Analysis of nonlinear conservation laws of mixed type and related equations.

守恒定律是发散形式的非线性偏微分方程（PDEs）系统。简单地说，这些方程断言，一个区域内包含的量的量的时间变化率等于该量通过该区域边界的通量率。这些系统中存在的非线性常常导致溶液中出现跳变不连续，即所谓的冲击。在高速流体流动中自然可以找到激波的例子，例如经过超音速或近音速飞机的流动，在那里产生的激波可以被称为“音爆”。这些流动通常由可压缩流体的欧拉方程或其变体控制，因此这些方程的数学分析可以促进飞机设计的改进，或者更普遍地说，空气动力学的进步。一维空间守恒定律的数学理论有很好的文献记载，但令人惊讶的是，高维（二维和三维与空气动力学最相关）的理论进展甚微，许多长期存在的开放性问题仍然存在。其中一个开放性问题就是跨音速激波问题。当激波击中障碍物（如飞机，或更简单的楔子或锥体）时，会发生激波反射衍射，在某些情况下，这会产生被激波分开的超音速和亚音速流动区域。跨声速激波问题则需要求解控制这种流动的方程，并可表述为涉及混合双曲-椭圆型非线性偏微分方程的自由边界问题。Chen和Feldman在2018年出版的《激波反射-衍射数学和冯·诺伊曼猜想》一书中表示，“要理解这些跨音速问题，需要对非线性混合偏微分方程的相应自由边界问题进行完整的数学解”，“这些问题是多维守恒定律数学理论的基础”。这本书提供了一个帐户的最新发展，在分析激波反射-衍射，特别是包含一个完整的解决方案，激波反射-衍射问题在两个维度。本研究旨在进一步发展这些新思想，并在混合型自由边界问题领域取得进展。该项目属于EPSRC数学分析研究领域。该研究将在陈桂强教授的指导下进行，目前没有计划涉及任何工业合作伙伴。

Conservation laws are systems of nonlinear partial differential equations (PDEs) in divergence form. Simply these equations assert that the time rate of change in the amount of a quantity contained within a region is equal to the rate of flux of this quantity through the boundary of that region. The nonlinearity present within these systems often leads to the formation of jump discontinuities in the solution, known as shocks. Examples of shocks are found naturally in high-speed fluid flows, such as the flow past supersonic or near-sonic aircraft where the resultant shock may be heard as a ``sonic boom''. These flows are typically governed by the Euler equations for compressible fluids, or variations thereof, and therefore the mathematical analysis of these equations can facilitate improvements in the design of aircraft, or more generally advancements in aerodynamics.The mathematical theory of conservation laws in one spatial dimension is well documented, but surprisingly little progress has been made with the theory in higher dimensions (with two and three dimensions being most relevant for aerodynamics), and many longstanding open problems remain. One such open problem is the transonic shock problem. When a shock hits an obstacle (such as an aircraft, or more simply a wedge or cone), shock reflection-diffraction occurs, and in certain situations this can produce regions of supersonic and subsonic flow respectively separated by the shock. The transonic shock problem is then to solve the equations governing this flow, and may be formulated as a free boundary problem involving nonlinear PDEs of mixed hyperbolic-elliptic type. In the 2018 book of Chen and Feldman The Mathematics of Shock Reflection-Diffraction and von Neumann's Conjectures, the authors express that ``the understanding of these transonic problems requires a complete mathematical solution of the corresponding free boundary problems for nonlinear mixed PDEs'', and that ``these problems are fundamental in the mathematical theory of multidimensional conservation laws''. The book provides an account of recent developments in the analysis of shock reflection-diffraction, and in particular contains a complete solution to the shock reflection-diffraction problem in two dimensions. This research aims to further develop these new ideas, and make progress in the area of free boundary problems of mixed type.This project falls within the EPSRC Mathematical Analysis research area. The research will be carried out under the supervision of Prof. Gui-Qiang Chen, and is not currently planned to involve any industrial partners.