Multidimensional Simple Wave and Monge-Ampère Equation: Theory and Applications

Abstract

This paper explores the properties and mathematical formulations of multidimensional simple waves, extending the well-established theory of one-dimensional simple waves to higher dimensions. The study focuses on the connection between simple waves and the Monge-Ampère equation, particularly in the context of gas dynamics and potential flows. Key aspects include the characterization of simple waves in unsteady and steady flows, the role of characteristic lines, and the application of Hodograph and Legendre transformations to derive solutions. The paper also addresses the challenges and open questions in extending simple wave theory to more complex systems, such as non-reducible systems, radiative heat transfer, and chemical reactions. The research highlights both theoretical advancements and practical applications, providing a foundation for future studies in this area.