Shock Diffraction/Reflection and Related Problems in Multi-Dimensional Conservation Laws

Description

The aim of this project is to develop the mathematical theory of multi-dimensionalconservation laws, particularly focusing on the rigorous mathematical analysis of shockdiffraction/reflection by convex or concave cornered wedges and related problems incompressible fluid flows governed by the potential flow equation.The problem of shock diffraction/reflection is not only a longstanding open challenge in thefluid mechanics, but also fundamental in the mathematical theory of multidimensionalconservation laws, since their solutions are building blocks and asymptotic attractors ofgeneral solutions to the multidimensional Euler equations for compressible fluids, which arethe oldest and still most prominent paradigm of a hyperbolic system of conservation laws.The research of the Euler equations contributes a lot to the fast expanding mathematicaltheories of nonlinear partial differential equations and related analysis.The mathematical analysis of entropy solutions of the Euler equations dates back to Riemannin 1860. He introduced and solved a celebrated and foundational problem named theRiemann problem, which plays a prominent role in the theory of conservation laws. Then byLax, Glimm, and many other mathematicians, extensive theoretical research has beenconducted in the mathematical analysis of entropy solutions in one space dimension over thepast fifty years. However the general theory of nonlinear hyperbolic systems of conservationlaws in several space dimensions is terra incognita. So far almost all the research is focusedon the problems which admit stationary or self-similar solutions such as the shockdiffraction/reflection problem. There are big challenges for these problems. One of thebiggest issues is that the nonlinear partial differential equations are hyperbolic-elliptic mixed-type.The type of the equations is unknown before they are solved.We have been working on the mentioned problems for several years. Recently, we haveestablished the global existence of solutions of the problem of the shock diffraction byconvex cornered wedges (the Lighthill problem) for the potential flow equation as well as thenonlinear wave system, and also a framework to prove the strict convexity of transonicshocks in the self-similar coordinates. In this project, we will be mainly focused on a furtherdevelopment to the existence and regularity of solutions of these problems involving othernonlinear waves, the rarefaction wave for example, and the uniqueness and stability of thesesolutions with respect to the initial data and wedge angle.So far, all the known results are on the global existence and regularity of solutions of theshock diffraction/reflection problem without other nonlinear waves. Hence the results andtechniques developed from this research project will also provide a better understanding ofthese mentioned problems.?