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

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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.