立一个前瞻性的世界级研究中心来响应国家对这一关键研究领域的需求，以便为英国该领域的基础研究提供更清晰的焦点，并提高其在数学界内外成功和持久影响的潜力。该中心将包括对非线性偏微分方程感兴趣的整个英国研究界，例如通过成立一个全国指导委员会来组织全国性的活动，如会议和讲习班。牛津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质量研究人员的核心，以及在数学相关领域和跨学科范围内非常强大的研究小组，这些研究小组依赖于对非线性偏微分方程的理解。此外，利用OCIAM的国际知名数学建模小组的专业知识和设施，将实现与工业界的双向知识转移，该小组通过与工业界的成功研究小组，与科学，工业，工程和商业的众多分支建立了牢固的联系。该大学致力于成立该中心，并将提供大量的财政捐助，特别是将epsrc资助的一位讲师提升为主席

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