Executive Summary : | This project aims to study initial and boundary value problems in physical phenomena governed by one and two-dimensional hyperbolic system of conservation laws. The nonclassical nature of solutions of one-dimensional hyperbolic systems of conservation laws is studied first, followed by the analysis and development of two-dimensional systems. The proposed methods include studying wave interactions and the existence of global solutions for Riemann problems in hyperbolic systems of conservation laws with nonconvex flux. The study also focuses on measure-valued solutions for certain classes of hyperbolic systems of conservation laws, focusing on Keyfitz-Kranzer type and triangular type. Nonclassical waves like delta shock waves are studied, and their interactions with classical waves are constructed to construct global solutions for different initial data. The transonic flow over a symmetric airfoil is considered for gas dynamics and relativistic hydrodynamics equations, where the system may change its behavior from supersonic to subsonic across the sonic boundary. The existence and regularity of global solutions for gas expansion through a sharp corner or wedge for different wall angles and general nonconvex equation of state are also investigated. Lastly, a new hyperbolic model governing two-phase surface tension driven thin film flow under the influence of surfactant is developed. This model is used to analyze Cauchy problems, particularly one and two-dimensional Riemann problems, for this flow. |