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

守恒律是散度形式的非线性偏微分方程组。简单地说，这些方程断言，一个区域内包含的一个量的量的时间变化率等于这个量通过该区域边界的流量速率。这些系统中存在的非线性经常导致解中跳跃不连续的形成，称为冲击。在高速流体流动中自然会发现激波的例子，例如流过超音速或近音速飞机时，所产生的激波可能被听到为“音速爆轰”。这些流动通常由可压缩流体的欧拉方程或其变体所控制，因此，对这些方程的数学分析可以促进飞机设计的改进，或者更广泛地促进空气动力学的进步。一维空间守恒定律的数学理论得到了很好的证明，但令人惊讶的是，该理论在更高维(二维和三维与空气动力学最相关)方面进展甚微，许多长期悬而未决的问题仍然存在。其中一个悬而未决的问题就是跨音速冲击问题。当激波击中障碍物(如飞机，或者更简单地说是楔形或锥体)时，激波反射-绕射就会发生，在某些情况下，这会产生分别被激波隔开的超音速和亚音速流动区域。跨音速激波问题就是求解控制这种流动的方程，并且可以表示为一个含有混合双曲-椭圆型非线性偏微分方程组的自由边界问题。在Chen和Feldman在2018年出版的《激波反射-绕射的数学和冯·诺依曼猜想》一书中，作者表示“要理解这些跨音速问题，需要非线性混合偏微分方程组相应的自由边界问题的完整数学解”，并且“这些问题是多维守恒律数学理论的基础”。这本书提供了激波反射-绕射分析的最新进展，特别是包含了激波反射-绕射问题在两个维度的完整解决方案。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。本项目属于EPSRC数学分析研究领域。这项研究将在陈桂强教授的监督下进行，目前没有计划让任何行业合作伙伴参与。

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.