This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The project is aimed at obtaining constructive proofs of existence of various particular solutions of the multi-dimensional compressible Euler and potential flow equations. Constructive proofs provide not only mathematical rigor, but also detailed information about flow structure and behavior under parameter changes. This information has allowed progress on important engineering questions, such as transition from regular to Mach reflection or weak versus strong shocks at wedges and ramps. Other open problems, like the nature of supersonic bubbles at wings or the von Neumann paradox for triple points, also require new techniques for constructing particular solutions revealing structural information. Even more importantly, recent examples show that particular solutions can reveal non-uniqueness and instability of the underlying model equations and the numerical schemes used to solve them. The project will study narrower function classes where well-posedness is more likely.
The Euler equations and related systems model the flow of gas or liquid, a central problem in physics and engineering. Applications in science and technology abound, from the theory of flight over combustion engines, hydraulics, magneto-hydrodynamic plasma, atmospheric and oceanic motion, to stars and nebulae in cosmology. Many applications rely on computer simulations of flows. However, results of computations are not perfectly reliable; sometimes spurious features are observed that do not occur in nature. The project relates these poorly understood problems with numerical simulations to previously unknown flaws of the underlying mathematical models and investigates possible remedies.
